Last class we talked about how light can
be considered a particle and the experiments
that kinda show us a lot of that, that
really kind of epitimizes both the
wave particle duality
but there's this other aspect that shows it as much more of a wave than just
the photoelectric effect really does
so the photoelectric effect does a great job
of showing both at once
this stuff that we're gonna talk about today
there's a little bit more of a job
showing it as a wave in a really
distinct way
that we could actually see if we did
so in order to talk about this
experiment effectively I need to give
you a little bit my background on what
waves are
and specifically what wave
interference is
so wave interference is something that you can see, you can see it in waves in a
pond or wherever if you drop a rock
into a pond you make waves right? You
can see the rippling effect going
around
and then if you drop two rocks into
uh... lake or pond something with
relatively stillwater
you can see these weird interference effects
you can do the same thing if you're out and
jet skis and yet you know
and on a relatively calm day and you can
watch the two waves kind of hit each
other.
This is we've interference, so this is
the kind you can see it and you can see
that these interesting patterns that
come out
That aren't just um...
aren't just simple you know
superimposition's of the two.
how do one of these look like when you when
you do superimpose the two on each other
so there's two different types of
interferences, there's constructive and destructive
 
and if you have a waves of a very
particular wavelength
um... and their interfering with each
other
you can overlap them right on top of each
other or exactly you know like cross
on each other
and if you overlap them right on top of each
other they add together
if you over top them exactly opposite each
other
they're going to subtract from each other. Now of course you could also do some sort of
you know
slight off
on it to
these are the two different ways
so think back to your algebra classes
whenever you have those
and you'll
you may have been forced to go through
and
these had little points on a graph and you had to 
add them together and if you added this
wave together you had at this point and
this point
so this is one and this was two
you add them together to be three
well this is zero plus zero and
so it adds together to be zero. This is
negative one and negative two so it would add together to be negative three
same thing over
here
now here only though you have a
negative one and let's say a plus two
we're giving these really arbitrary units
so if you were to add a negative one and a plus
two well that gives you about a plus one
and so you would get that and they would subtract from each other
you might have been forced to go through this
in painstaking detail in you're algebra
classes. We're not going to do that here but you need to kind of understand how it works
You need to see that ok well if they're over top of
each other they're gonna act
if they're opposite each other they're
gonna subtract
and that these are called constructive
and destructive interference
now this is gonna be important to understand
when we look at this next experiment
which is called the double split
experiment
because you're gonna get patterns of light
in these patterns of light are going to be 
really bright in some places and really
dim in others and that's because you get
constructive interference making it really
bright and destructive interference making
it very dim
and so that's how this is going to work
on this next experiment
the best way to show you this next experiment is to have a little video of
it
So this video does a great job
and you can watch it as much as you want on youtube
so I'm gonna go through to explain it
out and he does do the same thing
online with words and such
The way the double split experiment works
is you have this point source
and
you have on a place where you can let
light through
and so this is sort of showing you how
the light
would go through
and light expands out like a wave
now
okay we're gonna start over
now the other thing that you're
going to be able to do.
is you can make it the exact same thing
on the next one over
so instead of just having one point
source
you can have two point sources
so you have one point source here
and that's going to let light in a certain
amount
now you put another light source in
another place or just a hole
one of the east ways to do this is
to actually have a light source behind the
the cardboard or board or whatever and put a hole in it
now if you put these together
what happens is that they interact in
you get these interference patterns
and when you come over here
wherever the light interferes in a
constructive way
you're gonna get really bright shiny
parts
wherever interferes in a destructive way
you're not
you're gonna get very dim
and you can actually see this, this is a 
pretty
standard freshman physics lab if you end up taking any physics
and it's really, I mean it looks exactly
like this
so this is actually a picture of one
from one of these experiments that was
done
and really looks almost identical
using the right lasers and things like
this
you can actually do this at home
if, in a slightly different manner
if you take a piece of cardboard and you
cut out uh... a little slat
and you put two pieces of hair and you tape
it down and shine flashlights on it
you can get the same sort of situation
where you can get little interference
patterns
uh... I used to do it in class but it was honestly too
hard to see in a big lecture hall so I
quit
so
you'll go home and do that, it's fun
you just take two pieces of hair
tape it down onto a piece of cardboard and
take a
laser pointer and shine it around and you can
see interference patterns too
now let's look at this a little bit more
um... laid out
so here's the experiment that young
did
he had a...
something that was light blocking
here and he cut one little hole in it to
make sure the light was coherent
and then he shined a light over here
so that it came through here
at which point it's spread out
like this
now what that did is it allowed him to have two light sources
but as what they showed in the other video
but have it come through here at exactly the
same way
and then at this point now he had this
double split
or this double slit
and it comes through here, it comes through
here
and now you have the exact same wave
pattern
for each slit
but now because they're close to each other
they're gonna be interacting with each
other
now if you draw- if you then have
something over here that detects it
a human eye works just fine to detect it, but you can also actually detect
with you know something a little bit fancier
you can see these patterns
so where these interfere completely
you get this really bright peak
and then you get these sort of
peaks that get more or less
intense as you go through
but you still get these interference
patterns
so this is how
young's double slit experiment or split experiment worked
so again now that we kinda looked at both
of these experiments and you see how
they have properties above
they can act, light can act like
both the wave and a particle
and it does it at the exact same time, it's called wave particle duality
we saw that with the photoelectric
effect
you actually did see a little bit of both right, but you definitely the sort of
particle like uh... ways that it
worked
Where the double split experiment
most certainly amplified the wave-like effects
you can't look at that and say okay it's
not acting like a wave
after you've seen that and you've seen those interference patterns
and you can relate that to real life
for you see ripples in a pond and you see those
interact
and it's very much the same idea
now this is very true small of
particles
so, the last time that I had
looked
they were able to measure these wave-like properties and things up to 1610 amu
which is actually pretty big
if you think about it um... 
a thousand gram's per mole
if you can think of it
so,
so when we say that this is true 
for very small matter
that that's about the size that it goes
up to it
now that's just because that's exactly when
we can measure
this kind of bridges us into the idea of
well.. wha--
Does everything have wave particle
duality?
I mean we see it in the small things
but why would an electron have
wave particle duality
but I don't
why can't I act like a wave?
that's what DeBroglie said
in not quite so
colloquial terms
so he took wave particle duality from
light
and said that's true of matter too
and that's true of larger matter to some
extent but we'll see why doesn't quite work.
 
