PROFESSOR: So what
are we trying to do? We're going to try to
write a matter wave. We have a particle with
energy e and momentum p. e is equal to h bar omega. So you can get the
omega of the wave. And p is equal to h bar k. You can get the k of the wave. So de Broglie has told you
that's the way to do it. That's the p and the k. But what is the wave? Really need the phase to--
how does the wave look like? So the thing is that I'm
going to do an argument based on superposition and very basic
ideas of probability to get-- to find the shape of the wave. And look at this possibility. Suppose we have plane waves-- plane waves in the
x plus direction. A particle that is moving
in the plus x direction. No need to be more general yet. So what could the wave be? Well, the wave could be
sine of kx minus omega t. Maybe that's the
de Broglie wave. Or maybe the de Broglie wave
is cosine of kx minus omega t. But maybe it's
neither one of them. Maybe it is an e to
ikx minus i omega t. These things move to the right. The minus sign is there. So with an always an e
to the minus i omega t. Or maybe it's the
other way around. It's e to the minus
ikx plus i omega t. So always an e to the i omega t. And then you have to change
the sine of the first term in order to get a wave
that is moving that way. And now you say, how am I ever
going to know which one is it? Maybe it's all of them, a
couple of them, none of them. That's we're going
to try to understand. So the argument is going to be
based on superposition and just the rough idea that
somehow this has to do with the existence
of particles having a wave. And it's very strange. In some sense, it's
very surprising. To me, it was very
surprising, this argument, when I first saw it. Because it almost seems
that there's no way you're going to
be able to decide. These are all waves, so
what difference can it make? But you can decide. So my first argument
is going to be, it's all going to be
based on superposition. Use superposition-- --position. Plus a vague notion
of probability-- --bility. So I'm going to try to
produce with these waves a state of a particle that has
equal probability to be moving to the right or to the left. I'm going to try to
build a wave that has equal probability
of doing this thing. So in case 1, I would have to
put a sine of kx minus omega t. That's your wave that
is moving to the right. I have to change one sine here. Plus sine of kx. Say, plus omega t. And that would be a wave
that moves to the right. Just clearly, this is the
wave that moves to the left. And roughly speaking, by
having equal coefficients here, I get the sense
that this would be the only way I could produce a
wave that has equal probability to move to the left
and a particle that moves to the right. On the other hand,
if I expand this you get twice sine
of kx cosine omega t. The fact is that this
is not acceptable. Why it's not acceptable? Because this wave function
vanishes for all x at t omega t equal to pi over
2, 3pi over 2, 5pi over 2. At all those times, the
wave is identically 0. The particle has disappeared. No probability of a particle. That's pretty bad. That can't be right. And suddenly, you've proven
something very surprising. This sort of wave just
can't be a matter particle. Again, in the way we're trying
to think of probabilities. Same argument for
2 for same reason. 2-- So this is no good. No good. The wave function cannot
vanish everywhere at any time. If it vanished everywhere,
you have no particle. You have nothing. With 2, you can
do the same thing. You have a cosine
plus another cosine. Cosine omega t minus-- kx minus omega t plus
cosine of kx plus omega t. That would be 2 cosine
kx cosine omega t. It has the same problems. Let's do case number 3. Case number 3 is based
on the philosophy that the wave that we have--
e to the ikx minus i omega t always has an e to the
minus i omega t as a phase. So to get a wave that moves
in the opposite direction, we have to do minus
ikx minus i omega t. Because I cannot
change that phase. Always this [INAUDIBLE]. Now, in this case, we can
factor the time dependence. You have e to the ikx
e to the minus ikx e to the minus i omega t. And be left with 2 cosine
kx e to the minus i omega t. But that's not bad. This way function never
vanishes all over space. Because this is now a phase, and
this phase is always non-zero. The e to the minus i
omega t is never 0. The exponential of
something is never 0, unless that something
is real and negative. And a phase is never 0. So this function never
vanishes for all x-- vanishes for all x. So it can vanish at
some point for all time. But those would be points where
you don't find the particle. The function is nonzero
everywhere else. So this is good. Suddenly, this wave,
for some reason, is much better behaved than
these things for superposition. Let's do the other
wave, the wave number 4. And wave number 4 is
also not problematic. So case 4, you would do
an e to the minus ikx e to the i omega t plus an e to
the ikx e to the i omega t. Always the same exponential. This is simply 2 cosine
of kx e to the i omega t. And it's also good. At least didn't get in trouble. We cannot prove it is
good at this point. We can only prove that you
are not getting in trouble. We are not capable of producing
a contradiction, so far. So actually, 3 and 4 are good. And the obvious question
that would come now is whether you can
use both of them or either one at the same time. So the next claim is that both
cannot be true at the same time. You cannot use both of
them at the same time. So suppose 3 and 4 are good. Both 3 and 4-- and 4 are both good-- both right, even. Then remember that
superimposing a state to itself doesn't change the state. So you can superimpose 3 and 4-- e to the ikx minus i omega t. That's 3. You can add to it 4, which
is e to the minus ikx-- minus omega t. I factor a sine. And that's 4. And that should still
represent this same particle moving to the right. But this thing is twice
cosine of kx minus omega t. So it would mean
that this represents a particle moving to the right. And we already know
that if this represents a particle moving to the
right, you get in trouble. So now, we have to
make a decision. We have to choose one of them. And it's a matter of convention
to choose one of them, but happily, everybody
has chosen the same one. So we are led, finally,
to our matter wave. We're going to make a choice. And here is the choice. Psi of x and t equal to
the ikx minus i omega t. The energy part will
always have a minus sign. Is the mother wave
or wave function for a particle with p
equal hk and e equal h bar omega according to de Broglie. You want to do 3
dimensions, no problem. You put e to the i k vector,
x vector, minus i omega t. On p, in this case,
is h bar k vector. So it's a plane wave
in 3 dimensions. So that's the beginning
of quantum mechanics. You have finally found
the wave corresponding to a matter particle. And it will be a
deductive process to figure out what
equation it satisfies, which will lead us to
the Schrodinger equation.
