Hello everybody. My name is Victor Galitski and I gave the
first 8 lectures in this course Exploring Quantum Physics.
Today it is my pleasure to welcome those of you who survived this course to the
last concluding lecture of Exploring Quantum Physics.
So this lecutre is going to be on the very interesting subject of time
dependent quantum mechanics and we're going to talk about it a little later but
first I would like to take a couple of minutes to talk about the final exam
coming up. So the format for the final exam is going
to be slightly different than that of our regular homeworks.
So, there going to be, sixteen problems. Roughly one problem for every lecture and
these problems intend to cover all the material, sort of cover, that we have
studied in this course including this lecture and starting from the very first
lecture. So the final exam will be due, two weeks
from now. Now being the time, when we posted these,
last lectures. and the deadline for the final is
going to be the same as the hard deadline for the last posted homework.
So you will notice that there is no homework for this last week, no regular
homework but there is a final exam. Also you will have just one attempt to
take the final unlike the homeworks where you have multiple attempts.
And once you start there will be a time limit of six hours for you to complete
the final. But you shouldn't worry too much about
it, because the final exam should be a little bit on the easier side as compared
to the last homeworks you've seen in this course.
So if you've got this far, you certainly should take the final and get your
certificate. Or probably certificate of distinction
I'm sure many of you will get those. So in other words if you have so far
completed most homeworks, all the homeworks, you definitely should take the
final, don't even think about it. It's going to be pretty straightforward
and you will do well. Now going to the scientific part, well to
the lecture. So I've been sort of thinking A lot about
the possible subject for the last lecture.
And of course there's are many possibilities because there's a lot of
material we haven't had the time to study unfortunately, due to the time
constraints and some constraints on the format of these lectures.
But I decided to talk about this subject, time-dependent quantum mechanics for two
reasons. So the first reason is that this eh,
topic of time-dependent quantum mechanics is among a very few topics in theoretical
physics that is still being developed. So it's still a very active subject, or a
research topic in theoretical physics.
So this is different by the way from most other things we have seen in this course
where pretty much everything we want to know we do know so
of course people are still arguing about foundations of quantum physics, some
philosophical interpretations of quantum theory and et cetera, so these
discussions will probably continue forever.
But once you accept the axioms of quantum mechanics and look at it from the
particle point of view. So, we can pretty much explain or predict
the result of any conceivable experiment that you know, that can be performed with
quantum systems. So the problem won't say not with the.
mysteries of quantum physics, but with lack of mysteries at this stage.
We, we actually understand pretty well what we see.
So, but, this subject of time-dependent quantum mechanics is a bit different in
that, there's some open, open questions. I wouldn't call necessarily This
question's grand mysteries but they're definitely open problems and frontiers
where which we are still exploring. So this is one reason.
So the second reason I have chosen this subject is because I myself have been
working on it quite extensively for the last few years and this is among my sort
of chief research interests. So, of course what I just told you sort
of implies that well, I hardly can tell you all this interesting things about
time-dependent quantum mechanics in one lecture.
So it's a very active research topic. It could be the subject of a separate
course. But at least I can sort of give you a
flavor. What time-dependent quantum mechanics is
about and also maybe one message of this lecture will be to, for you to understand
why this subject is difficult. Why unlike in most other subfields of
quantum physics, here we still sort of struggling to
understanding the tehcnical law of what's going on.
So, this will be another sort of message of this lecture.
So, what we're actually going to be studying in today's lecture is truly time
dependent showing your equation. So, we actually derived, while in quotes,
this equation, in the very first lecture of the course.
but we never truly used the time-dependent part.
So we in some sense, we got rid of it the first moment we got the opportunity to do
so, because in most cases, we have, we have looked at so the potential, well
actually null cases we have looked at the potential was not time-dependent, the
potential was constant. So what were going to be studying today
is the situation where the Hamiltonian H is a function, explicit function of time.
So it's not a constant it's a functions of time.
So of course even if it were not to be a function of time, even if the Hamiltonian
were a constant. So the wave function would still have
been time-dependent and the wave function would be the.
We have talked little bit about this when we were driving the Feynman path integral
so the wave function is a function of time in this case would have been the
evolution operatior, e to the power minus i over h-bar HT.
You're acting on an initial condition the question is no.
Now if the hamiltonian is a function of time.
So this of course doesnt work. So this in this case doesnt work.
And basically we dont know our priority. What the wave function is now.
But this is exactly what we want to know. So basically to solve the time dependant
Schrodinger equation means either to find the wave function, Si of T explicitly.
Or even better, to find the evolution operator U of T that would relay the
initial condition to the to define a wave function.