So what DeBroglie says is well wavelength is
equal to H
Planck's constant over P
so depending on how much physics you have
you may or may not know what that is
that's momentum
otherwise known as m times v
this is where font gets us into trouble
again because most of the equation
editors like to make it the v kind of
like that which looks too much like a
"nu"
but that is a v, that is velocity
so "p" is
equal to "mv"
with this being "v"
so just said one more time that's quite important to make sure you notice
uh... and your also, this is more sort of a general problem solving thing
that you're gonna run into
a lot
you a lot of times going to be
givien kinetic energy
and asked to find the DeBroglie wavelength
well you can do that by finding velocity
from the kinetic energy
and then plugging that into here
so this takes the wave particle duality
and says ok well
that's great we have it for light
why can't matter have it too
and it gives us an equation that we can
actually use for it
 
so quick reminder there on your equation
for kinetic energy
okay so best way to go through and do some DeBroglie wavelengths is to do some problems
on it
so for the first one
I say let's find the DeBroglie wavelength
of an electron
so something small
and see I give you the
kinetic energy
so since I give you the kinetic energy we can't just directly fill in the
DeBroglie wavelength
because the DeBroglie wavelength
is "h" over "mv"
so we need to find "v"
so how do we find "v"
we know that kinetic energy
is equal to one-half
and v squared
so when we go through and do this we solve
for velocity
which is two times the kinetic energy
multiple that up
divide by the mass
take the square root
so we can fill in from the problem
now we gets mass and we say well I don't know the mass of an electron
luckily this is a homework type problem so you can look it up
on an exam it would just be given to
you on the tables
so we get this
and we get our velocity
something to be kind of careful of here with the
math
doesn't come up here
or in my other example actually but it will come up your our homework and it will come up on exams
 
if I had a mass on a electron, or proton, or a neutron or anything like that
I'm going to have to give you that mass
If I ask you for
the DeBroglie wavelength of a nitrogen atom
Do I have to give you the mass?
Well no it's on the periodic table right? You have mass
so if you think about nitrogen
are you just going to go through and
fill in the mass of let's say just an atom?
are you just going to go fill in
fourteen?
well that in grams per mole.
and if you look at this unit
we have joules
and we need to get meters per second
so remember what a joule is equal to
right?
joule
little side-bar here
joule is equals to
kilogram
meters squared over second squared
so we can't
have
something that, we can't just put grams per mole in there.
so you have to go through and you
would have to convert
anything from the periodic table
into kilograms
so you know make a huge note of that in
your notes because it comes up and
people tend to mess it up
that if you're looking for an atom
if you're looking for the DeBroglie wavelength of an
atom
you're going to be tempted to fill
in that grams per mole from the periodic table
You can't do that
you have to convert to kilograms
and not just kilo- and that doesn't just
mean you divide by thousand either right?
kilograms per mole isn't what you're
looking for, you are looking for kilograms
per atom
so you have to convert from grams
per mole to kilograms per atom or
molecule depending on whichever
so make sure you make a note of that you know
by your mass
continuing on though
so now we have a velocity
and we can fill that in
you'll notice I sort of changed my units for Planck's constant around
that's just so I can watch all the units
cancel
and you get this
and you end up with that for your answer
now I always get
a few people who ask, well...
Can't I just take this equation and fill
it into here and plug it all at once
that is perfectly fine, go for it
probably is how I would do it if I were doing the problem
but I think probably 80% of you or so prefer this way
where you do it out separately so
whichever way you prefer you can do
It doesn't really make too much
difference either way so
if you prefer to just fill this equation
into here you are welcome to do that
ok so that's the equation for an
electron
now
if we move on a little bit
That's small right? an electron small you can
definitely assume
it's this pretty tiny considering you know
it's an electron sub-atomic particle and all
 
