Lecture 03. More Postulates, Superposition, Operators and Measurement
Shaka, Athan J.
 This course provides an introduction to quantum mechanics and principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. It also examines the Schrödinger equation and study how it describes the behavior of very light particles, the quantum description of rotating and vibrating molecules is compared to the classical description, and the quantum description of the electronic structure of atoms is studied. 
 Welcome back to lecture 3, Chem 131A. We're going to continue where we left off. Today it's more postulate, superposition, operators, and measurement. Where we last left our hero, we had decided that the derivative operator is linear, but it was not Hermitian. And then I introduced this very ornate relationship to describe what I meant by Hermitian. And you might wonder what it means. But what it means is basically, suppose we had complex numbers and most numbers were complex. And then we wanted to say that a number was real, but we only had complex numbers. Well, one trick we could use is we could say, if z is equal to z star, then the number is real. Because the only imaginary part that can be equal to opposite itself is zero. And zero imaginary part means the number is real. And so really, Hermitian is just making sure that when we measure something, we get a real number. We still do believe that probability doesn't have an imaginary part and neither does energy. When we measure it, it has units of joules and so forth. And so we want to make sure that these things that we measure are Hermitian. And this formula with these integrals and stars and the operator in is just a very fancy way of saying z is equal to z star. Nothing more than that. Okay, let's show that the derivative operator then is not Hermitian. Well, here's what we have to do. We have to do an integral of f star d by dx g. And we have to show that that is or is not equal to the integral of g star d by dx f whole thing star. When you see an integrand that has a derivative in it, the first thing you think is I bet I can integrate that by parts. If you recall, integrating by parts is basically doing the opposite as doing the derivative of uv is udv plus vdu. We turn that around and we move the uv, one of them, to the other side and then we set the integral equal to that. Now the limits on this integration are plus and minus infinity but I won't always put them in because it may get a little bit messy. But whatever it is, the wave functions have to vanish at plus or minus infinity. And the argument as to why they have to vanish is if they had any amplitude out there, way out there, then we couldn't normalize them. They would get too big. And so the only way we can have the area under the curve crank down to some number is that it finally dies out when we get far enough away. So let's try integration by parts. The formula is the integral of u d by dx of v of x is equal to uv minus the integral the other way around v d by dx u. And this is going to be convenient because the Hermitian thing had them the other way around. And now let's let u, the function u of x conventionally and calculus, let's let that be f star. And the v, let's let that be g. And let's try it. So our equation becomes this, fairly intimidating looking but not too bad. The integral of f star d by dx g is equal to f star g evaluated at plus or minus infinity minus the integral of g d by dx f star. And that is equal to minus the integral of g star d by dx f whole thing star. But that's not equal to what we want. And so what we have, because we have a minus sign and we want it to be a plus sign. So it's not equal and therefore the derivative operator is not Hermitian. It's called anti-symmetric for obvious reasons. When you swap them it changes sign. But it's not Hermitian. How can we have it be Hermitian? Well interestingly enough we have to use our friend the square root of negative one again. And if we multiply the derivative operator by minus i h bar that's enough to do the job because minus i star is plus i. And so therefore that gets rid of the minus sign that we got stuck with with the regular derivative. And then we just follow everything else through. And it works. You say well what, why is the h bar there? And the answer is this is quantum mechanics. Of course there's an h bar there because we're going to have to have that in almost everything we use. And in fact the momentum operator p hat x which tells us the momentum when it operates on a wave function it tells us the momentum in the x direction is just given by minus i h bar d by dx and it is a linear Hermitian operator. Its eigenfunctions are very closely related to those of the derivative operator because after all, all it has is just an extra thing out in front. But we want to make sure that the eigenvalues are real and so these eigenfunctions are exponentials but we put in an i and here we realize that p is real, x is real, h bar is real and so this is e to the i px upon h bar. We know that we have to have the units go away if we take an exponential because an exponential is a power series, 1 plus x plus x squared over 2. And if it has units we're adding feet and feet squared and feet cubed and that doesn't make any sense. And so with a little bit of dimensional analysis we come to the idea that these functions here e to the i px upon h bar are very good candidates. So let's do another practice problem and have a look. So let's show that these are the eigenfunctions in fact of the derivative operator. Well, let's take p hat x on our function phi of x. Let's put in what p hat is minus i h bar d by dx on the function. Let's put in the function which we assume is e to the plus i px upon h bar and the derivative of anything times x is the thing, excuse me, anything times e to the a x is a e to the a x so we bring down the i p upon h bar and now I think you can see why we want h bar out in front. The h bars fold up minus i times plus i is minus i squared but minus i squared is plus 1. That goes away and that leaves us with p and that's p e to the i px upon h bar and that's p times the eigenfunction and therefore we've shown that the operator p hat returns the eigenvalue p which has the units of momentum. So the complex exponential is the eigenfunction of the momentum operator and the eigenvalue is p. In the language of linear algebra, the eigenfunctions of a linear Hermitian operator form a basis. So if I take a point in a two-dimensional plane and I want to figure out where I am, I know that if I go a certain unit out on x along the x axis and then up or down by y that I can get to the point and furthermore any point anywhere can be expressed as a combination of some distance this way and some distance up or down and there's no point that can escape. So if we have a vector, any vector, any point x naught, y naught that's equal to x naught times the coordinate unit along the x axis plus y naught times the coordinate unit along the y axis and just like that we can write any wave function as a linear combination of eigenfunctions of a Hermitian operator. They form a basis. No function can escape and that's important because if some functions could escape that would mean there were certain values that we couldn't measure anything and that would be very bad because what would happen to the probability? Particles would be disappearing then. Opostulate 5 is this. It's quite a mouthful but we'll get to it. When a wave function is not an eigenfunction of the measured observable, the result of the measurement is still an eigenvalue but now the probability is given by the square modulus of the expansion coefficient of the eigenfunctions of the operator. So if I have a wave function psi is some constant, could be complex number, it doesn't matter because all these functions can be complex. So let's just call it c1 phi 1 plus c2 phi 2. Then the probability of obtaining the first eigenvalue is the square of c1 with the absolute value so if there's an imaginary part you take c1 star c and the probability of obtaining the second one is the square of c2. Those are the two probabilities and if there are only two parts making up the wave function, those are the only two values you can get. Usually a wave function is made up of a whole bunch of different eigenfunctions and so there are a lot of different possibilities that you can get. The eigenfunctions themselves have to be normalized. That means if you happen to be in an eigenfunction, your chance of being somewhere in the universe and having that eigenvalue, let's say, of momentum is equal to 1. And so the basis functions themselves are normalized and we always assume that they are normalized without comment. And likewise for the wave function to be normalized once the basis functions are normalized, that means that the probabilities in those coefficients have to add up. So the sum of the squares of all the coefficients always have to add up to 1. It's as if we have a unit circle and we're some point on the circle and we have an x component and a y component and the Pythagorean theorem says x squared plus y squared is equal to 1 and that's how it works and it works the same way in higher dimensions. Second comment, the best way to think of these eigenfunctions is not as things spraying around in space. Think of them as vectors. Think of one eigenfunction pointing this way, telling you the amount on this side. The other one points this way. A third one points up. If I've got more I have to have an imagination. But basically they're all at right angles to each other and they're all telling the amount of this special state that is in there to begin with. And eigenfunctions of a linear Hermitian operator corresponding to different eigenvalues are orthogonal and that's another reason why it's good to think of them like vectors because if I have an eigenfunction here and this has one eigenvalue and I have another eigenfunction here and it has a different eigenvalue, then those two functions have nothing to do with each other. They are as different as different can be. They're in different directions. They have no influence on each other. And to see this normally suppose we have x and y, we can tell they're at right angles because we can look. But suppose I put my arms out some way and then I say, well are those orthogonal? Well you could try to mentally rotate and see if it comes back to x and y but that's a very, very slow and labor intensive way to do it. Instead what you do is you take the dot product. You take the product of the first two components, the second two, the third two, you add them all up and you see if that's zero. And if it's zero that means they're orthogonal. If it's not zero that means that they aren't orthogonal. So for three real components, let's say two vectors in 3D space, I just take ax, bx plus ay, by plus az, bz. And if that sum that comes to zero, it doesn't matter what the individual terms are. If that sum comes to zero that means that the vector a and the vector b are at right angles to each other. And that's much, much easier to compute. More generally if we've got lots of dimensions, then we need to expand our sum. So it could be a1, b1 because we don't want to have x, y, and z if we've got let's say 5 or 6 or 20. We run out a letter so we switch to numbers where we won't run out. A1, b1 plus a2, b2 plus a3, b3 and we just write that in a shorthand as the sum over n of an, bn. And that goes to however far we want it to go, including in some cases to infinity. And that should be zero. And the same idea holds as functions except one, the sum becomes an integral. Because when you multiply the functions by each other, they both depend on x. And so adding up, you can't just add up, you have to integrate to get the answer. And number two, because the functions can be complex, we have to take the complex conjugate of the first function. Let's suppose we've got two functions f and g. Then our orthogonality condition is as follows. The integral of f star times g dx is zero. Now we can show based on this and the definition of Hermitian that the eigenfunctions with different eigenvalues are orthogonal. So here's what we do. If it's an eigenfunction, it has an eigenvalue that's a real number. So let's put omega on phi one. And we get omega on phi one. We get phi one back because it's an eigenfunction. We put omega on phi two, we get omega two. And the only thing we need to know is that omega one is not equal to omega two. There are different numbers. They're real and they're unequal. And the operator, big omega hat, is Hermitian. Let's take the first eigenvalue equation and make a series of operations to both sides of it. That's always what you do when you simplify equations. You do the same thing to both sides methodically. And if you do that, you never get mixed up and nothing ever goes wrong. And if you do some shorthand of cross multiplying this and that and you don't know what you're doing exactly, you'll oftentimes get it wrong. So let's take this equation, omega phi one equals omega one, phi one, and let's first multiply on the left. We have to make sure we multiply on the same side when we do this by phi two star. Okay? So now we've got phi two star omega phi one is equal to phi two star little omega one, phi one. And then since omega one's a constant, I can pull it out and say that's little omega one, phi two star, phi one. Now I'm going to put an integral on both sides because if two things are equal, then if I multiply them both by phi two star, they're still equal. And if I integrate them both over dx, they're still equal. They don't become unequal. And so I integrate over phi two star x omega phi one dx. And that's the integral of omega one. And since omega one is a constant, I pull it out and I end up with omega one times the integral of phi two star, phi one dx. And we can do the same series of operations exactly. But instead of having omega hat phi one, we take omega hat phi two and we get omega two. And we just go through the same. Only we just swap the rolls of one and two. We multiply by phi one star. And if we do that, I've just not done every step here, but omega hat phi two is equal to omega two times the integral of phi one star, phi two dx. Now let's take the complex conjugate of both sides of the first equation. So on the left hand side, I have the complex conjugate of the whole thing, omega two, omega hat, omega one integrated. And on the other side, I have little omega one times the integral of