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PROFESSOR: You first are facing
the calculation of the energy
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eigenstate with some
arbitrary potential.
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You probably want to know some
of the key features of the wave
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functions you're
going to calculate.
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So in fact, all
of today's lecture
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is going to be devoted to this
intuitive, qualitative insights
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into the nature of
the wave function.
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So we will discuss
a few properties
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that help us think clearly.
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And these are two
of those properties.
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I want to begin with them.
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Then we'll do a third one
that we have already used,
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and we will prove it completely.
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And then turn to the classical
and semi-classical intuition
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that lets us figure out how
the wave function will look.
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And that's a great help for you.
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Even if you're solving for
your wave function numerically,
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you always need to know what
the answer should look like.
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And it's ideal if before you
calculate, you think about it.
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And you realize, well, it
should have this t properties.
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And if you find out
that those are not true,
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well, you will learn
something about your intuition
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and see what was wrong with it.
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So we're talking about one
dimensional potentials, time
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independent potentials.
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And a first statement
that is very important,
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and you will prove in an
exercise after spring break,
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and that is the fact that one
dimensional potentials, when
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you look at what are
called bound states,
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you never find degeneracies,
energy degeneracies.
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And this is when x extends from
minus infinity to infinity.
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You've seen already, in the
case of a particle in a circle,
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there are degenerate
energy eigenstates.
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But if the potential
extends to infinity,
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there is no such things.
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Now what is a bound state?
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A bound state sounds like
a complicated concept.
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But it is not.
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It really means an
energy eigenstate
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that can be normalized.
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Now if an energy eigenstate
can be normalized
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and you live in
the full real line,
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that the wave function
must go to 0 at infinity.
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Otherwise you would never
be able to normalize it.
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And if the wave function
goes 0 at infinity,
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the bound state is some sort
of bump in the middle region
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or something like that.
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And it eventually decays.
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So this is bound by the
potential in some way.
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And that's basically what we
use to define a bound state.
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We'll take it to
be that generally.
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So this is something,
this property,
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which is very important, is
something you will prove.
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But now we go to
another property.
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We've emphasized forever
that the Schrodinger equation
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is an equation with
complex numbers.
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And the solutions
have complex numbers.
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And suddenly, I wrote
a few lectures ago
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a wave function was real.
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And I was asked, well,
how can it be real?
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Well, we've discussed stationary
states in which the full wave
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function, capital PSI, is
equal to a little psi of x
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times the exponential of e
to the minus i et over h bar.
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And there in that
exponential, there
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is complex numbers
on this little psi
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of x in front of that
exponential, which
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is what we called basically
those energy eigenstates.
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The e to the minus
i et over h bar,
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it's understood
that little psi of x
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is the thing we've
been looking for.
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And this psi of x solves the
time independent Schrodinger
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equation h psi equal e psi.
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And that equation has
no complex number in it.
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So little psi of x can be real.
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And there's no contradiction.
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Because the full
solution to the time
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dependent Schrodinger
equation is complex.
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But here is a statement.
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With v of x real, the
energy eigenstates
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can be chosen to be real.
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And the words can be chosen
are very important here.
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It means that you may find
a solution that is complex,
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but you need not stick
to that solution.
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There is always a possibility
to work with real solutions.
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And what is the
way you prove this?
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This I will put
this in the notes.
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You don't have to
worry about the proof.
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You consider the Schrodinger
equation for psi.
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And you show that psi star,
the complex conjugate of psi,
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solves the same equation
that psi solves.
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And therefore, if
psi is a solution,
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psi star is a solution
with the same energy.
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That part is very important.
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Therefore, if you have
two energy eigenstates
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with the same energy,
you can form the sum.
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That's still an
energy eigenstate
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with the same energy.
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Even formed in
difference, that's
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still an energy eigenstate
with the same energy.
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And the sum of psi
plus psi star is real.
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And the difference psi minus
psi star, if you divide by 2i,
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is real as well.
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Therefore you can construct two
solutions, the real part of psi
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and the imaginary part of psi.
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And both are solutions to
the Schrodinger equation.
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So I've said in words what is
the proof of the first line.
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It's that if you have a psi,
psi star is also a solution.
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Therefore, psi plus psi
star and psi minus psi star
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are solutions.
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So given a complex
psi, then psi psi of x.
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Then psi real of x that
we define to be psi
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of x plus psi star of x over 2.
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And the imaginary
part of the wave
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function 1 over 2i
psi of x minus psi
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star of x are both solutions
with the same energy
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as this one has.
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So these are the two solutions.
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So far so good.
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You don't like to
work with complex psi?
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No need to work
with complex psi.
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Work with real psi.
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But here comes the second
part of the argument,
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the second sentence.
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I want you to be alert
that the second sentence is
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very powerful.
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It says that if you have
a bound state of a one
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dimensional potential,
more is true.
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There are no genuinely complex
solutions in this case.
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Any solution that you will
find, it's not that it's complex
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and then you can find the
real and imaginary part.
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No, any solution that you will
find will be basically real.
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And how can it fail to be real?
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It just has a complex
number in front of it
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that you can ignore.
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So it is a very
strong statement.
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That the wave function, it's not
that you can choose to work it.
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You're forced to do
it up to a phase.
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So how is that possible?
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How is that true?
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And here is the argument
for the second line.
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If we're talking bound
states, then these two
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are real solutions
with the same energy.
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So now suppose these
are bound states.
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There is a problem if there
are two real solutions
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with the same energy.
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They would be degenerate.
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And property number
1 says there's
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no such thing as degenerate
energy bound states.
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So they cannot be degenerate.
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So if you start with a complex
psi, and you build these two,
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they must be the same solution.
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Because since there are no
degenerate bound states,
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then psi, I will write
it as psi imaginary,
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of x must be proportional
to psi real of x.
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And both are real, so
the only possibility
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is that they are equal
up to a constant,
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where the constant
is a real constant.
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You see there cannot be
degenerate bound states.
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So the two tentative
solutions must be the same.
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But that means that the
original solution, psi,
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which is by definition
the real part plus i
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times the imaginary
part, is now equal to psi
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r plus i times c
times psi r again,
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which is 1 plus ic times psi r.
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And that is basically the
content of the theorem.
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Any solution is up to a
number, just the real solution.
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So you're not going to
find the real solution has
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non-trivial different
real imaginary parts here.
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No, just the real solution
and a complex number.
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Now if you want,
you can just write
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this as e to the i argument of
1 plus ic times square root of 1
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plus c squared psi r.
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And then it's literally
the way it's said here.
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The wave function is
proportional to a real wave
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function up to a phase.
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So that's a very neat situation.
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And therefore, you
should not be worried
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that we are going to
have to assume many times
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in our analysis that
the bound states were
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trying to look for are real.
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And we plot real bound states.
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And we don't have to worry
about, what are you plotting?
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The real part?
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The imaginary part?
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Many times we can just
work with real things.