In part a we developed the Schrodinger equation.
In this video we want to visualize some of
its solutions to try and figure out what it
actually predicts.
We let the computer numerically solve the
general form in various two-dimensional situations.
The basic idea is that if we know the wave
function, psi, at some time, the equation
gives us its slope which we can follow to
find the wave function a short time in the
future.
We repeat this process for as long as desired.
Remember from part a that the wave function,
bizarrely, has both a real part and an imaginary
part.
Our animations will be of the sum of the squares
of these.
In a classical theory this would represent
a wave intensity – like the intensity of
light.
In quantum mechanics it turns out to represent
the probability of finding an electron at
some point in space.
We begin with one of the simplest situations
– the so-called particle in a box scenario.
We have a square box, and we place an electron
at its center.
The walls are impenetrable so the wave function
is zero outside the box.
Inside the potential energy is zero, meaning
no forces act on the electron.
Classically we would expect to be able to
place a particle at rest and have it stay
put.
But under wave-particle duality we know the
uncertainty principle will apply.
If the electron's position is uncertain to
delta x, its momentum will be uncertain to
delta p, which is no smaller than Planck's
constant over delta x.
Therefore, the electron cannot be perfectly
at rest at a precise location.
Let's look at the classical picture of an
electron at the box's center.
We'll represent momentum uncertainty by showing
several copies of the electron moving with
small velocity in different directions.
The result is a cloud of particles that slowly
expands.
Now let's look at the prediction of the Schrodinger
equation.
We'll represent the wave function probability
as the color-coded height of a surface, viewed
from directly above.
To show small details, we'll greatly amplify
the heights and chop off the large parts which
then appear white in our top view.
So, the white area is where the probability
is very large, and the dark blue area has
zero probability.
We start with a broad wave function, meaning
we don't constrain the electron location very
tightly.
This large uncertainty in position will produce
a small uncertainty in momentum.
So as the wave function evolves according
to the Schrodinger equation it slowly expands.
Eventually it starts interacting with the
walls modifying its initially circular shape.
Now suppose we tightly constrain the initial
location of the electron.
This should produce a large momentum uncertainty,
and classically we'd see a rapidly expanding
group of particles tha t would quickly bounce
off the walls.
For the quantum mechanical case we start with
a narrow wave function.
It rapidly spreads, reflects from the box
and then produces a kaleidoscopic sequence
of interference patterns.
Now the obvious question is: What are we looking
at here?
What do these images physically represent?
Schroedinger's proposal was that, quote, “material
points consist of, or are nothing but, wave
systems.”
Therefore, the wave pattern we are seeing
here is an electron.
Period.
Electrons, indeed all particles, are nothing
but matter waves.
Furthermore he proposed that, quote, the wave
function physically means and determines a
continuous distribution of electricity in
space, the fluctuations of which determine
the radiation by the laws of ordinary electrodynamics.
In the next video we'll see that Schroedinger
had good reasons to think this.
In fact he hoped that his equation provided
a classical, deterministic basis for atomic
theory in which there would be no wave-particle
duality, and no quantum jumps occurring at
random governed only by probabilities.
At the same time he realized that, quote,
this extreme conception may be wrong, indeed
it does not offer as yet the slightest explanation
of why only such wave systems seem to be realized
in nature as correspond to mass points of
definite mass and charge.
And this was the objection raised by many
other physicists.
If the electron is a wave, it should be possible
for it to spread out and for us to detect
small pieces of it.
But all we ever detect, as in the Millikan
experiment, are whole electrons.
It was Max Born who proposed what has become
the standard interpretation of Schroodinger's
wave function.
The squared magnitude of the wave function
represents the probability of finding the
electron at some point in space at a given
time.
What Schroedinger's equation determines is
not the behavior of the electron itself but
the evolution of the amplitude of this probability.
When we experimentally detect it, the electron
appears as a particle at a definite place,
and the wave function will have collapsed.
In a sense the wave function represents not
an electron but our knowledge of the state
of an electron.
[a13] Let's mix things up by curving the bottom
of our box, or equivalently, using a spring
to attach the electron to the middle of the
box.
Everything is as before except that the potential
is now proportional to the square of the electron's
distance from the center of the box.
If it moves a small distance it feels a small
force pulling it back towards the center.
If it's a large distance the force is large.
This is classically the nature of the force
produced by a spring or by gravity on a particle
sliding on a curved surface.
The force is always pulling the particle towards
the center of the box.
In the classical case a particle released
at rest simply oscillates back and forth across
the box.
If there's some initial range of momenta the
possible paths spread out but then re-converge
on the other side of the box.
The greater the range of momenta the greater
the spreading, but the paths always re-converge.
Now let's see what Schroodinger's equation
predicts for the quantum mechanical case.
We start with a broad wave function and we
see it behaves more-or-less like a classical
particle with a bit of spreading and contracting
as we'd expect given the uncertainty principle.
For each oscillation we narrow the initial
wave function and we observe greater spreading.
Eventually the wave function starts interacting
with the sides of the box and wave interference
effects start showing up as distortion in
its initially round shape.
When wave effects become extreme we've transcended
the picture of a classically moving particle.
We can see Bohr's correspondence principle
manifested in these calculations.