phi two star phi one dx star. And I can simplify that. I leave the other side alone because that's going to be the definition of Hermitian. The right hand side, I turn to omega one star. And then the integral of phi two star star times phi one star. Well, the star of the star, let's see, I change i to minus i back to i, so that goes away. And I can then write the phi one in front of the phi two. It doesn't matter. I'm multiplying those. There's no operator. So I finally come to the following. The integral of phi two star omega hat phi one is equal to omega one times the integral of phi one star phi two. What does that get us? Well, the observable is Hermitian. And so when I put in omega hat phi two x to give the eigenvalue omega two and I do the same thing, I find that I get the same thing backwards. So omega two star omega one phi one is equal to omega one star, sorry, phi one star omega hat phi two. And so using our two series of equations, here's what we come to finally. Omega two times the integral of phi one star phi two is equal to omega one times the integral of phi one star phi two. But omega two is not equal to omega one. So let's subtract omega one times the integral from both sides. Then we find out that omega two minus omega one times the integral is equal to zero. But since omega two is not equal to omega one, it must be that the other thing zero. Because if I have any number times something, the only way I can make the whole thing zero is if the other thing zero. And that means that the integral of phi one star phi two dx is zero. And that means that they are orthogonal. So that's the proof. You'll have to go over it a couple of times to get it down. But that's kind of a standard thing that's done in quantum mechanics to show that eigenfunctions for different values of eigenvalues are in fact orthogonal. Now suppose we make a measurement on a quantum system. And it's represented by a wave function psi that's not an eigenstate of the operator in question. Then what happens? Well, we express psi as a linear combination of eigenfunctions. And that we know we can do that because there is no function that can escape us. And we know our eigenfunctions can be made normalized, so we assume they're normalized. And then the probability of obtaining a particular eigenvalue, let's say eigenvalue k out of the totality from one to n is the absolute value of Ck squared, where Ck is the coefficient of the kth eigenfunction. Now suppose we then make the measurement again right away. The question is, do we get a different result? And the answer is kind of surprising, but the answer is no. It turns out if we make the measurement again, we get the same result. And if we keep measuring the same observable over and over, we keep getting the same result. And now sort of mysteriously, in a way, it's 100% certain that we're going to get that result. There is no other result that we're going to get. And I've tried to encapsulate this in this kind of pseudo equation. We start out with probabilities. It could be any of these eigenstates. And then we make a measurement. And somehow one of them is chosen. And we can't say how, even in an ideal experiment, but we can say what the probability is. Let's say 25% of the time we get this result. Now if we measure again and nothing's intervened, we haven't done anything, we get the same result again. And again, and again, and again. And now there's no probability at all. It's always 100%. So it's exact certainty. That is, measurement is kind of like a filter. All the other possibilities are filtered out, leaving the one that's actually observed. If you sort coins with a coins order, you roll them down. And when they fit the size of the tube, they drop in. And if you don't know which tube is going to drop into it first, then that's like being uncertain. But then if you drop the thing in and it drops into the third tube, and then you empty it out of there and put it back in, it's going to drop into the third tube again. And that's kind of an analogy for what's going on here. So if we look at the probability of a coin dropping into the third tube, or we could say, for example, suppose we flip a coin until it hits and stops spinning and lies down, we assume it's 50% probability heads and 50% probability tails. Then we see its heads, then if we don't do anything, we don't flip it again. Just sit there. It's heads. It's heads again. It's heads again and so forth. And it's heads as many times as we want to keep looking at it. And that's what this theory of measurement is saying. In other words, when you make a measurement, you rule out certain possibilities. They're now gone. Now the measurement's made. It came up. It happened to come up this. If you make it again, it comes up this. If you make it again, it comes up this. And that's assuming you don't have any interaction in between. But this is an idealized experiment. We aren't talking about how we practically would implement it. And likewise, in quantum mechanics, it's just like looking at that coin. If we make the same ideal measurement again and again, after filtering out all the other possibilities, we just get the one result that we got, the same result each time. But before we made the measurement, it seemed like there were other possibilities. And if we start all over, not with the one we've measured, but with an identical particle coming through that we haven't measured, then we might get a different answer. And then if we measure that again, we'll get that different answer again, and so on and so forth. And so it seems as if the measurement itself took this very fragile thing, this wave function, and it made it collapse onto a particular eigenfunction. Said, right, this is it. And then all the other possibilities vanished forever. If I decide I'm going to give a lecture, I turn up and do the lecture, but if I decide I'm going to the beach instead and I go to the beach, then the lecture is not a possibility, and it's now vanished forever. It's gone and I'm at the beach. And so by making that choice, I've narrowed down the possibilities. Before I did that, I could say, well, 50-50, I give the lecture, go to the beach. And that's important because it means that measurement affects quantum systems, and that means that there is no such thing as a property without measuring it. We usually think that things have properties independent of measuring them because they seem to. This pointer, for example, has a mass, whether I have it on a scale or not, and I assume it's the same. And for big objects that are always being bombarded by all kinds of things and never have a chance to let the wave function sneak around, that's certainly true. But for small things, we have to be very wary about assuming that something has a property if we have not measured it, because the measurement will change. And so it could be that it was in some superposition of mixture and when we measure, we picked out one of them. But that doesn't mean it was like that before. It means we might have changed it. So if we had obtained, let's say, go back to the coin, if we obtained tails instead of heads on the first throw, then if we keep looking at it, it's tails. And so we get 50% probability, and it collapses onto the other side. It collapses onto a particular choice, half the time, and once it has collapsed onto that particular choice, it remains there for any number of repeated measurements. It does not change. Now, the question is this. What happened to the uncertainty principle? Because now I'm claiming that we can get measured results with certainty. We're saying we always get the same result. We just measure it once, then the uncertainty goes away. And that's kind of interesting because it's not so simple. Because the uncertainty principle, which we quoted for position and momentum, applies to measuring two things, position and momentum at once, or one right after the other, not just one observable. There is no uncertainty about measuring one thing as well as you like. The problem is if you want to measure what you think of as everything that you could measure, then there will be some problems, some blurring perhaps that you didn't anticipate. So a deeper analysis shows us that not all properties need to be uncertain. In fact, if the two operators have the same set of eigenfunctions, this is why it's very important mathematically for us to be able to determine the eigenfunctions of an operator. Because we might have this operator and that operator representing this and that. And if it turns out that the two operators mathematically have the same set of eigenfunctions, even if they have different eigenvalues, usually they will, because they have different units and so on. Then we can measure both of them and we get exact results for both. So we may measure this and that and we get a certain value. And if we measure this and that again, we get the same for both. And there's no uncertainty. But unfortunately, position and momentum, which are two things that people like to determine, are not compatible in that way. Oh, there's this idea called complementarity. Observables that are incompatible cannot be measured to arbitrary precision. So here, what I've shown is a real coin. I flipped it and it happened to come up heads. It's a penny. And you can see that it's heads because you can see Lincoln in profile on the face of the coin. And you can even read other things on it, like the year it's minted. But let's just say we can tell that it's heads for sure. Now, suppose instead of trying to measure heads, and if I leave it there, it's going to obviously measure heads, heads, heads. I'm not going to flip because I'm not allowed to do anything to it, except look at it, measure it. If on the other hand, we're interested in the exact thickness of the coin, in that case, we have to orient it like this. And this was tricky to do, but the coin did balance on its edge. It was thick enough and the surface was flat enough. And my collaborator had a steady enough hand. And now, if you have the coin oriented like this, you can see exactly how thick it is. Whereas when it was down with the head pointing, you had no idea how thick it was. Imagine you're looking straight down on it so you can get the best possible view. Now, you can try, now if the coin's on edge, it's obviously unstable. And so anything I try to do to look at it could have it drop. But the question is, when it's like that, which side is heads? And the answer is because I'm looking at it edge on, I have no idea which side is heads. And quantum systems are very much like that if we have complementary variables. If I try to zero in on one of them, it means the other one fades out. And I can't get both at once because they're interfering with each other. And I just, there's no possibility of doing that. We could try to cheat. Here's the coin balanced on a pen with an eraser to keep it steady. And we could look at the coin on an angle like this. And the way it's angled here, I can pretty much tell its heads. It's not as clear as it was before. But I can pretty much tell its heads. But what I can't do now is measure the thickness very well. Because I'm seeing the thickness from an angle and it gets smaller and smaller and smaller as I get the heads and then I can't measure it very well. And basically in order to get the thickness better, I have to turn the coin toward me like that. And then finally at some point, I can't see whether it's heads or tails. Now with a coin, a macroscopic thing, I can look at it heads, I can orient it and say, well that side's heads. But with small things, you can forget that. That's not possible. Unless you can see its heads, you don't know what it is. And that's the problem. So the uncertainty principle really makes this numerically rigorous. It says exactly how well you could measure the thickness and or tell its heads when it's a small thing and when you know how the different variables, the things you're trying to measure interact with each other. That's basically what it's making much more rigorous. Let's talk now about classical atoms. In a classical atom, we have Maxwell's equations. And this was another problem actually at the turn of the century is that it was fairly easy to work out that an accelerating charge would radiate energy in accordance with Maxwell's equations. But if the electron, which is certainly accelerating, if we imagine it going in a circle around a positively charged nucleus, then it has to radiate energy, then it has to slow down because the energy doesn't come from nowhere. And what that means is that the electron would spiral in toward the proton and it would eventually condense onto it. And if that happened, there wouldn't be any electrons around to make bonds and so there wouldn't be any molecules, there wouldn't be any life, there wouldn't be any atoms even. There would just be like a neutron star or something with just all this condensed matter. Somehow, the electron is not like in a planetary orbit and it's not behaving according to the way a charge would in Maxwell's equations. And the reason it doesn't partly is the uncertainty principle. Because suppose the electron starts slowing down and spiraling in and coming in and in and in smaller and smaller and smaller. Well, we just did a calculation that showed if the electrons within 200 picometers that the minimum uncertainty in velocity is like 100,000 meters per second. And what that means is that it is impossible for the electron to spiral in and be on top of the proton in that itsy bitsy space and be stationary because that violates the uncertainty principle. And quantum mechanics says it's not possible to measure position of momentum that accurately. And therefore, the electron may start spiraling in and then may just get tossed out suddenly and go in a different direction. And so that saves us. So if the electron is not spiraling around like a planetary model of an atom in an orbit, then what is it doing? It certainly maintains a stable probability distribution because we look at atoms and we see that they have a cloud of negative charge around and it doesn't change if we don't disturb the atom if we just leave it alone. But we don't know where the electron is because the electron behaves like a wave. Unless we try to measure its position by which we would have to use a very energetic photon and blow the electron clean out of the atom basically. And then of course, we've lost our thread. We were trying to figure out what it looked like when