PROFESSOR: interpretation
of the wave function. --pretation-- the wave function. So you should look at
what the inventor said. So what did Schrodinger say? Schrodinger thought
that psi represents particles that disintegrate. You have a wave function. And the wave function is
spread all over space, so the particle has
disintegrated completely. And wherever you find more psi,
more of the particle is there. That was his interpretation. Then came Max Born. He said, that doesn't
look right to me. If I have a particle, but I
solve the Schrodinger equation. Everybody started solving
the Schrodinger equation. So they solved it for a particle
that hits a Coulomb potential. And they find that the
wave function falls off like 1 over r. OK, the wave function
falls off like 2 over r. So is the particle
disintegrating? And if you measure, you get
a little bit of the particle here? No. Max Born said, we've
done this experiment. The particle chooses
some way to go. And it goes one way,
and when you measure, you get the full particle. The particle never
disintegrates. So Schrodinger hated
what Max Born said. Einstein hated it. But never mind. Max Born was right. Max Born said, it
represents probabilities. And why did they hate it? Because suddenly you
lose determinism. You can just talk
about probability. So that was sort of funny. And in fact, neither
Einstein nor Schrodinger ever reconciled themselves
with the probabilistic interpretation. They never quite liked it. It's probably said that
the whole Schrodinger cat experiment was a
way of Schrodinger to try to say how ridiculous the
probability interpretation was. Of course, it's not ridiculous. It's right. And the important
thing is summarized, I think, with one sentence here. I'll write it. Psi of x and t does not tell
how much of the particle-- is at x at time t. But rather-- what is the probability-- probability-- --bility-- to find it-- at x at time t. So in one sentence,
the first clause is what Schrodinger
said, and it's not that. It's not what fraction
of the particle you get, how much of the
particle you get. It's the probability of getting. But that requires-- a little more precision. Because if a particle
can be anywhere, the probability of being at one
point, typically, will be 0. It's a continuous
probability distribution. So the way we think of this
is we say, we have a point x. Around that point x, we
construct a little cube. d cube x. And the probability--
probability dp, the little probability to find
the particle at xt in the cube, within the cube-- the cube-- is equal to the value of the
wave function at that point. Norm squared times
the volume d cube x. So that's the probability
to find the particle at that little cube. You must find the square
of the wave function and multiply by the
little element of volume. So that gives you the
probability distribution. And that's, really, what
the interpretation means. So it better be, if you
have a single particle-- particle, it better be that
the integral all over space-- all over space-- of psi
squared of x and t squared must be equal to 1. Because that particle
must be found somewhere. And the sum of the probabilities
to be found everywhere must add up to 1. So it better be
that this is true. And this poses a
set of difficulties that we have to explore. Because you wrote the
Schrodinger equation. And this Schrodinger
equation tells you how psi evolve in time. Now, a point I want to emphasize
is that the Schrodinger equation says, suppose
you know the wave function all over space. You know it's here
at some time t0. The Schrodinger equation
implies that that determines the wave function for any time. Why? Because if you know the
wave function throughout x, you can calculate
the right hand side of this equation for any x. And then you know how
psi changes in time. And therefore, you can
integrate with your computer the differential equation
and find the wave function at a later time all over space,
and then at a later time. So knowing the wave
function at one time determines the wave
function at all times. So we could run into a
big problem, which is-- suppose your wave
function at some time t0 satisfies this at
the initial time. Well, you cannot force the wave
function to satisfy it at any time. Because the wave function now
is determined by the Schrodinger equation. So you have the possibility that
you normalize the wave function well. It makes sense at some time. But the Schrodinger equation
later, by time evolution, gives you another
thing that doesn't satisfy this for all times. So what we will have
to understand next time is how the Schrodinger
equation does the right thing and manages to make
this consistent. If it's a probability at
some time, at a later time it will still be a
probability distribution.