So again in the case of time-independent Hamiltonian, so this evolution operator
is written here, so this is this exponential e to the power of minus i
over h-bar h t. So but if the
Hamiltonian's implicit function of time? Well there is no simple closed equation
for this. One thing I should eh, reiterate about
this evolution operator which is true both in the case of time-independent,
Hamiltonian, and in the case of time-dependents.
Is that actually an evolutionary operator is independent of, of the initial
conditions. So it essentially describes a rotation of
the wave function in the Hilbert space. So let us recall that well wave function
describes well has probabilistic interpretation.
>> Okay and so if we take, let's say if we just treat spectral.
If we have let's say one particle in the system.
So the wave function must be normalized to one.
So and it's true both for the initial condition and for the time dependent
part. And it should remain.
>> one just because if we have a single particle in normal divistic quantum
mechanics is just going to stay there. It's not going to be not going to
disappear and a new particles are not going to appear out of the blue.
Well actually in quantum electrodynamics it is possible but were doing normal
divistic single particle quantum mechanics.
So this normalization condition should be preserved.
>> And so if you think about Hilbert space, again vetric space.
Let me just raise this arrow here. So if you think about Hilbert space as a
vetric space and which our wave funcions leave.
So what this condition implies in some sense is that the wave function exists on
a hyper sphere. Well, I cannot draw it in a, in a higher
than three dimensions but of course we should, sort of, take this picture with a
grain of salt. So some multidimential Hubert space, and
there is some sphere. Oh, it did not work out very well.
Let me just. We draw it.
Imagine it. So if we have a sphere.
Well this is not much better. Well anyways you understand what I'm
saying. So this sphere is essentially
manifistation of this condition. Basically means that my wave functions
exists inner most condition is presevered.
So this essentiatlly mean phi squared is equal to 1.
And if I start from some initial condition on this here.
So let's say this is my psi node. And the only thing which can happen with
this psi node is that it can sort of rotate in this Hilber space to a new psi
of t, somewhere else on this sphere. And the operator U essentially enforces
this rotation from Psi nought to Psi of T.
So we already discussed it, as a matter of fact, during the derivation of the
Feynman Path Integral, but it was a month, it was in the middle of, you know,
this optional technical derivation. And I'm not sure if all of you have
looked at it. But I think it's a very imporatnt
message, that actually U of T rotates the intial condition essentually by some
angle in this mulidimensional hilbert space.
The new time dependant wave functions were somewhere on this hypersphere whihc
is, which exits you to this normalization.
And the important thing is that if we were to choose.
Let's see a different initial condition somewhere well, let's say here on this
sphere. So, the and if we were to apply this same
time-dependant Hamiltonian or time-independent Hamiltonian.
So, in both cases, so the rotation would be sorted by end laws.
So, there would, it would, it would be different let's say Psi nought.
Prime and upside prime of t, okay? But the evolution operator, this guy
would be exactly the same. Okay?
And so this is that's why I say even better you know, so of course we want to
find in a particular case when we're dealing with a wave function.
So, we want to find the wave function is a function of time because it will
essentially describe everything we want to know about this system or the
probabilities or the transitions. Within various states and amplitudes.
But if we know the evolution operator. We in some sense can solve all possible
initial conditions in one shot. Just because the Evolution operator has
been enforced. This quantum of Evolution judges by
acting on an initial condition. But, well the problem.
With this I mentioned is that its very hard to find the evolution operator.
Now why is that? Why is it so difficult?
So let me sort of give you a very brief explanation why its difficult.
So if you look at this expression, this equation and if for a second, so let me
just write it here in quotes in some sense this initial conditioning for a
second we assume that the Hamiltonian is not really an operator but just some
function H of T without the hat. So then I can simply divide this
equation, let me put this in quotes since this is not true.
So if we divide by Si of T and integrate both sides of this equation over time, we
can ride the solution to this sort of sharing your equation down the down
version. Sharing your equation as E to the power
minus I over H bar, and then take it all from zero to T.
Age of well seven T prime and let's say DT prime on and here's going to be the
initial condition of sign up. So this would be a solution to an
ordinary differential equation. Time dependent differential equation to
first order. So if you look at it that's what it is,
so it seems very simple. So why don't we just write that and put a
hat here. So it turns out that we cannot do that.
And well, those of you who are interested should take the time and well, stare at
this eh, expression for a little while and well, work with it little bit to see
why it is the case. Well, why it is not the case, actually.
Why this is not true, that in the case of a true operator Schrodinger equation.