we didn't look at it. The problem is that's not allowed. You can only talk about the things you can measure. You can't talk about things that you can imagine what they might be. Stable distributions of the electron density have to be standing waves. Standing wave, you can think of a guitar string. If I put my finger, I have a fret here. You can't move here. You can't move at the other end in between. It can vibrate and it just makes a stable pattern and sits there doing the same thing. And the electron then has to do something very much like that when it's in an atom and it has a very interesting wave property. We couldn't understand it at all if we thought of it as a rock or a BB moving around in there. An orbit of course is a periodic trajectory, but electrons don't have trajectories. And so instead of an orbit, we speak of orbit-oles, which is the wave analogy of a stable orbit. Now it's the wave function that has to somehow, like the guitar string, can only play a certain note if I hit that fret. The wave function can only play certain notes in the atom. It has to come around and match itself and give a stable standing wave. And that means the wavelength of the wave function has to match into the space into which the electron is confined. If it doesn't, you won't find the electron in that wave function. So there is destructive interference and the wave function vanishes. And if the wave function vanishes, then the chance of seeing the electron at that energy also vanishes because the wave function tells us the probability. Let's, a 3D thing is kind of hard to visualize, but we can certainly do it on an electron on a 2D sphere, a ring, and that makes it much easier for us to draw. So let's have a look then on an electron on a ring. Here is an electron going around as a wave. I say going around, but I don't know where it is because I haven't measured its position. But I have a standing wave. It's equal everywhere in space. I'm just showing the real part. When the real part's zero, the imaginary part is big. And that's why, another reason why we have to have complex waves sometimes. This one goes round and round and round. And every time it comes around, it's back in the same place. Goes around, comes back in the same place. Round and round. And so therefore, it's going to make a stable repeating pattern. It's going to sit there. And in fact, we can't see anything going round and round. I imagined it was doing that as if it were a little rock going around and around. But in fact, all it is is just this pattern. Stable repeating pattern. Because it exactly matches the condition that it met, meet itself when it hooks up again. But if I have a slightly different wavelength so that it doesn't match when it comes around, but it's a little bit off, it goes around, instead of matching perfect, it's a little higher, then if it goes around again, it's a little worse. If it goes around again, it's a little worse. Waves go up and down. Finally, it's coming around as the opposite sign. And let's go around then another time on this thing. Second time around, it's a worse mismatch. And if I go around several times, it appears when I draw the thing that the wave is up and down and up and down and up and down and up and down everywhere. And that means that it cancels itself out. There is no wave there if it's up and down and up and down. It's just canceled itself out. The only position that can have a wave is the one that perfectly matches. Now there could be a higher one, instead of three lobes, it has four. It goes around four. So that perfectly matches too. But there's not three and a half and there's not 3.1 and there's not pi lobes. There's exactly an integer number of lobes and that means there are a certain number of integer energy levels that the electron can be in and it can't just be anywhere. That's not allowed. And that's very important because it explains the spectroscopic observation of atoms where they didn't just irradiate any old light, but every element gave certain characteristic lines which depended where it was in the periodic table and so forth and so on. And of course it's very important for chemical analysis. That's one way you can tell what's in an unknown sample as you do atomic emission spectroscopy. Confined systems, atoms are confined systems because the electron has to stay there. But there are many other confined systems, nanoparticles, the so-called particle in a box which is a model problem we're going to do which has very easy mathematics compared to real problems which is why we do it. Wherever you have a confined system, it doesn't matter how it's confined. The wave function has to somehow fit. It has to fit into the space available. If it goes round or bounces back and forth or does anything and it comes back different, that means that particular wave function is going to cancel itself out and it's just gone. This matching condition really restricts the wave function to a certain set of values and gives us allowed energy states such as those that are observed in atoms and molecules and form the basis of all kinds of chemical analysis that we're going to do. If we've got much shorter wavelength light, we learned that short wavelength equals high energy and that long wavelength light equals low energy. And De Bruyley said, well, particles have a wave associated with them and now we've given this thing a name, the wave function. And it's a function and we can plot it if we have a functional form for it, which we sometimes do, and we can look at it. And what we find is that if the wave function has higher curvature, it's going up and down, up and down a lot like crazy. That's sort of like a photon with a short wavelength and that means that that state, that quantum state is higher energy than one that's all spread out and just kind of moping around and not really very many up and down parts, up and down nodes in the things. I'm going to close here with the position operator and we'll pick this up next time. We found that the momentum operator was to take minus i h bar times the derivative. The position operator is, in fact, I gave it before as an example of an operator. X hat on psi is just equal to the number x on psi. And the number x is the position of the particle. The momentum eigenfunction is e to the i p x upon h bar. And to make sure that the eigenfunction is normalized, we should include a normalization factor, which I'll just put n, some number here, to make sure it's normalized. And the question is, what does the probability density look like for a momentum eigenstate? Well, we just take phi of p times, phi star of p times phi of p. We get n e to the minus i p x, n e to the plus i p x, e to the plus, e to the minus, that's 1, because that's e to the 0. It doesn't matter whether it's imaginary or real, that still works. We just get n squared. But that's weird because it says the probability density doesn't depend on x. It's just some number. And what that means then is for a momentum eigenstate, the particle has equal probability of being anywhere at all, basically from plus or minus infinity, equally likely. And so position eigenstates, where the particle is definitely within 10 to the minus, no matter how small you want to think, at that point, a momentum eigenstates, where the momentum is exactly determined, are two completely different aspects of measurement. And they're completely at odds. The best you can do is you can get the position to within a certain limit, and then simultaneously you can get the momentum to within a certain limit. But if you try to get too aggressive with one, you just like squeeze this in, then the other one gets wide, because somehow this area between the two of them is like pushing on jello or something, it squeezes out if you try to get too aggressive with it. And there's no way you can minimize that effect except to have the very best uncertainty on the inequality, but you can't make it zero the way you'd like to. To have a position eigenstate, on the other hand, instead of this corkscrew, you should think of e to the i p x as a corkscrew. It is corking this way, the particle is going that way. It is corking the other way, it's going this way. But in either case, it's just a constant thing corking around like a corkscrew driving through a wine bottle cork. It's going to go a certain direction. On the other hand, position shouldn't be like that at all, but it should be like is it should be, the wave function should be piled up like a big pile of sand in this position. And then it should be zero elsewhere because we know when we take the wave function and square it, that tells us the probability of finding the particle. So if we take some function and pile it up like the Eiffel Tower real steep, then the particle is going to be there. And then it might have some uncertainty, it might be out here. But we could imagine piling it up very steep and very high, and that would be a position eigenfunction. Next time what I'm going to do is I'm going to take a model position function that's localized, and I'm going to expand it in terms of momentum functions and show you that the momentum of such a function like that becomes more and more uncertain as we make the position sharper. And then finally, we'll finally introduce, after the first week of class is over, we're going to introduce the wave equation, that's what we didn't have so far, that tells us exactly how these wave functions move forward in time and how they have certain energy and other properties. And that allows us then to discover what these wave functions actually are. So we'll pick it up there next time.
 Welcome back to lecture 3, Chem 131A.  We're going to continue where we left off.  Today it's more postulate, superposition,  operators, and measurement.  Where we last left our hero, we had decided  that the derivative operator is linear,  but it was not Hermitian.  And then I introduced this very ornate relationship  to describe what I meant by Hermitian.  And you might wonder what it means.  But what it means is basically, suppose we had complex numbers  and most numbers were complex.  And then we wanted to say that a number was real,  but we only had complex numbers.  Well, one trick we could use is we could say,  if z is equal to z star, then the number is real.  Because the only imaginary part that can be equal  to opposite itself is zero.  And zero imaginary part means the number is real.  And so really, Hermitian is just making sure  that when we measure something, we get a real number.  We still do believe that probability doesn't have  an imaginary part and neither does energy.  When we measure it, it has units of joules and so forth.  And so we want to make sure that these things  that we measure are Hermitian.  And this formula with these integrals and stars  and the operator in is just a very fancy way  of saying z is equal to z star.  Nothing more than that.  Okay, let's show that the derivative operator then  is not Hermitian.  Well, here's what we have to do.  We have to do an integral of f star d by dx g.  And we have to show that that is or is not equal  to the integral of g star d by dx f whole thing star.  When you see an integrand that has a derivative in it,  the first thing you think is I bet I can integrate  that by parts.  If you recall, integrating by parts is basically doing the  opposite as doing the derivative of uv is udv plus vdu.  We turn that around and we move the uv, one of them,  to the other side and then we set the integral equal to that.  Now the limits on this integration are plus and minus infinity  but I won't always put them in because it may get a little bit  messy.  But whatever it is, the wave functions have to vanish  at plus or minus infinity.  And the argument as to why they have to vanish is  if they had any amplitude out there, way out there,  then we couldn't normalize them.  They would get too big.  And so the only way we can have the area under the curve crank  down to some number is that it finally dies  out when we get far enough away.  So let's try integration by parts.  The formula is the integral of u d by dx of v of x is equal  to uv minus the integral the other way around v d by dx u.  And this is going to be convenient  because the Hermitian thing had them the other way around.  And now let's let u, the function u of x conventionally  and calculus, let's let that be f star.  And the v, let's let that be g. And let's try it.  So our equation becomes this, fairly intimidating looking  but not too bad.  The integral of f star d by dx g is equal  to f star g evaluated at plus or minus infinity minus the integral  of g d by dx f star.  And that is equal to minus the integral of g star d  by dx f whole thing star.  But that's not equal to what we want.  And so what we have, because we have a minus sign  and we want it to be a plus sign.  So it's not equal and therefore the derivative operator is  not Hermitian.  It's called anti-symmetric for obvious reasons.  When you swap them it changes sign.  But it's not Hermitian.  How can we have it be Hermitian?  Well interestingly enough we have  to use our friend the square root of negative one again.  And if we multiply the derivative operator by minus i h bar  that's enough to do the job because minus i star is plus i.  And so therefore that gets rid of the minus sign  that we got stuck with with the regular derivative.  And then we just follow everything else through.  And it works.  You say well what, why is the h bar there?  And the answer is this is quantum mechanics.  Of course there's an h bar there because we're going to have  to have that in almost everything we use.  And in fact the momentum operator p hat x which tells us the  momentum when it operates on a wave function it tells us the  momentum in the x direction is just given by minus i h bar d  by dx and it is a linear Hermitian operator.  Its eigenfunctions are very closely related to those  of the derivative operator because after all,  all it has is just an extra thing out in front.  But we want to make sure that the eigenvalues are real  and so these eigenfunctions are exponentials but we put in an i  and here we realize that p is real, x is real,  h bar is real and so this is e to the i px upon h bar.  We know that we have to have the units go away if we take an  exponential because an exponential is a power series,  1 plus x plus x squared over 2.  And if it has units we're adding feet and feet squared  and feet cubed and that doesn't make any sense.  