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PROFESSOR: We're
finished with WKB.

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In recitation, you saw some
transmission across the barrier

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and that's also
included in the notes.

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That's an important application
of WKB and should look at it.

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And today we're going to
start with a new topic.

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It's time-dependent
perturbation theory.

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And time-dependent
perturbation theory

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is going to keep us busy
for a number of lectures.

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There's a lot of
applications of these ideas

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and the methods
are rather general.

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Here is the place where we
will learn about Fermi's Golden

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Rule.

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The kind of rule that
is useful for radiation

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problems, ionization
problems, transitions.

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It's all very applied.

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Nevertheless, we have to develop
the theory carefully and see

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what are the main concepts.

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So time-dependent perturbation
theory is our subject.

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Time-dependent
perturbation theory.

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Again, we will begin
with a Hamiltonian

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that is time independent.

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And we know about it.

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We call this Hamiltonian H zero.

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The same name we used for
time-independent perturbation

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theory.

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We have H zero.

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This time, however, we
will have a perturbation.

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The perturbation will
also be called delta H.

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But the big difference is this
time the perturbation will

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be time dependent.

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And this will be our
whole Hamiltonian.

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So this is the subject
we're trying to understand.

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Whenever you have a Hamiltonian,
this time independent,

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we already know how we're
supposed to deal with it.

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We're supposed to find
the energy eigenstates.

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And then we'll have
the whole collection

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of energy eigenstates.

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We can solve any problem.

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In the initial condition
of the wave function,

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you expand it in
energy eigenstates.

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You evolve the state in time.

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Everything is nice and simple.

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In a sense, for the first
time in your studies

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of quantum mechanics
at MIT, we're

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going to face very directly
the difficulties of time

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dependence.

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And the first difficulty of a
time dependent Hamiltonian is

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that you cannot define
energy eigenstates anymore.

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The whole concept is gone.

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That's pretty radical.

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But it's unfortunately
the truth.

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When you first learn
in 804, how to work

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with energy
eigenstates, it was all

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dependent on a factorization.

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A possible factorization
of the solution

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in the factor that depends on
position and a function that

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depends on time.

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And that time-dependence
was always very simple.

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E to the minus iEt
over h-bar, where

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E was the energy
which was a solution

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of the time-independent,
spatial part of the problem.

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So separating the
differential equation

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was possible because, for
example, the potentials never

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depended on time.

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If the potential
depends on time,

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imagine the equation
H psi equal e psi.

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This supposed to be
time independent.

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But H has a time
dependent potential.

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It's just impossible.

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So we don't have energy
eigenstates, any more,

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for this potential,
for the new potential,

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for the new Hamiltonian.

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And we'll have to
think how we're going

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to face these difficulties.

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So no energy
eigenstates for H of t.

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Of course, there are
energy eigenstates for H0.

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H0 is supposed to be your
time-independent Hamiltonian.

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So how do we think of this?

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We will think of
time, evolving here,

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and typically we will
have an initial time t0.

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And throughout this region, the
Hamiltonian, H is equal to H0,

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for time less than 10 0.

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Then we imagine that the
perturbation turns on,

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and suddenly things start
happening, up to time tf.

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Here the Hamiltonian
is H0 plus delta H t.

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And after, the time tf, the
Hamiltonian, is back to H0.

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So that's a nice way of thinking
about the problem, in which we

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imagine, OK,
perturbation is localized

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that some time t
begins there before you

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have the known Hamiltonian,
after you have the known

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Hamiltonian.

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That allows you to rephrase
questions in a clearer way,

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because there are
energy eigenstates here.

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If you sit at that
energy eigenstate

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before time equal t0, you will
remain in an energy eigenstate.

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There's no reason
why it changes.

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That's what time-independent
Hamiltonians do.

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The energy eigenstate
changes by a phase.

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And that's all it does.

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So here, we can speak
about energy eigenstates,

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and after time tf, we can
speak about energy eigenstates.

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So we can ask the
question, suppose

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you were sitting on this
particular energy eigenstate,

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here.

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Then the world shakes
for a few moments.

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Which state are you
going to find yourself,

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after this process?