but what's to say we can't do this
for a baseball, so let's do it, let's calculate the BeBroglie wavelength for something big
for something we can see in real life
because I'm telling you that an electron has a wavelength
which by that same token your chair
should have a wavelength, I should have a wavelength
my water bottle should have a
wavelength
so why don't we see that why can't we, why can't we look at that and see
something there
well the easiest way to figure out why
we can't see it is to
you know calculate it
so will do that so I give you the mass of the
baseball
and I give you
an approximate meters per second
obviously this is going to change alot if I throw a baseball or if
some famous pitcher throws a
baseball
same rules apply
we're going to fill into here
again just for the sake of watching
my units I'm gonna just
fill-in
my Planck's constant units in slightly
different way
so we go through we do the exact same
thing we did before, it's a little easier
this time because I gave you all the
values out right, and we get that
so see baseball does have a wavelength
now why don't we see it
well think about what a baseball looks like right, it's you know
a decent size you can see that size thing
now look at it's wavelength
to the -34 meters
that's not something we can see that's
not something that we would calculate
it's not something that we would
you know, be able to look at
and so sure maybe by this definition
macroscopic matter does have a
wavelength
but its so tiny that it doesn't matter
and if we look at this equation again
this means that mass is in the numerator
right? or excuse me, denominator
So the bigger that the mass gets
what's gonna happen to the wavelength
this gets bigger and bigger and bigger
it makes this whole thing smaller and
smaller and smaller
so the bigger that something gets
the smaller that it's wavelength gets
so you're making something larger you're
making it's wavelength smaller
and proportionally it end's up just not mattering anymore
when you had
an electron
we had a wavelength of this
for something that is very very tiny
now we have a wavelength of this for something that is relatively large
and so the wavelengths just start to
not matter
so that's what I meant when I said the
largest thing that has
sort of shown that we been able to measure
wave particle duality in
it's that point where our
measurements are really good enough and
it doesn't really make any difference
anymore
something like an electron or proton or a neutron it definitely matters for
something like a baseball not so much
so now we get to move on to something called the
heisenberg uncertainty principle
what the eisenberg uncertainty principle
says, on one level
is that it's impossible to
simultaneously know
both the momentum and the position
with absolute certainty
so it puts a limit on how well we were
able to know this
it says that if you take
the uncertainty in X, so notice that's not X
its the uncertainty in X
the part that we don't know about where
the position is
so in really crude terms
you know maybe if I take a guess and I say
you know I'm three meters away from that
door
I'm not very good with estimations so my
delta X might be a meter
Maybe I measure it now and I say well
you know now my uncertainty I measure it
and its really close and I'm actually 3.5 meter away
and I can say, ok well
I'm pretty sure of that within plus or minus
4 centimeters because my measuring isn't very good, I use, you know a cheap measuring tape
now maybe I measaure in standing one place and I use calipers and I'm really careful
and I know it within a millimeter where
I am
that's what an uncertainty would be
it's how well
you know where something is
this comes into play much more, it is
really more talked about with small things
you know electrons protons and things of that sort
but that's how you want to think of it
momentum is going to be the same way
assuming that it keeping in mind that
momentum is mass times velocity
a lot of times you'll just see this as
"mv"
because you assume you know what the
mass of the particle is you assume you can
look that up and you can weight that and
that's pretty perfect
what happens if your momentum and
uncertainty decreases, let's say your
using really really good equipment
because do notice this is greater than
or equal to
meaning that it's gotta be greater than
that, that's the lower limit but it might
be more
it might be that your uncertainty in both
are huge because you' aren't using very good
equipment
but according to this even the best
equipment in the world
which arguably there maybe some people
who have
published things saying that they
made it, broke it, we'll see how
that works
but according to this you can't
and if this happens, so let's say best
instruments
you are right at that heisenberg
uncertainty principle limit
if your momentum uncertainty
decreases
well what does that mean for your
position?
if the uncertainty in your momentum
decreases
if you measure that better and better
and better
that means that your uncertainty and your
position will increase
everytime you improve one
you lose the other
so you improve this, you can know
where that particles is at with a little
bit more certainty
but you're not going to know it's momentum
very well
and you can get better and better better
knowing its momentum
but you're not gonna know its position very
then
so what happens if your uncertainty
and position
decreases
well than your position your other
one is going to increase so it's always this
back and forth
You decrease one you increase the other,
you decrees that one you increase the
other
so this problem is based off a
commonly told and really bad chemistry
joke
so that the joke goes that hiensburg is
driving along and he's pulled over by a
police officer
and police officer comes up to him and he
says
"do you do you how fast
you are going?"
and he looks at the police officer and says, "Nope but I know exactly where I am."
so the joke there being that you only know
one of the other
so
I want to try to prove whether or not we
can use this, you know I'm driving
to work arguably too fast and I get pulled over
can I argue with the cop on the
thing that, well he knows where I
am
so you can't know how fast I was going.
Obviously there is you know some sort
of quantum
system here that he can't write me a ticket
and I don't think I would suggest this
argument anyways
but let's see if it actually valid let's
see if it holds up
We're starting from this equation on the
slide
which is to say delta X times delta p
has to be greater
then H over 4 pi
or equal to
so that's our limit now course there can be more but it's gotta be equal to that
So let's rewrite this a little bit
differently so that we have velocity there
 