And so with a little bit of dimensional analysis we come  to the idea that these functions here e to the i px upon h bar  are very good candidates.  So let's do another practice problem and have a look.  So let's show that these are the eigenfunctions in fact  of the derivative operator.  Well, let's take p hat x on our function phi of x. Let's put  in what p hat is minus i h bar d by dx on the function.  Let's put in the function which we assume is e  to the plus i px upon h bar and the derivative of anything times  x is the thing, excuse me, anything times e to the a x is a e  to the a x so we bring down the i p upon h bar  and now I think you can see why we want h bar out in front.  The h bars fold up minus i times plus i is minus i squared  but minus i squared is plus 1.  That goes away and that leaves us with p and that's p e  to the i px upon h bar and that's p times the eigenfunction  and therefore we've shown that the operator p hat returns the  eigenvalue p which has the units of momentum.  So the complex exponential is the eigenfunction  of the momentum operator and the eigenvalue is p.  In the language of linear algebra, the eigenfunctions  of a linear Hermitian operator form a basis.  So if I take a point in a two-dimensional plane  and I want to figure out where I am, I know that if I go a certain  unit out on x along the x axis and then up or down by y  that I can get to the point and furthermore any point anywhere  can be expressed as a combination of some distance this way  and some distance up or down and there's no point  that can escape.  So if we have a vector, any vector, any point x naught,  y naught that's equal to x naught times the coordinate unit  along the x axis plus y naught times the coordinate unit  along the y axis and just like that we can write any wave  function as a linear combination of eigenfunctions  of a Hermitian operator.  They form a basis.  No function can escape  and that's important because if some functions could escape  that would mean there were certain values  that we couldn't measure anything and that would be very bad  because what would happen to the probability?  Particles would be disappearing then.  Opostulate 5 is this.  It's quite a mouthful but we'll get to it.  When a wave function is not an eigenfunction  of the measured observable, the result  of the measurement is still an eigenvalue  but now the probability is given by the square modulus  of the expansion coefficient of the eigenfunctions  of the operator.  So if I have a wave function psi is some constant,  could be complex number, it doesn't matter  because all these functions can be complex.  So let's just call it c1 phi 1 plus c2 phi 2.  Then the probability of obtaining the first eigenvalue is the  square of c1 with the absolute value  so if there's an imaginary part you take c1 star c  and the probability of obtaining the second one is the square  of c2.  Those are the two probabilities  and if there are only two parts making up the wave function,  those are the only two values you can get.  Usually a wave function is made up of a whole bunch  of different eigenfunctions and so there are a lot  of different possibilities that you can get.  The eigenfunctions themselves have to be normalized.  That means if you happen to be in an eigenfunction,  your chance of being somewhere in the universe  and having that eigenvalue, let's say,  of momentum is equal to 1.  And so the basis functions themselves are normalized  and we always assume that they are normalized without comment.  And likewise for the wave function to be normalized once the  basis functions are normalized, that means that the probabilities  in those coefficients have to add up.  So the sum of the squares of all the coefficients always have  to add up to 1.  It's as if we have a unit circle and we're some point  on the circle and we have an x component and a y component  and the Pythagorean theorem says x squared plus y squared is equal  to 1 and that's how it works and it works the same way  in higher dimensions.  Second comment, the best way to think  of these eigenfunctions is not as things spraying around  in space.  Think of them as vectors.  Think of one eigenfunction pointing this way,  telling you the amount on this side.  The other one points this way.  A third one points up.  If I've got more I have to have an imagination.  But basically they're all at right angles to each other  and they're all telling the amount of this special state  that is in there to begin with.  And eigenfunctions of a linear Hermitian operator corresponding  to different eigenvalues are orthogonal and that's another  reason why it's good to think of them like vectors  because if I have an eigenfunction here  and this has one eigenvalue and I have another eigenfunction here  and it has a different eigenvalue,  then those two functions have nothing to do with each other.  They are as different as different can be.  They're in different directions.  They have no influence on each other.  And to see this normally suppose we have x and y,  we can tell they're at right angles because we can look.  But suppose I put my arms out some way and then I say,  well are those orthogonal?  Well you could try to mentally rotate and see if it comes back  to x and y but that's a very, very slow  and labor intensive way to do it.  Instead what you do is you take the dot product.  You take the product of the first two components,  the second two, the third two, you add them all up  and you see if that's zero.  And if it's zero that means they're orthogonal.  If it's not zero that means that they aren't orthogonal.  So for three real components, let's say two vectors in 3D space,  I just take ax, bx plus ay, by plus az, bz.  And if that sum that comes to zero,  it doesn't matter what the individual terms are.  If that sum comes to zero that means that the vector a  and the vector b are at right angles to each other.  And that's much, much easier to compute.  More generally if we've got lots of dimensions,  then we need to expand our sum.  So it could be a1, b1 because we don't want to have x, y,  and z if we've got let's say 5 or 6 or 20.  We run out a letter so we switch to numbers  where we won't run out.  A1, b1 plus a2, b2 plus a3, b3 and we just write that  in a shorthand as the sum over n of an, bn.  And that goes to however far we want it to go,  including in some cases to infinity.  And that should be zero.  And the same idea holds as functions except one,  the sum becomes an integral.  Because when you multiply the functions by each other,  they both depend on x.  And so adding up, you can't just add up,  you have to integrate to get the answer.  And number two, because the functions can be complex,  we have to take the complex conjugate of the first function.  Let's suppose we've got two functions f and g.  Then our orthogonality condition is as follows.  The integral of f star times g dx is zero.  Now we can show based on this and the definition of Hermitian  that the eigenfunctions  with different eigenvalues are orthogonal.  So here's what we do.  If it's an eigenfunction, it has an eigenvalue  that's a real number.  So let's put omega on phi one.  And we get omega on phi one.  We get phi one back because it's an eigenfunction.  We put omega on phi two, we get omega two.  And the only thing we need to know is  that omega one is not equal to omega two.  There are different numbers.  They're real and they're unequal.  And the operator, big omega hat, is Hermitian.  Let's take the first eigenvalue equation and make a series  of operations to both sides of it.  That's always what you do when you simplify equations.  You do the same thing to both sides methodically.  And if you do that, you never get mixed up  and nothing ever goes wrong.  And if you do some shorthand of cross multiplying this  and that and you don't know what you're doing exactly,  you'll oftentimes get it wrong.  So let's take this equation, omega phi one equals omega one,  phi one, and let's first multiply on the left.  We have to make sure we multiply on the same side  when we do this by phi two star.  Okay? So now we've got phi two star omega phi one is equal  to phi two star little omega one, phi one.  And then since omega one's a constant, I can pull it out  and say that's little omega one, phi two star, phi one.  Now I'm going to put an integral on both sides  because if two things are equal, then if I multiply them both  by phi two star, they're still equal.  And if I integrate them both over dx, they're still equal.  They don't become unequal.  And so I integrate over phi two star x omega phi one dx.  And that's the integral of omega one.  And since omega one is a constant, I pull it out and I end  up with omega one times the integral of phi two star,  phi one dx.  And we can do the same series of operations exactly.  But instead of having omega hat phi one, we take omega hat phi two  and we get omega two.  And we just go through the same.  Only we just swap the rolls of one and two.  We multiply by phi one star.  And if we do that, I've just not done every step here,  but omega hat phi two is equal to omega two times the integral  of phi one star, phi two dx.  Now let's take the complex conjugate of both sides  of the first equation.  So on the left hand side, I have the complex conjugate  of the whole thing, omega two, omega hat, omega one integrated.  And on the other side, I have little omega one times the integral  of phi two star phi one dx star.  And I can simplify that.  I leave the other side alone because that's going  to be the definition of Hermitian.  The right hand side, I turn to omega one star.  And then the integral of phi two star star times phi one star.  Well, the star of the star, let's see, I change i to minus i back  to i, so that goes away.  And I can then write the phi one in front of the phi two.  It doesn't matter.  I'm multiplying those.  There's no operator.  So I finally come to the following.  The integral of phi two star omega hat phi one is equal  to omega one times the integral of phi one star phi two.  What does that get us?  Well, the observable is Hermitian.  And so when I put in omega hat phi two x to give the eigenvalue omega two  and I do the same thing, I find that I get the same thing backwards.  So omega two star omega one phi one is equal to omega one star,  sorry, phi one star omega hat phi two.  And so using our two series of equations,  here's what we come to finally.  Omega two times the integral of phi one star phi two is equal  to omega one times the integral of phi one star phi two.  But omega two is not equal to omega one.  So let's subtract omega one times the integral from both sides.  Then we find out that omega two minus omega one times the integral is equal to zero.  But since omega two is not equal to omega one, it must be that the other thing zero.  Because if I have any number times something,  the only way I can make the whole thing zero is if the other thing zero.  And that means that the integral of phi one star phi two dx is zero.  And that means that they are orthogonal.  So that's the proof.  You'll have to go over it a couple of times to get it down.  But that's kind of a standard thing that's done in quantum mechanics to show  that eigenfunctions for different values of eigenvalues are in fact orthogonal.  Now suppose we make a measurement on a quantum system.  And it's represented by a wave function psi that's not an eigenstate  of the operator in question.  Then what happens?  Well, we express psi as a linear combination of eigenfunctions.  And that we know we can do that because there is no function that can escape us.  And we know our eigenfunctions can be made normalized,  so we assume they're normalized.  And then the probability of obtaining a particular eigenvalue, let's say eigenvalue k  out of the totality from one to n is the absolute value of Ck squared,  where Ck is the coefficient of the kth eigenfunction.  Now suppose we then make the measurement again right away.  The question is, do we get a different result?  And the answer is kind of surprising, but the answer is no.  It turns out if we make the measurement again, we get the same result.  And if we keep measuring the same observable over and over, we keep getting the same result.  And now sort of mysteriously, in a way, it's 100% certain that we're going to get that result.  There is no other result that we're going to get.  And I've tried to encapsulate this in this kind of pseudo equation.  We start out with probabilities.  It could be any of these eigenstates.  And then we make a measurement.  And somehow one of them is chosen.  And we can't say how, even in an ideal experiment,  but we can say what the probability is.  Let's say 25% of the time we get this result.  Now if we measure again and nothing's intervened, we haven't done anything,  we get the same result again.  And again, and again, and again.  And now there's no probability at all.  It's always 100%.  So it's exact certainty.  That is, measurement is kind of like a filter.  All the other possibilities are filtered out, leaving the one that's actually observed.  If you sort coins with a coins order, you roll them down.  And when they fit the size of the tube, they drop in.  And if you don't know which tube is going to drop into it first,  then that's like being uncertain.  But then if you drop the thing in and it drops into the third tube,  and then you empty it out of there and put it back in,  it's going to drop into the third tube again.  And that's kind of an analogy for what's going on here.  So if we look at the probability of a coin dropping into the third tube,  or we could say, for example, suppose we flip a coin until it hits  and stops spinning and lies down, we assume it's 50% probability heads  and 50% probability tails.  Then we see its heads, then if we don't do anything, we don't flip it again.  Just sit there.  It's heads.  It's heads again.  It's heads again and so forth.  And it's heads as many times as we want to keep looking at it.  And that's what this theory of measurement is saying.  In other words, when you make a measurement, you rule out certain possibilities.  They're now gone.  Now the measurement's made.  It came up.  It happened to come up this.  If you make it again, it comes up this.  If you make it again, it comes up this.  And that's assuming you don't have any interaction in between.  But this is an idealized experiment.  We aren't talking about how we practically would implement it.  