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And this is a question of going
from one energy aide and state

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to some collection of
energy eigenstates.

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By the time the lecture
will be finished,

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we will have solved this
problem in principle,

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and set up how you would do
it, in practice, for any case.

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This is a very physical
way of thinking, as well.

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You can imagine, you
have a hydrogen atom

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in its ground state.

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So the original system
is a Hamiltonian

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for a particle, an
electron, and a proton.

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And it happens to be
in a ground state.

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Then you send in an
electromagnetic wave, something

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we will do next lecture.

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No, in a couple of lectures.

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And then, it's possible
that the atom gets ionized,

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or that the electron is
kicked up to a higher level.

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You will be able to calculate
those transition functions.

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You will be able to calculate
the probability of ionization.

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In which, after the perturbation
is all said and done, you ask,

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what is the
probability that you're

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in an eigenstate, that
this a higher excited state

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of the hydrogen atom?

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And with a little bit of
flexibility in your mind,

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you can think of
the hydrogen atom

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as a system that
has bound states,

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and continuum
states, in which you

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have a proton and an
electron traveling a wave.

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If you your electron is
far enough from the proton,

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it's like a plane wave.

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If it's not that
far, it can travel.

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And its wave is
deformed a little.

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Those are the continuum
states of the hydrogen atom.

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So you could ask, what is the
probability that it's ionized,

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and it's a transition to
a continuum eigenstate

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in the hydrogen atom?

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This sounds more complicated.

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It's a little more complicated.

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Why?

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Because when you go from
one state to another,

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you can count it.

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But when you ask what is the
probability that the electron

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goes into the continuum
of plane wave,

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you'll have infinitely
many continuum states.

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And we will have
to deal with that.

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Where we will know--

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we will not be able to
avoid this complication.

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And that complication makes
for a very interesting result,

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transitions to the continuum.

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So this is what we want to do.

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And if given, that
in general, this

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is not a problem of
finding energy eigenstates,

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the energy
eigenstates are known,

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we want to find the
wave function, psi of t.

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That's our real unknown.

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And to do that, we
will use something

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called the interaction picture.

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Just like we have Heisenberg
picture, Schrodinger picture,

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we'll have an
interaction picture.

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Don't worry.

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It's not more complicated
than anything you've seen.

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It's, in fact, a
very sensible way

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of doing things, in
which you combine

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good things from the
Heisenberg picture,

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and nice things from the
Schrodinger picture, together.

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So remember, a little
of what was going on

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with the Heisenberg picture.

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It all began by saying that,
if you have the expectation

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value of a Schrodinger operator,
that was the same thing.

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A s for A Schrodinger, that
was possible to compute

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as the expectation value of the
Heisenberg operator on the time

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equals 0 states, in the
states that don't vary.

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You don't need to compute
expectation values of operators

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used in the time
involved states.

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You can think of time
involved operators

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and compute in this
expectation value

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in the time equals 0 states.

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For that, you use
the unitary operator,

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that does time
evolution, u of t psi

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at 0 u dagger of t psi at 0.

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And this unitary
operator, it's in general

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difficult to calculate.

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It's a unitary operator
that does time evolution

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and, in our case, it's hard
because the Hamiltonian

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is complicated.

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But in general, we call this
the Heisenberg operator.

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A Heisenberg of t.

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That's the definition of
the Heisenberg operator.

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Another concept that
is kind of useful,

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is the idea of operators
that brings states to rest.

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So suppose you have the
state psi at sum time t.

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I want to act with an operator
that will bring it to rest.

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That means that this
is time-dependent.

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I want to act with
something that

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will make it time-independent.

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So the answer is U dagger.

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U dagger is a unitary operator,
so it's the inverse of U.

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So look at this expression,
U dagger on this state

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00:14:03,200 --> 00:14:05,850
gives you U dagger.

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00:14:05,850 --> 00:14:11,540
The state is U on the
state at time equals 0.

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00:14:11,540 --> 00:14:14,990
U dagger U, being
unitary, it's just

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00:14:14,990 --> 00:14:20,060
the unit matrix, and your 2 0.

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00:14:20,060 --> 00:14:27,380
So this operator, U dagger,
removes the time-dependence.