we're going to assume I weighed
my car and I know exactly how much it weighs
since this is a little bit of a back-of-the-envelope type calculation
so
I tell you that you know I think that
officer should be able to measure where my car is
by about half a meter
and I know the mass of my car to be
hundred kilograms
we need to solve for delta "v"
so we can see whether I can argue myself
out of a ticket this way
so let's first solve for delta "v"
and then fill in everything that we need
so if we do this
we get a delta "v"
our delta "v" has to be less then or great to
according to the quantum mechanics
this
so now can I argue to the cop that shouldn't get a  ticket?
no right?
my delta "v" here is tiny
so according to quantum mechanics he
knows how fast I was going
times ten the negative thirty eight
meters per second
I promise you I was speeding by more
than that
so this quantum mechanics limit
isn't an issue here
and so this, you know it's a silly
example but it does show how you would
go through and do it
for any other particles and you would
just have a different mass
 
so same rules apply here. Has to be kilograms right?
so grams per mole isn't gonna cut it, you
need to change it to kilograms per atom.
If I give you the mass in grams you need to
make sure to convert it to kilograms
so same rules apply here
for uh...
any sort of sub-atomic particle
but it's a little more fun to try to
calculate this way
so this is not a good way to get out of
the ticket when you get pulled over
even if your particular police officer does has a strong
background in quantum mechanics
so next thing that are going to
talk about then
uh...
This seems to be every freshmens
favorite topic in um... chemistry
so wave functions energy levels and
particle in the box
so this sort of sets up this next big section
of this chapter. We sort of talking about
all
the history and the basics
to get us to this point where we can start
building up a molecule
we can start building up a hydrogen atom and then we can move on to multi-electron systems
and we can move on to
comics
and at that point will kind of skip some steps
and just have molecules
so
what wavefunction is
it's represented by the "psi"
which is another greek letter
and it's describe the movement of a
particle
It's just a symbol for mathematical
function
so I'm gonna show you those functions in a little bit
we're never really gonna have to work
with them
I'm gonna give you the solutions that
come out of them, I'm going to give you the pictures
 
They pull out of them
you need to know the general way in
which these work though
you need to know that it describes 
the movement of a particle
you need to know that it is a function
that you can actually graph
that they solve these functions and
they can put them into a computer and they
can graph them, and that's what gives us
our orbitals
Different particles are gonna have
different wave functions so a particle
in a one "s" orbital
isn't gonna have the same
wave function as a particle in a two "s" orbitial
 