And likewise, in quantum mechanics, it's just like looking at that coin.  If we make the same ideal measurement again and again,  after filtering out all the other possibilities,  we just get the one result that we got, the same result each time.  But before we made the measurement, it seemed like there were other possibilities.  And if we start all over, not with the one we've measured,  but with an identical particle coming through that we haven't measured,  then we might get a different answer.  And then if we measure that again, we'll get that different answer again,  and so on and so forth.  And so it seems as if the measurement itself took this very fragile thing,  this wave function, and it made it collapse onto a particular eigenfunction.  Said, right, this is it.  And then all the other possibilities vanished forever.  If I decide I'm going to give a lecture, I turn up and do the lecture,  but if I decide I'm going to the beach instead and I go to the beach,  then the lecture is not a possibility, and it's now vanished forever.  It's gone and I'm at the beach.  And so by making that choice, I've narrowed down the possibilities.  Before I did that, I could say, well, 50-50, I give the lecture, go to the beach.  And that's important because it means that measurement affects quantum systems,  and that means that there is no such thing as a property without measuring it.  We usually think that things have properties independent of measuring them  because they seem to.  This pointer, for example, has a mass, whether I have it on a scale or not,  and I assume it's the same.  And for big objects that are always being bombarded by all kinds of things  and never have a chance to let the wave function sneak around, that's certainly true.  But for small things, we have to be very wary about assuming  that something has a property if we have not measured it,  because the measurement will change.  And so it could be that it was in some superposition of mixture  and when we measure, we picked out one of them.  But that doesn't mean it was like that before.  It means we might have changed it.  So if we had obtained, let's say, go back to the coin,  if we obtained tails instead of heads on the first throw,  then if we keep looking at it, it's tails.  And so we get 50% probability, and it collapses onto the other side.  It collapses onto a particular choice, half the time,  and once it has collapsed onto that particular choice,  it remains there for any number of repeated measurements.  It does not change.  Now, the question is this.  What happened to the uncertainty principle?  Because now I'm claiming that we can get measured results with certainty.  We're saying we always get the same result.  We just measure it once, then the uncertainty goes away.  And that's kind of interesting because it's not so simple.  Because the uncertainty principle, which we quoted for position and  momentum, applies to measuring two things, position and momentum at once,  or one right after the other, not just one observable.  There is no uncertainty about measuring one thing as well as you like.  The problem is if you want to measure what you think of as everything  that you could measure, then there will be some problems,  some blurring perhaps that you didn't anticipate.  So a deeper analysis shows us that not all properties need to be uncertain.  In fact, if the two operators have the same set of eigenfunctions,  this is why it's very important mathematically for us to be able to  determine the eigenfunctions of an operator.  Because we might have this operator and that operator representing this and  that.  And if it turns out that the two operators mathematically have the same set of  eigenfunctions, even if they have different eigenvalues, usually they will,  because they have different units and so on.  Then we can measure both of them and we get exact results for both.  So we may measure this and that and we get a certain value.  And if we measure this and that again, we get the same for both.  And there's no uncertainty.  But unfortunately, position and momentum,  which are two things that people like to determine, are not compatible in that way.  Oh, there's this idea called complementarity.  Observables that are incompatible cannot be measured to arbitrary precision.  So here, what I've shown is a real coin.  I flipped it and it happened to come up heads.  It's a penny.  And you can see that it's heads because you can see Lincoln in  profile on the face of the coin.  And you can even read other things on it, like the year it's minted.  But let's just say we can tell that it's heads for sure.  Now, suppose instead of trying to measure heads, and if I leave it there,  it's going to obviously measure heads, heads, heads.  I'm not going to flip because I'm not allowed to do anything to it, except look at it, measure it.  If on the other hand, we're interested in the exact thickness of the coin,  in that case, we have to orient it like this.  And this was tricky to do, but the coin did balance on its edge.  It was thick enough and the surface was flat enough.  And my collaborator had a steady enough hand.  And now, if you have the coin oriented like this, you can see exactly how thick it is.  Whereas when it was down with the head pointing, you had no idea how thick it was.  Imagine you're looking straight down on it so you can get the best possible view.  Now, you can try, now if the coin's on edge, it's obviously unstable.  And so anything I try to do to look at it could have it drop.  But the question is, when it's like that, which side is heads?  And the answer is because I'm looking at it edge on, I have no idea which side is heads.  And quantum systems are very much like that if we have complementary variables.  If I try to zero in on one of them, it means the other one fades out.  And I can't get both at once because they're interfering with each other.  And I just, there's no possibility of doing that.  We could try to cheat.  Here's the coin balanced on a pen with an eraser to keep it steady.  And we could look at the coin on an angle like this.  And the way it's angled here, I can pretty much tell its heads.  It's not as clear as it was before.  But I can pretty much tell its heads.  But what I can't do now is measure the thickness very well.  Because I'm seeing the thickness from an angle and it gets smaller and smaller and smaller  as I get the heads and then I can't measure it very well.  And basically in order to get the thickness better, I have to turn the coin toward me like that.  And then finally at some point, I can't see whether it's heads or tails.  Now with a coin, a macroscopic thing, I can look at it heads, I can orient it and say,  well that side's heads.  But with small things, you can forget that.  That's not possible.  Unless you can see its heads, you don't know what it is.  And that's the problem.  So the uncertainty principle really makes this numerically rigorous.  It says exactly how well you could measure the thickness and or tell its heads when it's a small thing  and when you know how the different variables, the things you're trying to measure interact  with each other.  That's basically what it's making much more rigorous.  Let's talk now about classical atoms.  In a classical atom, we have Maxwell's equations.  And this was another problem actually at the turn of the century is  that it was fairly easy to work out that an accelerating charge would radiate energy  in accordance with Maxwell's equations.  But if the electron, which is certainly accelerating, if we imagine it going in a circle  around a positively charged nucleus, then it has to radiate energy,  then it has to slow down because the energy doesn't come from nowhere.  And what that means is that the electron would spiral in toward the proton  and it would eventually condense onto it.  And if that happened, there wouldn't be any electrons around to make bonds  and so there wouldn't be any molecules, there wouldn't be any life,  there wouldn't be any atoms even.  There would just be like a neutron star or something with just all this condensed matter.  Somehow, the electron is not like in a planetary orbit and it's not behaving according  to the way a charge would in Maxwell's equations.  And the reason it doesn't partly is the uncertainty principle.  Because suppose the electron starts slowing down and spiraling in and coming in and in  and in smaller and smaller and smaller.  Well, we just did a calculation that showed if the electrons within 200 picometers  that the minimum uncertainty in velocity is like 100,000 meters per second.  And what that means is that it is impossible for the electron to spiral in and be on top  of the proton in that itsy bitsy space and be stationary  because that violates the uncertainty principle.  And quantum mechanics says it's not possible to measure position of momentum that accurately.  And therefore, the electron may start spiraling in and then may just get tossed  out suddenly and go in a different direction.  And so that saves us.  So if the electron is not spiraling around like a planetary model of an atom in an orbit,  then what is it doing?  It certainly maintains a stable probability distribution because we look at atoms  and we see that they have a cloud of negative charge around and it doesn't change  if we don't disturb the atom if we just leave it alone.  But we don't know where the electron is because the electron behaves like a wave.  Unless we try to measure its position by which we would have to use a very energetic photon  and blow the electron clean out of the atom basically.  And then of course, we've lost our thread.  We were trying to figure out what it looked like when we didn't look at it.  The problem is that's not allowed.  You can only talk about the things you can measure.  You can't talk about things that you can imagine what they might be.  Stable distributions of the electron density have to be standing waves.  Standing wave, you can think of a guitar string.  If I put my finger, I have a fret here.  You can't move here.  You can't move at the other end in between.  It can vibrate and it just makes a stable pattern and sits there doing the same thing.  And the electron then has to do something very much like that when it's in an atom  and it has a very interesting wave property.  We couldn't understand it at all if we thought of it as a rock or a BB moving around in there.  An orbit of course is a periodic trajectory, but electrons don't have trajectories.  And so instead of an orbit, we speak of orbit-oles,  which is the wave analogy of a stable orbit.  Now it's the wave function that has to somehow, like the guitar string,  can only play a certain note if I hit that fret.  The wave function can only play certain notes in the atom.  It has to come around and match itself and give a stable standing wave.  And that means the wavelength of the wave function has to match into the space  into which the electron is confined.  If it doesn't, you won't find the electron in that wave function.  So there is destructive interference and the wave function vanishes.  And if the wave function vanishes, then the chance of seeing the electron  at that energy also vanishes because the wave function tells us the probability.  Let's, a 3D thing is kind of hard to visualize,  but we can certainly do it on an electron on a 2D sphere, a ring,  and that makes it much easier for us to draw.  So let's have a look then on an electron on a ring.  Here is an electron going around as a wave.  I say going around, but I don't know where it is  because I haven't measured its position.  But I have a standing wave.  It's equal everywhere in space.  I'm just showing the real part.  When the real part's zero, the imaginary part is big.  And that's why, another reason why we have to have complex waves sometimes.  This one goes round and round and round.  And every time it comes around, it's back in the same place.  Goes around, comes back in the same place.  Round and round.  And so therefore, it's going to make a stable repeating pattern.  It's going to sit there.  And in fact, we can't see anything going round and round.  I imagined it was doing that as if it were a little rock going around and around.  But in fact, all it is is just this pattern.  Stable repeating pattern.  Because it exactly matches the condition that it met, meet itself when it hooks up again.  But if I have a slightly different wavelength so that it doesn't match when it comes around,  but it's a little bit off, it goes around, instead of matching perfect, it's a little higher,  then if it goes around again, it's a little worse.  If it goes around again, it's a little worse.  Waves go up and down.  Finally, it's coming around as the opposite sign.  And let's go around then another time on this thing.  Second time around, it's a worse mismatch.  And if I go around several times, it appears when I draw the thing  that the wave is up and down and up and down and up and down and up and down everywhere.  And that means that it cancels itself out.  There is no wave there if it's up and down and up and down.  It's just canceled itself out.  The only position that can have a wave is the one that perfectly matches.  Now there could be a higher one, instead of three lobes, it has four.  It goes around four.  So that perfectly matches too.  But there's not three and a half and there's not 3.1 and there's not pi lobes.  There's exactly an integer number of lobes and that means there are a certain number  of integer energy levels that the electron can be in and it can't just be anywhere.  That's not allowed.  And that's very important because it explains the spectroscopic observation of atoms  where they didn't just irradiate any old light, but every element gave certain characteristic lines  which depended where it was in the periodic table and so forth and so on.  And of course it's very important for chemical analysis.  That's one way you can tell what's in an unknown sample as you do atomic emission spectroscopy.  Confined systems, atoms are confined systems because the electron has to stay there.  But there are many other confined systems, nanoparticles, the so-called particle in a box  which is a model problem we're going to do which has very easy mathematics compared to real problems  which is why we do it.  Wherever you have a confined system, it doesn't matter how it's confined.  The wave function has to somehow fit.  It has to fit into the space available.  