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00:14:27,380 --> 00:14:29,120
The [INAUDIBLE]
uses the expression,

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00:14:29,120 --> 00:14:33,110
brings the operator to rest.

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00:14:33,110 --> 00:14:39,710
So we can think of doing
something of this kind.

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00:14:39,710 --> 00:14:45,490
You see, the whole Hamiltonian
is H0 plus delta H.

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00:14:45,490 --> 00:14:47,900
H0 you understand well.

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00:14:47,900 --> 00:14:50,120
Delta H is complicated.

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00:14:50,120 --> 00:14:58,430
So how about doing the time
evolution through Heisenberg,

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with H0, with what you know.

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00:15:01,160 --> 00:15:04,580
You don't know the
full U, but you

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00:15:04,580 --> 00:15:07,430
know the U that
would do the time

220
00:15:07,430 --> 00:15:11,120
evolution, for the
time-independent Hamiltonian,

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00:15:11,120 --> 00:15:12,200
H0.

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00:15:12,200 --> 00:15:18,830
So let's attempt to do
the part that is easy.

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00:15:18,830 --> 00:15:20,960
You see, there's going
to be time evolution

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00:15:20,960 --> 00:15:23,150
as you go from t0 to tf.

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Some of that is going
to be generated by H0,

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some of that by delta H. Let's
let Heisenberg do the work

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00:15:32,390 --> 00:15:40,040
for H0, and Schrodinger
do the work for delta H.

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That's basically the idea. .

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00:15:43,130 --> 00:15:47,120
You see, you know that what is
difficult about this problem

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is the delta H. So you solve
the Schrodinger equation.

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00:15:50,940 --> 00:15:55,400
Let's solve the Schrodinger
equation that just has delta H,

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00:15:55,400 --> 00:15:59,480
doesn't have H anymore.

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So motivated by this,
we'll do exactly that.

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00:16:03,680 --> 00:16:18,310
Think of for H0, the operator
U is e to the minus i H0

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00:16:18,310 --> 00:16:19,900
t over h-bar.

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00:16:19,900 --> 00:16:22,180
This is the operator
that generates

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00:16:22,180 --> 00:16:26,890
time evolution for H0.

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00:16:33,460 --> 00:16:43,400
So we will take
the state psi of t

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00:16:43,400 --> 00:16:53,810
and remove the time-dependence
associated to H0.

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00:16:53,810 --> 00:16:57,140
So try to bring
the state to rest.

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00:16:57,140 --> 00:17:00,650
So we'll put here,
e to the minus--

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I'm sorry.

243
00:17:03,760 --> 00:17:05,540
I'm supposed to put U dagger.

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00:17:05,540 --> 00:17:14,069
So I'll put e to the
I H0 t over H-bar.

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00:17:14,069 --> 00:17:17,450
And look at that.

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00:17:17,450 --> 00:17:19,010
This is U dagger.

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00:17:19,010 --> 00:17:22,849
That's the kind of thing that
brings the state to rest.

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00:17:22,849 --> 00:17:29,510
If the Hamiltonian had
only been H0, only H0,

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00:17:29,510 --> 00:17:32,480
this would be time-independent.

250
00:17:32,480 --> 00:17:35,420
H0 brings it to rest.

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00:17:35,420 --> 00:17:39,140
Because the Hamilton
is not just H0,

252
00:17:39,140 --> 00:17:42,800
this will not be, in
general, time-independent.

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00:17:42,800 --> 00:17:44,790
But it will depend on time.

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00:17:44,790 --> 00:17:49,370
So this is a kind of a nice
wave function in which you

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00:17:49,370 --> 00:17:55,850
sort of have removed the time
evolution having to do with H0.

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00:17:55,850 --> 00:18:02,360
So we will define this as our
auxiliary variable, psi of t.

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00:18:05,180 --> 00:18:06,590
That's the definition.

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00:18:06,590 --> 00:18:21,310
It's motivated by the idea that,
if delta H was equal to 0, then

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00:18:21,310 --> 00:18:27,665
psi tilde is constant in time.

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00:18:31,750 --> 00:18:34,930
Because if delta
H was equal to 0,

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00:18:34,930 --> 00:18:37,810
all the evolution
is created by H0.