or a 2 p-orbital or all of those
now we have something called "psi" squared
or the probability density
now what this is this is gonna give us the
probability of finding a particle in a
particular region
so if we say we want to find the particle from
here to here what is the probability that it's
gonna be there
and that's what the probability density
what we can gain from it
keep in mind if wave function or "psi"
is a mathematical function what is "psi"
squared?
Well it's that function squared so that's why
it's called "psi" squared
so um... technically the probability desnity is a probability of finding a particle
divided by the volume so that's just sort of
a technical thing to make the numbers work
that I wanted to add here.
but don't worry about that too much
let's go through in
and talk this out would like a super
super simple example uh... not something
that would actually be a wave function
um...but it give us a little bit more of a
clear way in which how these bigger
functions that I'm going to show you work
so
this is more understanding purposes than
being able to replicate it purposes
so let's take an example where "psi" is a number
I do this to make it simple right? How
can you get a simpler function than a number?
So we're gonna say that "psi" is just .44 cm inverse
so your book walks you through this sort of idea too
uh... but I wanted to go over it again
so we graph this function
.44 cm
inverse we get this
so what "psi" squared going to be?
well its just .44 times .44
so we can graph that too
again really simple examples these wouldn't
be wavefunctions assuming we'd actually
calculate
but it gets our point across
this would describe the movement of the
particle
this would describe the probability of us finding it in any particular place
the wavefunction, this is sort of 
nomenclature how we would say this,
the wave function is equal to .44
centimeters cubed -er inverse.
the probability density is equal to .2 
centimeters cubed
So if I wanted to know
what is the probability of a particle
being
within 0 to 3 centimeters cubed
I say what is the probability of
me finding this particular particle
between 0 and 3 centimeters cubed
in other words between here and then
3 centimeters out from it in a circle
well the probability of that you just
multiply this together right?
It's effectively taking an integral but it's a
simple integral because it's a box
it's just
the three
times by the two
so our Y access times by our X access
since it's a box
and that would give us our probability
density
Now this is a very simple example with a
very simple function
when we do this with bigger functions
we're going to get much more complicated
probability densities
and uh... different shapes and things of
that sort but this is how they do it they
just do it with a computer instead of by
hand
let's look at
some things about this so some questions
for you
"psi" is a function
so it any point can we determine it sign?
meaning positive or minis
does it have to be positive does it have to
be negative, can it be both?
Well it's just a function right?
So sure the function I showed you in the last one was always positive, we agree
that was sort of a simple example
wave function it's just a function so it 
can be both, it doesn't matter
The sign doesn't matter
so if I show you a graph like the one on
the other slide
like this
and I say well which ones the
wavefunction you would know that it's
the one that can be negative
this one
now if we move on to "psi" squared
and I say well the square of that
function right? it's just taking function
squaring it
so if you square something can that ever
be negative?
assuming that were in the the realm of
um...
real numbers
that's gotta always be positive
So if that's always positive
if I were to give you a graph like this
one
and I say I have both the probability
and the prob-- and I have the wave function
and
"psi" squared graph you could pick out
which one is which could say well
this one's negative so it's gotta be
"psi"
this one's always positive so that one
well be "psi" squared
next thing to talk about then is nodes
so nodes occur when "psi" is equal to zero
now if "psi" is equal to zero what does it
mean about "psi" squared?
that was going to mean that "psi" squared also
equals zero right? 0 times 0 is
most certainly 0
so wherever
wave function is equal to zero
"psi" squared is equal to zero
as well
now think about what "psi" squared is
though it's a probability density
so it's where you're likely to
find something so add a node.
what's the likelihood that the particle
will be there?
If "psi" squared is 0
it's gonna be 0
because if the "psi" is 0 that means
"psi" squared has to be 0 and "psi" squared
is probability density
so if the probability of finding a
particle someplace is 0 it's not
gonna be there
wave functions
Have "n" minus one nodes that may not 
mean a lot to you right now
but when we start talking about energy
levels
that'll mean a little bit more to you
so if we have a wave function in a
particular level for particle in a box
it's going to have "n" minus one nodes
so here we don't really have a node
um... technically this is it
asymptotic
so it just approaches zero and never quite hits it
you talk about nodes just being in the
center like right here
so this is asymptotically going to zero
it's not a node
this is a node
so now schrodinger equation, this is where we
really get into all of your guy's favorite things
so what schrodinger equation does
is it allows us to take the wavefunction
and come up with an energy level
now we talked energy levels already
right? We talked with the Bohr model
we talked about having an "n" equals one
and "n" equals two and an "n" equals three
it might be an oversimplification of energy levels but it does work to help you kind of think about it
 