If it goes round or bounces back and forth or does anything and it comes back different,  that means that particular wave function is going to cancel itself out and it's just gone.  This matching condition really restricts the wave function to a certain set of values  and gives us allowed energy states such as those that are observed in atoms and molecules  and form the basis of all kinds of chemical analysis that we're going to do.  If we've got much shorter wavelength light, we learned that short wavelength equals high energy  and that long wavelength light equals low energy.  And De Bruyley said, well, particles have a wave associated with them  and now we've given this thing a name, the wave function.  And it's a function and we can plot it if we have a functional form for it, which we sometimes do,  and we can look at it.  And what we find is that if the wave function has higher curvature, it's going up and down,  up and down a lot like crazy.  That's sort of like a photon with a short wavelength and that means that that state,  that quantum state is higher energy than one that's all spread out and just kind of moping  around and not really very many up and down parts, up and down nodes in the things.  I'm going to close here with the position operator and we'll pick this up next time.  We found that the momentum operator was to take minus i h bar times the derivative.  The position operator is, in fact, I gave it before as an example of an operator.  X hat on psi is just equal to the number x on psi.  And the number x is the position of the particle.  The momentum eigenfunction is e to the i p x upon h bar.  And to make sure that the eigenfunction is normalized,  we should include a normalization factor, which I'll just put n, some number here,  to make sure it's normalized.  And the question is, what does the probability density look like for a momentum eigenstate?  Well, we just take phi of p times, phi star of p times phi of p. We get n e to the minus i p x,  n e to the plus i p x, e to the plus, e to the minus, that's 1, because that's e to the 0.  It doesn't matter whether it's imaginary or real, that still works.  We just get n squared.  But that's weird because it says the probability density doesn't depend on x.  It's just some number.  And what that means then is for a momentum eigenstate,  the particle has equal probability of being anywhere at all,  basically from plus or minus infinity, equally likely.  And so position eigenstates, where the particle is definitely within 10 to the minus,  no matter how small you want to think, at that point, a momentum eigenstates,  where the momentum is exactly determined, are two completely different aspects of measurement.  And they're completely at odds.  The best you can do is you can get the position to within a certain limit,  and then simultaneously you can get the momentum to within a certain limit.  But if you try to get too aggressive with one, you just like squeeze this in,  then the other one gets wide, because somehow this area between the two of them is like pushing  on jello or something, it squeezes out if you try to get too aggressive with it.  And there's no way you can minimize that effect except to have the very best uncertainty  on the inequality, but you can't make it zero the way you'd like to.  To have a position eigenstate, on the other hand, instead of this corkscrew,  you should think of e to the i p x as a corkscrew.  It is corking this way, the particle is going that way.  It is corking the other way, it's going this way.  But in either case, it's just a constant thing corking around like a corkscrew driving  through a wine bottle cork.  It's going to go a certain direction.  On the other hand, position shouldn't be like that at all, but it should be like is it should be,  the wave function should be piled up like a big pile of sand in this position.  And then it should be zero elsewhere because we know when we take the wave function and square it,  that tells us the probability of finding the particle.  So if we take some function and pile it up like the Eiffel Tower real steep,  then the particle is going to be there.  And then it might have some uncertainty, it might be out here.  But we could imagine piling it up very steep and very high,  and that would be a position eigenfunction.  Next time what I'm going to do is I'm going to take a model position function that's localized,  and I'm going to expand it in terms of momentum functions and show you that the momentum  of such a function like that becomes more and more uncertain as we make the position sharper.  And then finally, we'll finally introduce, after the first week of class is over,  we're going to introduce the wave equation, that's what we didn't have so far,  that tells us exactly how these wave functions move forward in time  and how they have certain energy and other properties.  And that allows us then to discover what these wave functions actually are.  So we'll pick it up there next time.
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Welcome back to lecture 3, Chem 131A. We're going to continue where we left off. Today it's more postulate, superposition, operators, and measurement. Where we last left our hero, we had decided that the derivative operator is linear, but it was not Hermitian. And then I introduced this very ornate relationship to describe what I meant by Hermitian. And you might wonder what it means. But what it means is basically, suppose we had complex numbers and most numbers were complex. And then we wanted to say that a number was real, but we only had complex numbers. Well, one trick we could use is we could say, if z is equal to z star, then the number is real. Because the only imaginary part that can be equal to opposite itself is zero. And zero imaginary part means the number is real. And so really, Hermitian is just making sure that when we measure something, we get a real number. We still do believe that probability doesn't have an imaginary part and neither does energy. When we measure it, it has units of joules and so forth. And so we want to make sure that these things that we measure are Hermitian. And this formula with these integrals and stars and the operator in is just a very fancy way of saying z is equal to z star. Nothing more than that. Okay, let's show that the derivative operator then is not Hermitian. Well, here's what we have to do. We have to do an integral of f star d by dx g. And we have to show that that is or is not equal to the integral of g star d by dx f whole thing star. When you see an integrand that has a derivative in it, the first thing you think is I bet I can integrate that by parts. If you recall, integrating by parts is basically doing the opposite as doing the derivative of uv is udv plus vdu. We turn that around and we move the uv, one of them, to the other side and then we set the integral equal to that. Now the limits on this integration are plus and minus infinity but I won't always put them in because it may get a little bit messy. But whatever it is, the wave functions have to vanish at plus or minus infinity. And the argument as to why they have to vanish is if they had any amplitude out there, way out there, then we couldn't normalize them. They would get too big. And so the only way we can have the area under the curve crank down to some number is that it finally dies out when we get far enough away. So let's try integration by parts. The formula is the integral of u d by dx of v of x is equal to uv minus the integral the other way around v d by dx u. And this is going to be convenient because the Hermitian thing had them the other way around. And now let's let u, the function u of x conventionally and calculus, let's let that be f star. And the v, let's let that be g. And let's try it. So our equation becomes this, fairly intimidating looking but not too bad. The integral of f star d by dx g is equal to f star g evaluated at plus or minus infinity minus the integral of g d by dx f star. And that is equal to minus the integral of g star d by dx f whole thing star. But that's not equal to what we want. And so what we have, because we have a minus sign and we want it to be a plus sign. So it's not equal and therefore the derivative operator is not Hermitian. It's called anti-symmetric for obvious reasons. When you swap them it changes sign. But it's not Hermitian. How can we have it be Hermitian? Well interestingly enough we have to use our friend the square root of negative one again. And if we multiply the derivative operator by minus i h bar that's enough to do the job because minus i star is plus i. And so therefore that gets rid of the minus sign that we got stuck with with the regular derivative. And then we just follow everything else through. And it works. You say well what, why is the h bar there? And the answer is this is quantum mechanics. Of course there's an h bar there because we're going to have to have that in almost everything we use. And in fact the momentum operator p hat x which tells us the momentum when it operates on a wave function it tells us the momentum in the x direction is just given by minus i h bar d by dx and it is a linear Hermitian operator. Its eigenfunctions are very closely related to those of the derivative operator because after all, all it has is just an extra thing out in front. But we want to make sure that the eigenvalues are real and so these eigenfunctions are exponentials but we put in an i and here we realize that p is real, x is real, h bar is real and so this is e to the i px upon h bar. We know that we have to have the units go away if we take an exponential because an exponential is a power series, 1 plus x plus x squared over 2. And if it has units we're adding feet and feet squared and feet cubed and that doesn't make any sense. And so with a little bit of dimensional analysis we come to the idea that these functions here e to the i px upon h bar are very good candidates. So let's do another practice problem and have a look. So let's show that these are the eigenfunctions in fact of the derivative operator. Well, let's take p hat x on our function phi of x. Let's put in what p hat is minus i h bar d by dx on the function. Let's put in the function which we assume is e to the plus i px upon h bar and the derivative of anything times x is the thing, excuse me, anything times e to the a x is a e to the a x so we bring down the i p upon h bar and now I think you can see why we want h bar out in front. The h bars fold up minus i times plus i is minus i squared but minus i squared is plus 1. That goes away and that leaves us with p and that's p e to the i px upon h bar and that's p times the eigenfunction and therefore we've shown that the operator p hat returns the eigenvalue p which has the units of momentum. So the complex exponential is the eigenfunction of the momentum operator and the eigenvalue is p. In the language of linear algebra, the eigenfunctions of a linear Hermitian operator form a basis. So if I take a point in a two-dimensional plane and I want to figure out where I am, I know that if I go a certain unit out on x along the x axis and then up or down by y that I can get to the point and furthermore any point anywhere can be expressed as a combination of some distance this way and some distance up or down and there's no point that can escape. So if we have a vector, any vector, any point x naught, y naught that's equal to x naught times the coordinate unit along the x axis plus y naught times the coordinate unit along the y axis and just like that we can write any wave function as a linear combination of eigenfunctions of a Hermitian operator. They form a basis. No function can escape and that's important because if some functions could escape that would mean there were certain values that we couldn't measure anything and that would be very bad because what would happen to the probability? Particles would be disappearing then. Opostulate 5 is this. It's quite a mouthful but we'll get to it. When a wave function is not an eigenfunction of the measured observable, the result of the measurement is still an eigenvalue but now the probability is given by the square modulus of the expansion coefficient of the eigenfunctions of the operator. So if I have a wave function psi is some constant, could be complex number, it doesn't matter because all these functions can be complex. So let's just call it c1 phi 1 plus c2 phi 2. Then the probability of obtaining the first eigenvalue is the square of c1 with the absolute value so if there's an imaginary part you take c1 star c and the probability of obtaining the second one is the square of c2. Those are the two probabilities and if there are only two parts making up the wave function, those are the only two values you can get. Usually a wave function is made up of a whole bunch of different eigenfunctions and so there are a lot of different possibilities that you can get. The eigenfunctions themselves have to be normalized. That means if you happen to be in an eigenfunction, your chance of being somewhere in the universe and having that eigenvalue, let's say, of momentum is equal to 1. And so the basis functions themselves are normalized and we always assume that they are normalized without comment. And likewise for the wave function to be normalized once the basis functions are normalized, that means that the probabilities in those coefficients have to add up. So the sum of the squares of all the coefficients always have to add up to 1. It's as if we have a unit circle and we're some point on the circle and we have an x component and a y component and the Pythagorean theorem says x squared plus y squared is equal to 1 and that's how it works and it works the same way in higher dimensions. Second comment, the best way to think of these eigenfunctions is not as things spraying around in space. Think of them as vectors. Think of one eigenfunction pointing this way, telling you the amount on this side. The other one points this way. A third one points up. If I've got more I have to have an imagination. But basically they're all at right angles to each other and they're all telling the amount of this special state that is in there to begin with. And eigenfunctions of a linear Hermitian operator corresponding to different eigenvalues are orthogonal and that's another reason why it's good to think of them like vectors because if I have an eigenfunction here and this has one eigenvalue and I have another eigenfunction here and it has a different eigenvalue, then those