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00:18:37,810 --> 00:18:42,585
You would put here, oh, this is
e to the minus i H0 times psi

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00:18:42,585 --> 00:18:45,790
of t equal 0, the
exponentials would cancel,

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00:18:45,790 --> 00:18:47,920
and everything would be simple.

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00:18:47,920 --> 00:18:51,550
So this is a wave
function that is

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00:18:51,550 --> 00:18:53,080
going to be our new variable.

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00:18:53,080 --> 00:18:56,390
We wanted to find psi of t.

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00:18:56,390 --> 00:19:02,330
Now you can say your task
is find psi tilde of t.

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00:19:05,030 --> 00:19:06,920
That's your new task.

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00:19:06,920 --> 00:19:10,310
And it's an equally good
task, because if you

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00:19:10,310 --> 00:19:16,340
find psi tilde of t, then
you can write psi of t

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00:19:16,340 --> 00:19:19,820
as, from this equation,
e to the minus

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00:19:19,820 --> 00:19:26,740
i H0 t over H-bar
psi tilde of t.

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00:19:31,150 --> 00:19:37,260
So our task now will
be to find psi tilde.

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00:19:37,260 --> 00:19:40,350
If we have psi tilde,
we have psi, as well.

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00:19:40,350 --> 00:19:44,170
So we haven't lost
any information.

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00:19:44,170 --> 00:19:47,170
And this is all good.

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00:19:47,170 --> 00:19:51,510
So let's try to see what
equation is satisfied

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00:19:51,510 --> 00:19:55,050
by psi tilde, what kind
of Schrodinger equation

280
00:19:55,050 --> 00:19:57,050
is satisfied by it.

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00:19:57,050 --> 00:20:09,330
So what is the Schrodinger
equation for psi tilde.

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00:20:09,330 --> 00:20:20,270
I'll just take i H-bar d dt of
psi tilde and see what I get.

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00:20:20,270 --> 00:20:20,780
OK.

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00:20:20,780 --> 00:20:28,360
If I have to differentiate
this term, i H-bar d dt.

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00:20:28,360 --> 00:20:31,820
I have to differentiate
this exponential.

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00:20:31,820 --> 00:20:35,540
And the i is going to
give you a minus sign.

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00:20:35,540 --> 00:20:38,130
The H's are going to cancel.

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00:20:38,130 --> 00:20:42,320
And this is just going
to bring an H0 down.

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00:20:42,320 --> 00:20:50,720
So I'm going to get minus
H0 times that exponential.

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00:20:50,720 --> 00:20:55,740
And that exponential times
psi of t is psi tilde of t.

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00:21:03,600 --> 00:21:06,870
So the derivative
of the first term

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00:21:06,870 --> 00:21:08,670
gives me something with H0.

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00:21:08,670 --> 00:21:12,090
And the face times that
is still psi tilde.

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00:21:12,090 --> 00:21:16,050
Now I have to differentiate
the second one so I have plus

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00:21:16,050 --> 00:21:25,540
e to the i H0 t over H-bar.

296
00:21:25,540 --> 00:21:31,680
And i H-bar d dt of this cat.

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00:21:31,680 --> 00:21:35,430
But that's the Schrodinger
equation for the original cat.

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00:21:35,430 --> 00:21:42,150
So I should put here the full
Hamiltonian, H0 plus delta H

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00:21:42,150 --> 00:21:43,830
times psi of t.

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00:21:48,760 --> 00:21:53,890
So when the i dd
H hits the state,

301
00:21:53,890 --> 00:21:58,630
you get the full
Hamiltonian time psi of t.

302
00:21:58,630 --> 00:22:06,190
But I actually were right for
psi of t e to the minus i H0 t

303
00:22:06,190 --> 00:22:11,080
over H-bar psi tilde of t.

304
00:22:11,080 --> 00:22:16,480
Because I'm looking for
an equation for psi tilde.

305
00:22:16,480 --> 00:22:18,370
I'm going a little--

306
00:22:18,370 --> 00:22:20,935
I'm speaking slowly,
but going a little fast.

307
00:22:25,930 --> 00:22:27,100
Now what happens?