so the schrodinger equation gives us a
way to actually go through
and find these energy levels
now you don't have to worry about this
equation so much
um... I put it there mostly because
your book does and it's good to have seen
once if you go on in physical chemistry
you'll see a lot uh...
but the way
that this works, the way that this
equation works, This "h" is called an
operator
and what that means
is that you taking something in your
doing something to this function
an operator operates on a function
so for those of you who have had some
calculus which
is most of you
that means that for instance if you take
a derivative of something
if you take the derivative of sine
if you take the derivative of cosine
taking a derivative of
is an operator
Maybe it's something simpler maybe it's
+ 2
so if you take whatever function you have
and you add 2 and that's your operator
uh... It just means you're doing something
to that function
so this
happens to be
the h
the operator
this is what the hat means this happens
to be what we call a hamiltonian
and this is the hamiltonian for this
now if you care
This is one of those them for your info
sorts of things
this is a double derivative.
so that would mean you take the wavefunction
and you take a double derivative of it
you multiply it by this, this is just a
number right?
It's a big equation but this part is just a
number
it's "h" bar
which is "h" over 2 pi, its just a constant  squared
2m which is just a constant
so you have that and then taking the
double derivative of it
so again you don't need to worry about
this too much it just sort of for your info
part you do need to worry about is it
knowing that this is a Schrodinger
equation
that this means you have to do something
to the wavefunction
and then you get your energy back
one of the easiest ways to think about
how this would work
is if you take the double derivative of
a sine or a cosine
if you take a double derivative of
a sine.
you end up at back with sine but now it 
has a -1 in front of it
this would be like this, you get the
same equation back
but you have some number in front of it
so taking a double derivative of a sine
would give you
negative sine
and you would get the -1 back
if you don't have any calculus it's a
little bit harder to come up
with the way this would work
so we'll just kind of leave it at that
but you have to know you have an
operator
and operates on a function and you get an
energy level
that's sort of that
condensed version of what I want you to
know for this class
so the way we are going to use this is we can use this to find the wave function in "E"
now for this class we're just gonna use the results of this, we aren't
gonna go through and do this, we aren't gonna
calculate these
we are just going to say, okay and here
are the energy levels
here's the results
so we're gonna start with a system called a particle in a box
now this got started with because it's a
little bit simpler than something than something like
a hydrogen atom
hydrogen atom has sort of
three-dimensional space a particle in a
box is simpler
so the way that this works is you have a
box, you can think of it as a box
and the electron can't go through the
box
and it's in the middle
it has to be inside the box and its
going back and forth which is a
little bit not quite the way you want to
word it, but it's in the box
kind of the real life way you can
think about this since it's hard to
picture an electron being delocalized in
a box
is if I took a string and I stretched it from room to room
and then made it like a guitar string and plucked it, and you have this wave going back and forth
 
so go home steal someone's guitar and look at
you know pluck the strings and see what happens
You'll be able to see the wave kind off oscillating back and forth
you're allowed certain wavelengths
This is relatively easy to do with something
like a jump rope right? You can take a
jump rope and you can very slowly go like
that
and you can get
this pattern
or you can speed it up
when you speed it up
you can get a sort of pattern like this
and the faster you go the more times you
can get it to oscillate in one.
Eventually your arm won't be able to do
any more but that's not an issue with
the particle in a box system
so that's how you can kind of think about this working
now you'll notice only certain wave lengths
are allowed, same is true when we go through
and try do this with a jump rope right?
You're allowed the certain wave lengths
where you get this, the certain wavelengths
where you get this, but not in between
And when you try to, when you try to
switch from going to the slowly way
to a little bit faster
there's this weird point where you get this,
it's not a wave anymore
its not a normal wave
it's a transition point
so it kinda works there too
Here is where we are going to take the Schrodinger equation and is
gonna tell you what the result is
The "h" "psi" = "e" "psi"
 
this the  "h" "psi" = "e" "psi"
and told you that we're gonna just solve
for "e" I'm just gonna tell you what it is
 
