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PROFESSOR: We have two
equations now relating
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this number of states.
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And now you can say, oh,
OK, so I look at the k line.
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And I look at the
little piece of the k
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and say, oh, how many
states were there
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with 0 potential, some
number, first blackboard.
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How many states are there
now with some potential,
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some other number?
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It has changed.
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For every-- because these two
equations, the n for equal dk,
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the n is not equal
to the dn prime.
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In one case, the energy
levels or the momentum levels
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are more compressed
or more separated,
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but whatever it is,
whatever the sign is,
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there is a little discrepancy.
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So both of them are
giving me the total number
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of positive energy
states in the little dk.
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Case So if I take
the difference,
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I will get some information.
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So I would say the
following, if I
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want to calculate the number
of positive energy solutions
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and now I think the following,
I take the potential V equals 0
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and slowly but surely deepen
it, push it, and do things
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and create the potential V of
x slowly, slow the formation.
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In this process, I can look
at a little interval dk
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and tell how many states are
positive energy states I lost.
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So if, for example, dn is
bigger than dn prime, dn equal 5
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and dn prime is equal to 3, I
started with 5 positive energy
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states in this little
interval and by the time
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I change the potential
I ended up with 3.
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So I lost 2.
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So let me write here the
number of positive energy
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solutions lost in
the interval dk
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as the potential is turned
on is dn the original number
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minus the dn prime.
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If that's positive,
I've lost state.
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If it's negative, I gained
state, positive energy states.
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In this number, we can
calculate the difference.
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This is minus 1 over
pi d delta dk dk.
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I'll put it here.
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We'll we're not far.
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We'll this is what you lost.
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The number of positive
energy eigenstates
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that you lost in little dk.
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To see how many positive energy
states you lost over all,
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you must integrate over all
the dk's and see how much
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you lost in every little piece.
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So the number of positive energy
solutions lost, not in the dk,
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but lost as the
potential is turned on
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is equal to the integral over
k from 0 to infinity of minus 1
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over pi d delta dk is in
the way of that expression
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of that right coincide.
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But this is a total derivative.
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So this is minus 1 over
pi delta of k evaluated
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between infinity and 0.
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And therefore, the number
of states lost is 1 over pi,
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because of the sign down to
0 minus delta of infinity.
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So we're almost there.
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This is the number of positive
energy solutions lost.
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Now I want to emphasize that the
situation is quite interesting.
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Let me make a
little drawing here.
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So suppose here
is the case where
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you have the
potential equal to 0
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and here is energy equal to 0.
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Then you have all these states.
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Now even though
we've put the wall,
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the wall allows us
to count the states,
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but there are still going to be
an infinite number of states.
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The infinite square wall has
an infinite number of states.
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So that thing really continues,
but what happens by the time v
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is deferred from 0?
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Here is that the E
equals 0 line and here is
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the E equals 0 line.
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As we've discussed,
as you change
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the potential slowly, this
are going to shift a little
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and some are going
to go down here,
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are going to become
bound states.
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They're going to be a number of
bound states, N bound states,
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number of bound states
equal N. And then
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there's going to be
still sub states here
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that's also go to infinity.
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So you cannot quite
say so easily, well,
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the number of states here minus
the number of states here is
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the number lost.
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That's not true, because that's
infinite, that's infinite,
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and subtracting infinity is bad.
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But you know that you've lost
a number of finite number
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of positive energy solutions.
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So as you track here, the
number of states must--
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the states must go
into each other.
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And therefore, if these
four states are now here,
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before they were here, and
those were the positive energy
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solutions that were lost,
in going from here to here,
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you lost positive
energy solutions.
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You lost a finite number of
positive energy solutions.
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Even though there's infinite
here and infinite here,
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you lost some.
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And you did that by
keeping track at any place
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how much you lost.
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And therefore the states
lost are never really lost.
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They are the ones that
became the band states here.
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So the positive energy
states that got lost
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are the bound states.
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So the number bound states
is equal to the number
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of positive energy
solutions, because there
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are no lost states.
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So this is equal to a
number of bound states,
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because there are
overall no lost states.