At relatively large scales the behavior predicted
by quantum mechanics corresponds to the classical
prediction for a particle moving under a force.
Even though the classical and quantum theories
employ quite different physical concepts and
mathematical expressions, at large enough
scales they make similar predictions about
the behavior of particles.
It's only when we push down to very small
scales that quantum theory strongly diverges
in its predictions and wave-particle duality
becomes very apparent.
Of course it was behavior at these small scales
that motivated the development of the theory
in the first place.
So quantum mechanics and classical mechanics
are not two disjointed, unrelated theories.
Instead, they represent two ends of a continuum.
If the scale of a system being analyzed is
really big then quantum and classical mechanics
give the same predictions.
In this case there's no need for the extra
complexity of quantum mechanics, and we can
think of this as a purely classical realm.
On the other hand, if the scale is really
small then wave-particle duality becomes important
and classical mechanics fails to accurately
describe our observations.
This is the purely quantum realm where Schroodinger's
equation reign's supreme.
The uncertainty principle is what delineates
these extremes.
If the product of the ranges of position and
momentum is much bigger than Planck's constant
then quantum effects are negligible.
This is true in the macroscopic world of our
day-to-day experience.
If, on the other hand, the product is on the
order of Planck's constant then quantum effects
will be very important.
Recall from video 4c how Bohr proposed the
concept of stationary orbits to explain why
atoms have discrete energy levels and don't
suffer radiation collapse.
de Broglie proposed that these were orbits
in which fit a whole number of electron waves.
The stationary state concept naturally arises
in the solution of the Schroedinger equation
without any additional assumptions.
We can find certain solutions for which the
wave function intensity, the probability distribution
for finding the electron at a given place
in space, doesn't change with time.
As we discussed in part a, these are the solutions
where the electron has a well-defined energy.
Here are some of the stationary states for
our curved-bottom box.
This is the lowest energy state.
It's the quantum state that most nearly corresponds
to the electron being at rest in the middle
of the box.
The next-lowest energy state has two lobes.
We can have three-lobe, and six-lobe and so
on, stationary states where generally the
more lobes the higher the energy.
Let's look at the two-lobe stationary state
in a little more detail.
This image tells us that if we measure the
position of the electron it will most likely
be in one of the two reddish regions, but
we can't know which one.
Likewise, although presumably the electron
is oscillating back and forth, we can't know
which direction it's traveling in at a given
time.
In this pure energy state the electron is
equally likely to be on the left as on the
right, and equally likely to be traveling
left to right as right to left.
Following Schroedinger's interpretation of
this figure as representing the distribution
of electric charge in space, we conclude that
there is no movement of charge, hence no radiation
is generated.
But, suppose that at some time we somehow
determine that the electron cannot be on the
left side of the box.
So, let's chop off that part of the wave function.
Then Schroedinger's equation tells us that
the electron which we now know is somewhere
on the right, is moving towards the left.
We can see the oscillation.
And if this does represent the motion of electric
charge which produces radiation by the laws
of classical electrodynamics, then we expect
it to radiate away energy until it settles
down into a stationary state of lower energy
– the one-lobe state.
This is an example of the way in which Schroedinger
thought his equation could bring classical
determinism back into atomic physics.
However, just as electrons only show up as
discrete particles, so too does radiation
only show up as discrete photons.
Schroedinger's equation was a great triumph
indeed, but his classical interpretation of
the wave function failed to explain what is
actually observed.
Instead, it was Born's interpretation of the
wave function as a probability amplitude and
the corresponding picture of wave-particle
duality that was found to be consistent with
experimental results.
The double slit experiment is the quintessential
demonstration of photon wave-particle duality.
So let's see what Schroedinger's equation
predicts for a double slit experiment performed
with electrons.
We'll simulate the experiment inside our flat-bottom
box.
We're using a slightly different color scheme
here where black represents zero amplitude
up through blue, green, red and finally white
which represents large amplitude.
We start off with a wave function that is
wide in the horizontal dimension and narrow
in the vertical dimension.
It has an average momentum towards the top
of the box.
As it moves through the box we see very little
spreading horizontally, but significant spreading
vertically, consistent with the demands of
the uncertainty principle.
Eventually the wave function hits the upper
wall.
Now we repeat with a single-slit screen in
the middle of the box.
The wave function is reflected from the screen
but passes through the slit and rapidly spreads,
fairly uniformly, into the top half of the
box.
Moving the slit to the right we get a similar
result with the transmitted part of the wave
function simply shifted with the slit.
Now with both slits open we see what looks
basically like a composite of the two single-slit
cases until the two spreading lobes begin
to overlap.
At that point we see interference fringes
forming.
These fringes indicate that on average electrons
should arrange themselves on the top screen
in alternating bands of high and low probability,
hence of high and low density.
When the experiment is performed, remarkably,
that is indeed what is observed.
Schroedinger's equation provides us with a
precise description of the physical world.
But it's a description in terms of the probabilities
of observing non-deterministic physical events.
It's seems that we must heed the words of
Niels Bohr - It is wrong to think that the
task of physics is to find out how Nature
is.
Physics concerns what we can say about Nature.
And clearly, there seem to be fundamental
limits to what it is we are able to say about
Nature.