okay well here's our results
so this is "psi"
this describes the different equations that can relate this
electron in the box
and then this is our solutions to the Schrodinger equation
this is our energy
now don't let this equation bother you
there's a lot of things in it but none of
them are having a big deal
"n" that just
one-two-three-four
you thinking about your jump rope that's
you know which level of energy at that
how hard you had to move your arm to get
that
Now "h" is Planck s constant, right?
it's not any different then what we have been talking about
8, I think you're ok with the number 8 hopefully
and we have "m"
that's the mass so just whatever particle
we're talking about
and then "l" is the length of the box so
that's not a big deal either that's
just a number that we're gonna decide on
for any particular system it's just from
here to here
so what we can do is we can kind of use this to model
uh... a particle in the box
example
where we're given the wavelength an
electron
we wanna find the wave-length of an electron, of a theoretical atom
given an approximate diameter
if we say that the diameter is about a
hundred picometers and we can find
the wavelength of that theoretical atom
I've changed this example around a little bit
from what was on the slides
so this is what was on the worksheets
and as go ahead and we're gonna solve for this
actually I just decided to skip this
forgot about that we decided to skip that and
do that at a later time once we know a
little bit more about the way
quantum mechanics work
um... I think it will be a little easier
to understand then
so
in the meantime let's now take and go to the next step
so the particle box is a nice model
and the kind of lets us take uh... when
if you actually go through and do the
quantum mechanics
it's a little easier to work with than
something like a hydrogen atom
the equations are simpler
so that's kind of why it was
developed
but now we also have hydrogen atoms
and we can actually go through and we
can find the equation for that
now I'm going to talk about two different
ways about thinking of all of this
with the hydrogen atom
we are going to talk about the Rydberg equation
and then I'm gonna go through and do it with the schrodinger equation as well
now the results of these are exactly the
same
so what you're gonna find is regardless
of which we we do it you're gonna have
the same answer and we're gonna have the
same equation
so you can sort of pick the easier of
the equations when you're trying to solve
for actual number here
so
in my opinion I think the Rydberg equation
is probably easier to work with
now this both of these were
used to describe the hydrogen atom, but the
differences in how they would were
discovered
Rydberg did it experimentally, he looked
at a whole bunch of spectra coming in
and said
notice patterns and noticed that we
numbers work together and came up with
an equation that described it
and they had a constant in there that
we're calling the Rydberg constant
and it had all of the energy levels and
everything in it
now what Schrodinger did
is he said well I'm gonna start from you
know, the weight function
and I'm going to put the hamiltonian on it,
and then I'm going to solve for the energy
and then that's gonna give me all of the
energy levels
so one was done experimentally
one was done theoretically
either way you do it you get the exact same
results
so Rydberg's was done first and then
schrodinger's agreed which is always
great, in science that's what you want
you want
someone to do experimentally and someone
and/or someone to do it
theoretically and have the two
results agree because that means that your
theory is relatively sound
if your theory doesn't agree with your
experiment or your experiment doesn't agree with your theory
you're obviously not understanding
something, so this showed that you know,
they both kinda had it right
so this is going to be true for the hydrogen-like atoms
and what I mean by hydrogen-like atoms is one electron systems
so if you take some sort of atom and you take away all but one of its electrons
you can use this solution set to do it, with one little change
now I'm going to start backwards from reality
and start with schrodinger because I
think having the theory based first
and then working into the experimental
is a little bit easier to
understand
with Schrodinger, this is going to walk through and do the exact same thing that
we did the particle in the box
now remember I said I was just gonna give
you the results
so this is the results when you come out,
this is the energy level
now "Z"
is equal to
 
atomic number
"Z" is one of those letters that is is kind of universal in
chemistry so you want to recognize at that as
atomic number
"h", Planck's constant
We'll take about "R" in just a minute, we'll leave that alone for a second.
and then "N" is the energy level
 
now what "R" is this whole series
of equation-- or this whole
series of constants
none of these are a big deal
they're all just constants that you can look up
it just gets a little tedious to write
them all in
you just have a ton of constants here
and then the "Z" squared "h" over "n"
squared
so really
"h" is a constant too right?
So your really just have "Z" squared
over "n" squared times a whole bunch of
constants.
Keep that, that in mind when we go do Rydberg's equation, it's really just
"Z" squared over "n" squared times a whole
bunch of constants
Now
let's look at the bohr model of the atom
one more time
so remember we have our nucleus with all of our positive charges
and we have our rings of electrons going around the outside
this arose
because they knew that we had a proton
filled nucleus
uh... and we had the electrons going
around them
now the interesting thing here is that if
you look at the laws of physics
it says that the electron should
spiral into the nucleus
but it should be going around and around
and around and then just spiraling in and
everything colides together
well after Planck discovered quantization and
Planck said,
Hey these particles are emitting at
all different sorts of of wavelengths and
their very particular wavelengths and
not inbetween
bohr said, That's because the
energy levels are going to be quantized
that you can have this energy and you can have this energy but you can't have an
energy in between the two
now this particular model is only going
to work for one-electron systems it does
turn out that their quantized just in a
slightly different way than what Bohr thought
now the energy difference is given by "-Rh(Z^2/n^2)"
 