two functions have nothing to do with each other. They are as different as different can be. They're in different directions. They have no influence on each other. And to see this normally suppose we have x and y, we can tell they're at right angles because we can look. But suppose I put my arms out some way and then I say, well are those orthogonal? Well you could try to mentally rotate and see if it comes back to x and y but that's a very, very slow and labor intensive way to do it. Instead what you do is you take the dot product. You take the product of the first two components, the second two, the third two, you add them all up and you see if that's zero. And if it's zero that means they're orthogonal. If it's not zero that means that they aren't orthogonal. So for three real components, let's say two vectors in 3D space, I just take ax, bx plus ay, by plus az, bz. And if that sum that comes to zero, it doesn't matter what the individual terms are. If that sum comes to zero that means that the vector a and the vector b are at right angles to each other. And that's much, much easier to compute. More generally if we've got lots of dimensions, then we need to expand our sum. So it could be a1, b1 because we don't want to have x, y, and z if we've got let's say 5 or 6 or 20. We run out a letter so we switch to numbers where we won't run out. A1, b1 plus a2, b2 plus a3, b3 and we just write that in a shorthand as the sum over n of an, bn. And that goes to however far we want it to go, including in some cases to infinity. And that should be zero. And the same idea holds as functions except one, the sum becomes an integral. Because when you multiply the functions by each other, they both depend on x. And so adding up, you can't just add up, you have to integrate to get the answer. And number two, because the functions can be complex, we have to take the complex conjugate of the first function. Let's suppose we've got two functions f and g. Then our orthogonality condition is as follows. The integral of f star times g dx is zero. Now we can show based on this and the definition of Hermitian that the eigenfunctions with different eigenvalues are orthogonal. So here's what we do. If it's an eigenfunction, it has an eigenvalue that's a real number. So let's put omega on phi one. And we get omega on phi one. We get phi one back because it's an eigenfunction. We put omega on phi two, we get omega two. And the only thing we need to know is that omega one is not equal to omega two. There are different numbers. They're real and they're unequal. And the operator, big omega hat, is Hermitian. Let's take the first eigenvalue equation and make a series of operations to both sides of it. That's always what you do when you simplify equations. You do the same thing to both sides methodically. And if you do that, you never get mixed up and nothing ever goes wrong. And if you do some shorthand of cross multiplying this and that and you don't know what you're doing exactly, you'll oftentimes get it wrong. So let's take this equation, omega phi one equals omega one, phi one, and let's first multiply on the left. We have to make sure we multiply on the same side when we do this by phi two star. Okay? So now we've got phi two star omega phi one is equal to phi two star little omega one, phi one. And then since omega one's a constant, I can pull it out and say that's little omega one, phi two star, phi one. Now I'm going to put an integral on both sides because if two things are equal, then if I multiply them both by phi two star, they're still equal. And if I integrate them both over dx, they're still equal. They don't become unequal. And so I integrate over phi two star x omega phi one dx. And that's the integral of omega one. And since omega one is a constant, I pull it out and I end up with omega one times the integral of phi two star, phi one dx. And we can do the same series of operations exactly. But instead of having omega hat phi one, we take omega hat phi two and we get omega two. And we just go through the same. Only we just swap the rolls of one and two. We multiply by phi one star. And if we do that, I've just not done every step here, but omega hat phi two is equal to omega two times the integral of phi one star, phi two dx. Now let's take the complex conjugate of both sides of the first equation. So on the left hand side, I have the complex conjugate of the whole thing, omega two, omega hat, omega one integrated. And on the other side, I have little omega one times the integral of phi two star phi one dx star. And I can simplify that. I leave the other side alone because that's going to be the definition of Hermitian. The right hand side, I turn to omega one star. And then the integral of phi two star star times phi one star. Well, the star of the star, let's see, I change i to minus i back to i, so that goes away. And I can then write the phi one in front of the phi two. It doesn't matter. I'm multiplying those. There's no operator. So I finally come to the following. The integral of phi two star omega hat phi one is equal to omega one times the integral of phi one star phi two. What does that get us? Well, the observable is Hermitian. And so when I put in omega hat phi two x to give the eigenvalue omega two and I do the same thing, I find that I get the same thing backwards. So omega two star omega one phi one is equal to omega one star, sorry, phi one star omega hat phi two. And so using our two series of equations, here's what we come to finally. Omega two times the integral of phi one star phi two is equal to omega one times the integral of phi one star phi two. But omega two is not equal to omega one. So let's subtract omega one times the integral from both sides. Then we find out that omega two minus omega one times the integral is equal to zero. But since omega two is not equal to omega one, it must be that the other thing zero. Because if I have any number times something, the only way I can make the whole thing zero is if the other thing zero. And that means that the integral of phi one star phi two dx is zero. And that means that they are orthogonal. So that's the proof. You'll have to go over it a couple of times to get it down. But that's kind of a standard thing that's done in quantum mechanics to show that eigenfunctions for different values of eigenvalues are in fact orthogonal. Now suppose we make a measurement on a quantum system. And it's represented by a wave function psi that's not an eigenstate of the operator in question. Then what happens? Well, we express psi as a linear combination of eigenfunctions. And that we know we can do that because there is no function that can escape us. And we know our eigenfunctions can be made normalized, so we assume they're normalized. And then the probability of obtaining a particular eigenvalue, let's say eigenvalue k out of the totality from one to n is the absolute value of Ck squared, where Ck is the coefficient of the kth eigenfunction. Now suppose we then make the measurement again right away. The question is, do we get a different result? And the answer is kind of surprising, but the answer is no. It turns out if we make the measurement again, we get the same result. And if we keep measuring the same observable over and over, we keep getting the same result. And now sort of mysteriously, in a way, it's 100% certain that we're going to get that result. There is no other result that we're going to get. And I've tried to encapsulate this in this kind of pseudo equation. We start out with probabilities. It could be any of these eigenstates. And then we make a measurement. And somehow one of them is chosen. And we can't say how, even in an ideal experiment, but we can say what the probability is. Let's say 25% of the time we get this result. Now if we measure again and nothing's intervened, we haven't done anything, we get the same result again. And again, and again, and again. And now there's no probability at all. It's always 100%. So it's exact certainty. That is, measurement is kind of like a filter. All the other possibilities are filtered out, leaving the one that's actually observed. If you sort coins with a coins order, you roll them down. And when they fit the size of the tube, they drop in. And if you don't know which tube is going to drop into it first, then that's like being uncertain. But then if you drop the thing in and it drops into the third tube, and then you empty it out of there and put it back in, it's going to drop into the third tube again. And that's kind of an analogy for what's going on here. So if we look at the probability of a coin dropping into the third tube, or we could say, for example, suppose we flip a coin until it hits and stops spinning and lies down, we assume it's 50% probability heads and 50% probability tails. Then we see its heads, then if we don't do anything, we don't flip it again. Just sit there. It's heads. It's heads again. It's heads again and so forth. And it's heads as many times as we want to keep looking at it. And that's what this theory of measurement is saying. In other words, when you make a measurement, you rule out certain possibilities. They're now gone. Now the measurement's made. It came up. It happened to come up this. If you make it again, it comes up this. If you make it again, it comes up this. And that's assuming you don't have any interaction in between. But this is an idealized experiment. We aren't talking about how we practically would implement it. And likewise, in quantum mechanics, it's just like looking at that coin. If we make the same ideal measurement again and again, after filtering out all the other possibilities, we just get the one result that we got, the same result each time. But before we made the measurement, it seemed like there were other possibilities. And if we start all over, not with the one we've measured, but with an identical particle coming through that we haven't measured, then we might get a different answer. And then if we measure that again, we'll get that different answer again, and so on and so forth. And so it seems as if the measurement itself took this very fragile thing, this wave function, and it made it collapse onto a particular eigenfunction. Said, right, this is it. And then all the other possibilities vanished forever. If I decide I'm going to give a lecture, I turn up and do the lecture, but if I decide I'm going to the beach instead and I go to the beach, then the lecture is not a possibility, and it's now vanished forever. It's gone and I'm at the beach. And so by making that choice, I've narrowed down the possibilities. Before I did that, I could say, well, 50-50, I give the lecture, go to the beach. And that's important because it means that measurement affects quantum systems, and that means that there is no such thing as a property without measuring it. We usually think that things have properties independent of measuring them because they seem to. This pointer, for example, has a mass, whether I have it on a scale or not, and I assume it's the same. And for big objects that are always being bombarded by all kinds of things and never have a chance to let the wave function sneak around, that's certainly true. But for small things, we have to be very wary about assuming that something has a property if we have not measured it, because the measurement will change. And so it could be that it was in some superposition of mixture and when we measure, we picked out one of them. But that doesn't mean it was like that before. It means we might have changed it. So if we had obtained, let's say, go back to the coin, if we obtained tails instead of heads on the first throw, then if we keep looking at it, it's tails. And so we get 50% probability, and it collapses onto the other side. It collapses onto a particular choice, half the time, and once it has collapsed onto that particular choice, it remains there for any number of repeated measurements. It does not change. Now, the question is this. What happened to the uncertainty principle? Because now I'm claiming that we can get measured results with certainty. We're saying we always get the same result. We just measure it once, then the uncertainty goes away. And that's kind of interesting because it's not so simple. Because the uncertainty principle, which we quoted for position and momentum, applies to measuring two things, position and momentum at once, or one right after the other, not just one observable. There is no uncertainty about measuring one thing as well as you like. The problem is if you want to measure what you think of as everything that you could measure, then there will be some problems, some blurring perhaps that you didn't anticipate. So a deeper analysis shows us that not all properties need to be uncertain. In fact, if the two operators have the same set of eigenfunctions, this is why it's very important mathematically for us to be able to determine the eigenfunctions of an operator. Because we might have this operator and that operator representing this and that. And if it turns out that the two operators mathematically have the same set of eigenfunctions, even if they have different eigenvalues, usually they will, because they have different units and so on. Then we can measure both of them and we get exact results for both. So we may measure this and that and we get a certain value. And if we measure this and that again, we get the same for both. And there's no uncertainty. But unfortunately, position and momentum, which are two things that people like to determine, are not compatible in that way. Oh, there's this idea called complementarity. Observables that are incompatible cannot be measured to arbitrary precision. So here, what I've shown is a real coin. I flipped it and it happened to come up heads. It's a penny. And you can see that it's heads because you can see Lincoln in profile on the face of the coin. And you can even read other things on it, like the year it's minted. But let's just say we can tell that it's heads for sure. Now, suppose instead of trying to measure heads, and if I leave it there, it's going to obviously measure heads, heads, heads. I'm not going to flip because I'm not allowed to do anything to it, except look at it, measure it. If on the other hand, we're interested in the exact thickness of the coin, in that case, we have to orient it like this. And this was tricky to do, but the coin did balance on its edge. It was thick enough and the surface was flat enough. And my collaborator had a steady