308
00:22:29,800 --> 00:22:33,970
What you wanted to
happen, happened.

309
00:22:33,970 --> 00:22:36,060
H0 is here.

310
00:22:36,060 --> 00:22:37,900
And look, H0 is here.

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00:22:37,900 --> 00:22:40,480
Well it's accompanied by
these two exponentials,

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00:22:40,480 --> 00:22:42,130
but they have H0.

313
00:22:42,130 --> 00:22:45,040
So they commute through
and these two exponentials

314
00:22:45,040 --> 00:22:49,960
cancel as far as this
first term is concerned.

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00:22:49,960 --> 00:22:56,440
So nice consolation over here.

316
00:22:56,440 --> 00:23:05,090
And then we get the
following equation,

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00:23:05,090 --> 00:23:15,050
i H-bar d dt of psi tilde is
equal to, well, this delta H

318
00:23:15,050 --> 00:23:21,290
sandwiched in these
two operators.

319
00:23:21,290 --> 00:23:25,670
So I'll just copy it.

320
00:23:25,670 --> 00:23:36,970
e to the H0 t over H-bar
delta H e to the minus i H0 t

321
00:23:36,970 --> 00:23:42,310
over H-S bar times
pi tilde of t.

322
00:23:47,510 --> 00:23:51,440
So this equation makes
what we wanted manifest.

323
00:23:51,440 --> 00:23:58,310
If delta H vanishes, psi
tilde is time-independent.

324
00:23:58,310 --> 00:24:01,100
But it's more than that.

325
00:24:01,100 --> 00:24:07,285
This which we will
call delta H tilde.

326
00:24:10,170 --> 00:24:12,280
So tilde objects
are objects that

327
00:24:12,280 --> 00:24:17,290
have been acted by H0,
like the tilde state,

328
00:24:17,290 --> 00:24:20,200
it has an H0 with
respect to the other.

329
00:24:20,200 --> 00:24:24,970
This delta H tilde is because
it has been acted by similarity

330
00:24:24,970 --> 00:24:27,580
with those things.

331
00:24:27,580 --> 00:24:30,010
But even more, I
think you should

332
00:24:30,010 --> 00:24:47,340
realize that this thing
is really the Heisenberg 0

333
00:24:47,340 --> 00:24:55,830
version of delta H. A
Heisenberg operator is obtained

334
00:24:55,830 --> 00:24:59,490
by taking the Schrodinger
operator, putting U dagger

335
00:24:59,490 --> 00:25:02,850
and U. And that's exactly
what you've done here.

336
00:25:02,850 --> 00:25:06,790
You've taken the Schrodinger
operator and put U

337
00:25:06,790 --> 00:25:12,360
dagger with respect
to H0, and U for H0.

338
00:25:12,360 --> 00:25:18,210
So this is the Heisenberg
version of delta H

339
00:25:18,210 --> 00:25:21,890
relative to H0.

340
00:25:21,890 --> 00:25:29,180
So delta H has been
"Heisenberg-ed" using H0.

341
00:25:29,180 --> 00:25:37,300
And then this whole thing
looks like i H-bar d dt of psi

342
00:25:37,300 --> 00:25:43,220
tilde equal delta
H tilde psi tilde.

343
00:25:47,750 --> 00:25:50,285
And this is a
Schrodinger equation.

344
00:25:55,780 --> 00:26:01,950
So there it is, for you, the
so-called interaction picture.

345
00:26:01,950 --> 00:26:05,050
The interaction
picture says whatever

346
00:26:05,050 --> 00:26:08,830
is not an interaction
will make it Heisenberg.

347
00:26:08,830 --> 00:26:14,320
Whatever is purely interactive
will make it Schrodinger.

348
00:26:14,320 --> 00:26:18,140
And therefore, this
state varies in time.

349
00:26:18,140 --> 00:26:20,320
And there are some
operators that

350
00:26:20,320 --> 00:26:25,510
have acquired extra
time-dependence, as well, due

351
00:26:25,510 --> 00:26:28,270
to the Heisenberg process.

352
00:26:28,270 --> 00:26:32,736
So this is the situation we
are going to try to solve.