and that's what Rydberg went through and said
so before we talk about exactly what Rydberg said
I think it's kinda worth noting the stair and ball  analogy for quantization and energy levels
The idea here is that if you set a ball
down on you know this energy level or
this energy level or this energy level it
can be at any stair step energy level
never going to be in between
it can transfer between the energy
levels, it can go from here to here
but it cannot be like
held in the middle somewhere it needs to
be on one of these particular levels so
that sort of how you want to think of these,
is you can be here
you can be here you can never be inbetween
so now moving on to what Rydberg said
Rydberg
looked at these sorts of spectra that we're going to talk about in a minute
and said well
this is what I think the energy
levels are at
I think that it is at some constant that
I'm gonna call the Rydberg constant
because my name is Rydberg and so I like the way that sounds
times "Z" squared over "n" squared
keep in mind what did we have for
Schrodinger's equation?
We had a whole bunch of random constants that all had numbers associated
with them
times "Z" squared
over "n" squared
so the two aren't any different
the only difference is is that when
Bohr did it he just knew it was a number
he just knew because experimentally
this had to be a number
and he has called it one constant
where Schrodinger because he did it
theoretically.
was able to say well I know what all those
constant are
I know that they're-- it's all of
these combined together
if you multiply all of those together it
comes out to be the Rydberg constant
comes out to be the same number
and so that's how these two are related
I think that the sun's the easier one to
work with because we only have to worry
about one constant as opposed to those
like five or six different constants
so that's our energy level
don't forget "Z" is atomic number
that's were still always going to be talking about one electron systems but because we
can have ions
"Z" can change, we can have a helium 
1+ or a lithium 2+
or things of that sort that we're
calculating
so this is what the Rydberg constant turned
out to be
 
There are some other versions of the
Rydberg constant that you'll see
um... a lot of times you'll see it written in terms of hertz and you'll see
the Rydberg equation that we are going to
talk about in a minute written a different way
as 1 over lambda the way I'm doing it on
the slides are the ways that I found
the least amount of people to make
mistakes
on actually doing the problem
I would suggest doing it this way
your homework suggest doing it a different
way
I don't remember what your particular
book says-- there's like four
or five different ways to do all these
problems
I think this is the best way to not make
mistakes at your guy's level
"Z" is again here, atomic number, drive that
point home
so what is our energy difference?
We're going to care alot about these
transitions
how much energy it takes to go from this
level to this level
how much energy we get back if instead
it drops down to this level
and we were going be able to calculate all of
these
you can do this with both Schrodinger's 
and Rydbergs equation
but again Schrodinger is a little bit
complicated because of all these
different
constants
where Rydberg only has "R"
so it's a little bit easier to work with
so we're going to go through and we're going
to work with Rydberg
so if we find the difference an energy
levels
the difference that it takes for an electron to
go from here to here
or from here here's so on
it's a difference, so it's always final minus
initial
we're gonna go through and we're going to derive this as sort of our last thing of the day.
For a given energy level
it's gonna be defined by this right?
This is just from the last slide
So we know that we can find the difference in
energy
by taking the final and subtracting the
initial
that's just the definition of change
now we also know that we can define
both states
we can define the initial and we can define the final
so all I did was re-write this equation
two times.
One time  calling it arbitrarily initial and one time calling it final.
now we can fill these in
to this equation.So we are going to take the  final and fill it in
and take the initial and fill it in.
so change is equal to final minus initial
so we fill everything in
so again taking the energy levels
filling it
so we're trying to find the difference
in energy between two different levels
now we have "Rh" which doesn't
change and "Z" which isn't going to change either right?
because it's not switching atoms.It's
just switching within one atom so
we can just factor those out of the
equation. Set them aside for the sake of
algebra
so when we do that
we get these
where we take and we have the "Rh" and the "Z" squared.
You can factor the "Z" squared out too
some books will leave it in some books won't. You end up with this equation
now
something that gets talked about alot and students have issues with is the fact that this
is written two different ways depending
on where you happen to find it
sometimes you have a negative sign and
sometimes you don't
the difference here is that here you
have a full final minus initial
and you factored out the negative
Here you basically put the negative
inside but when you did that you ended
up with initial minus final
it doesn't really matter which one you
use on the final I'm gonna give you
one of them so just use that one but
when you're doing your homework be
careful if you're doing this off memory
not to mess this up
The one with the negative is final minus
initial, the one without the negatives
initial and final
and it's not that one of them is wrong or one of them is right
it's just different ways of writing the
exact same thing
keep that in mind
so next time will go through and will do
some examples using this and we're gonna
do a lot of talking about sign
convention because the signs for all of
these are really important and honestly
is one of the most commonly missed things
on the entire exam and if you mess up the
sign you're going to have problems
again going back to what I said before there's
different versions of this equation
you'll see it with 1 over lambda
uh... sapling does that
some books do that
I think it's easier to do it this way
and I'll show you why when we go to
worry about sign conventions and we
do some examples. You have to be really
careful about which direction these
transitions are going
Delta "E" is going to be
allowed to be positive or negative and
you're going to run into problems if you
mess that up if you mess up whether the
delta "E" is positive or negative
so we'll talk about all that next time
all this sign convention, examples, things
of that sort and really kinda finish
up the hydrogen atom, next time.