enough hand. And now, if you have the coin oriented like this, you can see exactly how thick it is. Whereas when it was down with the head pointing, you had no idea how thick it was. Imagine you're looking straight down on it so you can get the best possible view. Now, you can try, now if the coin's on edge, it's obviously unstable. And so anything I try to do to look at it could have it drop. But the question is, when it's like that, which side is heads? And the answer is because I'm looking at it edge on, I have no idea which side is heads. And quantum systems are very much like that if we have complementary variables. If I try to zero in on one of them, it means the other one fades out. And I can't get both at once because they're interfering with each other. And I just, there's no possibility of doing that. We could try to cheat. Here's the coin balanced on a pen with an eraser to keep it steady. And we could look at the coin on an angle like this. And the way it's angled here, I can pretty much tell its heads. It's not as clear as it was before. But I can pretty much tell its heads. But what I can't do now is measure the thickness very well. Because I'm seeing the thickness from an angle and it gets smaller and smaller and smaller as I get the heads and then I can't measure it very well. And basically in order to get the thickness better, I have to turn the coin toward me like that. And then finally at some point, I can't see whether it's heads or tails. Now with a coin, a macroscopic thing, I can look at it heads, I can orient it and say, well that side's heads. But with small things, you can forget that. That's not possible. Unless you can see its heads, you don't know what it is. And that's the problem. So the uncertainty principle really makes this numerically rigorous. It says exactly how well you could measure the thickness and or tell its heads when it's a small thing and when you know how the different variables, the things you're trying to measure interact with each other. That's basically what it's making much more rigorous. Let's talk now about classical atoms. In a classical atom, we have Maxwell's equations. And this was another problem actually at the turn of the century is that it was fairly easy to work out that an accelerating charge would radiate energy in accordance with Maxwell's equations. But if the electron, which is certainly accelerating, if we imagine it going in a circle around a positively charged nucleus, then it has to radiate energy, then it has to slow down because the energy doesn't come from nowhere. And what that means is that the electron would spiral in toward the proton and it would eventually condense onto it. And if that happened, there wouldn't be any electrons around to make bonds and so there wouldn't be any molecules, there wouldn't be any life, there wouldn't be any atoms even. There would just be like a neutron star or something with just all this condensed matter. Somehow, the electron is not like in a planetary orbit and it's not behaving according to the way a charge would in Maxwell's equations. And the reason it doesn't partly is the uncertainty principle. Because suppose the electron starts slowing down and spiraling in and coming in and in and in smaller and smaller and smaller. Well, we just did a calculation that showed if the electrons within 200 picometers that the minimum uncertainty in velocity is like 100,000 meters per second. And what that means is that it is impossible for the electron to spiral in and be on top of the proton in that itsy bitsy space and be stationary because that violates the uncertainty principle. And quantum mechanics says it's not possible to measure position of momentum that accurately. And therefore, the electron may start spiraling in and then may just get tossed out suddenly and go in a different direction. And so that saves us. So if the electron is not spiraling around like a planetary model of an atom in an orbit, then what is it doing? It certainly maintains a stable probability distribution because we look at atoms and we see that they have a cloud of negative charge around and it doesn't change if we don't disturb the atom if we just leave it alone. But we don't know where the electron is because the electron behaves like a wave. Unless we try to measure its position by which we would have to use a very energetic photon and blow the electron clean out of the atom basically. And then of course, we've lost our thread. We were trying to figure out what it looked like when we didn't look at it. The problem is that's not allowed. You can only talk about the things you can measure. You can't talk about things that you can imagine what they might be. Stable distributions of the electron density have to be standing waves. Standing wave, you can think of a guitar string. If I put my finger, I have a fret here. You can't move here. You can't move at the other end in between. It can vibrate and it just makes a stable pattern and sits there doing the same thing. And the electron then has to do something very much like that when it's in an atom and it has a very interesting wave property. We couldn't understand it at all if we thought of it as a rock or a BB moving around in there. An orbit of course is a periodic trajectory, but electrons don't have trajectories. And so instead of an orbit, we speak of orbit-oles, which is the wave analogy of a stable orbit. Now it's the wave function that has to somehow, like the guitar string, can only play a certain note if I hit that fret. The wave function can only play certain notes in the atom. It has to come around and match itself and give a stable standing wave. And that means the wavelength of the wave function has to match into the space into which the electron is confined. If it doesn't, you won't find the electron in that wave function. So there is destructive interference and the wave function vanishes. And if the wave function vanishes, then the chance of seeing the electron at that energy also vanishes because the wave function tells us the probability. Let's, a 3D thing is kind of hard to visualize, but we can certainly do it on an electron on a 2D sphere, a ring, and that makes it much easier for us to draw. So let's have a look then on an electron on a ring. Here is an electron going around as a wave. I say going around, but I don't know where it is because I haven't measured its position. But I have a standing wave. It's equal everywhere in space. I'm just showing the real part. When the real part's zero, the imaginary part is big. And that's why, another reason why we have to have complex waves sometimes. This one goes round and round and round. And every time it comes around, it's back in the same place. Goes around, comes back in the same place. Round and round. And so therefore, it's going to make a stable repeating pattern. It's going to sit there. And in fact, we can't see anything going round and round. I imagined it was doing that as if it were a little rock going around and around. But in fact, all it is is just this pattern. Stable repeating pattern. Because it exactly matches the condition that it met, meet itself when it hooks up again. But if I have a slightly different wavelength so that it doesn't match when it comes around, but it's a little bit off, it goes around, instead of matching perfect, it's a little higher, then if it goes around again, it's a little worse. If it goes around again, it's a little worse. Waves go up and down. Finally, it's coming around as the opposite sign. And let's go around then another time on this thing. Second time around, it's a worse mismatch. And if I go around several times, it appears when I draw the thing that the wave is up and down and up and down and up and down and up and down everywhere. And that means that it cancels itself out. There is no wave there if it's up and down and up and down. It's just canceled itself out. The only position that can have a wave is the one that perfectly matches. Now there could be a higher one, instead of three lobes, it has four. It goes around four. So that perfectly matches too. But there's not three and a half and there's not 3.1 and there's not pi lobes. There's exactly an integer number of lobes and that means there are a certain number of integer energy levels that the electron can be in and it can't just be anywhere. That's not allowed. And that's very important because it explains the spectroscopic observation of atoms where they didn't just irradiate any old light, but every element gave certain characteristic lines which depended where it was in the periodic table and so forth and so on. And of course it's very important for chemical analysis. That's one way you can tell what's in an unknown sample as you do atomic emission spectroscopy. Confined systems, atoms are confined systems because the electron has to stay there. But there are many other confined systems, nanoparticles, the so-called particle in a box which is a model problem we're going to do which has very easy mathematics compared to real problems which is why we do it. Wherever you have a confined system, it doesn't matter how it's confined. The wave function has to somehow fit. It has to fit into the space available. If it goes round or bounces back and forth or does anything and it comes back different, that means that particular wave function is going to cancel itself out and it's just gone. This matching condition really restricts the wave function to a certain set of values and gives us allowed energy states such as those that are observed in atoms and molecules and form the basis of all kinds of chemical analysis that we're going to do. If we've got much shorter wavelength light, we learned that short wavelength equals high energy and that long wavelength light equals low energy. And De Bruyley said, well, particles have a wave associated with them and now we've given this thing a name, the wave function. And it's a function and we can plot it if we have a functional form for it, which we sometimes do, and we can look at it. And what we find is that if the wave function has higher curvature, it's going up and down, up and down a lot like crazy. That's sort of like a photon with a short wavelength and that means that that state, that quantum state is higher energy than one that's all spread out and just kind of moping around and not really very many up and down parts, up and down nodes in the things. I'm going to close here with the position operator and we'll pick this up next time. We found that the momentum operator was to take minus i h bar times the derivative. The position operator is, in fact, I gave it before as an example of an operator. X hat on psi is just equal to the number x on psi. And the number x is the position of the particle. The momentum eigenfunction is e to the i p x upon h bar. And to make sure that the eigenfunction is normalized, we should include a normalization factor, which I'll just put n, some number here, to make sure it's normalized. And the question is, what does the probability density look like for a momentum eigenstate? Well, we just take phi of p times, phi star of p times phi of p. We get n e to the minus i p x, n e to the plus i p x, e to the plus, e to the minus, that's 1, because that's e to the 0. It doesn't matter whether it's imaginary or real, that still works. We just get n squared. But that's weird because it says the probability density doesn't depend on x. It's just some number. And what that means then is for a momentum eigenstate, the particle has equal probability of being anywhere at all, basically from plus or minus infinity, equally likely. And so position eigenstates, where the particle is definitely within 10 to the minus, no matter how small you want to think, at that point, a momentum eigenstates, where the momentum is exactly determined, are two completely different aspects of measurement. And they're completely at odds. The best you can do is you can get the position to within a certain limit, and then simultaneously you can get the momentum to within a certain limit. But if you try to get too aggressive with one, you just like squeeze this in, then the other one gets wide, because somehow this area between the two of them is like pushing on jello or something, it squeezes out if you try to get too aggressive with it. And there's no way you can minimize that effect except to have the very best uncertainty on the inequality, but you can't make it zero the way you'd like to. To have a position eigenstate, on the other hand, instead of this corkscrew, you should think of e to the i p x as a corkscrew. It is corking this way, the particle is going that way. It is corking the other way, it's going this way. But in either case, it's just a constant thing corking around like a corkscrew driving through a wine bottle cork. It's going to go a certain direction. On the other hand, position shouldn't be like that at all, but it should be like is it should be, the wave function should be piled up like a big pile of sand in this position. And then it should be zero elsewhere because we know when we take the wave function and square it, that tells us the probability of finding the particle. So if we take some function and pile it up like the Eiffel Tower real steep, then the particle is going to be there. And then it might have some uncertainty, it might be out here. But we could imagine piling it up very steep and very high, and that would be a position eigenfunction. Next time what I'm going to do is I'm going to take a model position function that's localized, and I'm going to expand it in terms of momentum functions and show you that the momentum of such a function like that becomes more and more uncertain as we make the position sharper. And then finally, we'll finally introduce, after the first week of class is over, we're going to introduce the wave equation, that's what we didn't have so far, that tells us exactly how these wave functions move forward in time and how they have certain energy and other properties. And that allows us then to discover what these wave functions actually are. So we'll pick it up there next time.
quantum mechanics. Of course there's an h bar eigenfunction of the momentum operator and the 1 and that's how it works and it works the same eigenvalue equation and make a series of with probabilities. It could be any of these In a classical atom, we have Maxwell's equations. explains the spectroscopic observation of atoms mathematics compared to real problems which is