So this solution doesn't work. And the chief reason for that is because
the Hamiltonian H of t does not commute with itself necessarily at different
moments of time. So if we take two different moments of
time with let's say H of t1 and H of t2 so the Hamiltonian is not going to be
the, come one, commutator, the Hamiltonian with itself, at different
moments of time is not zero. And this is major, major complication
which makes all quantum mechanical, time-dependent quantum mechanical
problems extremely difficult. And this even goes even the simplest
versions of this problem. Let's say with spin one half, with two
level systems. Even there, there is no generally no
close solution to a time-dependent problem.
So let me just leave it here. So now let me provide a couple of general
examples of time dependent and quantum, problems.
So the first example will be a the following.
So let's imagine we have a particle. Which is moving in a potential v of r.
So we just draw a v of x here. For the purpose of illustration let's
just assume it's one dimension. And let's imagine that this potential
itself is sort of breathing. So it's moving as a function of time.
For example it's changing as a function of time, and as time goes by, let's say
it may become different. So, the question that one may ask is what
happens with the particle? With the quantum particle moving in this
potential. So, how does this wave function change as
a function of time? So even if the potential were constant,
so we know that this is strictly speaking, a rather non trivial problem.
But, if you ground the energy levels in this quantum potential.
And at any moment of time we can define a sort of equivalent time-independent
problem and find the corresponding energy levels.
But if the Hamiltonian is time-dependent the energy itself is not conserved.
So the, the, this would be wrong question to ask in some sense.
What are the energy levels in the problem?
So we know that at any moment of time there is formally a time-independent sort
of eigenvalue problem but unless the potential is changing time very slowly,
these energy levels do not have a meaning because we're always sort of pumping in
energy in the system and the energy is not conserved.
So this is really one of the major complications.
But, the probabilistic interpretation of the wavefunction is still valid, and we
can still ask the questions about, well, the probability, let's say, of finding a
particle in a certain region in space. And it's still going to be given So this
probability is still going to be given by, so let me just write it here.
As an example, probability let's say to find a particle between points x1, and
x2, is going to be given, well it's a function of time, let me write it down
here. Is going to be equal to the integral from
x1 to x2 sin Of x, of t, everything squared integrated over dx.
So this is an example of a physical observable property that we can we can
calculate using a solution to the Schrininger equation if this solution is
no. Now the second kind of problem, well it's
actually the same. Time-dependent Shroedinger equation but
for a different type of a system is a let's say spin in a time-dependent
magnetic field. So this relates to the lecture of
Professor Appelbaum who introduced spin, and in particular he was talking about
nuclear Magnetic resonance. MRI imaging for example is where,
actually what's being done is that the spin is being rotated by time-dependent
field. So and in this case, the Hamiltonian well
the wave function first of all is a two-component spinner, and the
Hamiltonian here, so this is our H of t, so in this case the Hamiltonian is simply
B Dot sigma, sigma being the vector of poly matrices.
And this Hamiltonian, two by two matrix acts on these two components speed.
And essentially this is first order differential equation and the problem
here would be lets say to find the you know, this side of t.
And in this case for example the absolute values squared of this psy op of T.
Squared this would give us the probability of finding the spin in the
state up in the moment of time machine so this would be another observable
characteristic we can talk about. Now an amazing thing actually and I have
to admit I didn't know it until just a few years ago, is that this equation is
actually, well it's two by two matrix. Time dependent of the first differential
equation for a two by two matrix. So this equation appears not to be
solvable in elementary functions. You cannot solve this equation in
general, analytically at least. And so when I first looked at this
equation, I thought well I was just solving during my lunch break.
I didn't even want to look into the text books.
I just thought It was just a toy model. But it turns out that even this equation,
maybe the simplest quantum mechanical problem we can think of, you know, two by
two matrix, or two component, two dimensional complex Hillber, Hilburt
state. Even this problem is not exactly solvable
for a general >> H of T.
So this gives you something well some sort of insight into the complexity of
the subject. But as usual in theoretical physics if we
can't solve the problem exactly it doesn't stop us from solving it by some
approximation methods. Actually I should mention one major
mission in this course was Perturbation theory.
Which is a general theory of solving approximately various stationary quantum
problems. We didn't talk about it, but it's a very,
very important piece of quantum mechanics and you, and those of you who are
interested in professionally pursuing physics you should definitely, you should
definitely look into this option. But in this context, in the context of
non-equilibrium or time-dependent quantum mechanics there is a similar, there is,
there is a similar math as various approximation schemes.
And there are two clear situations. Two sort of asymptotically opposite cases
of either very fast perturbation, or very slow perturbations or so-called adabatic
perturbations. So let me just write it down then.
so so we're going to study actually in in the next lecture, we're going to study
fast or sudden perturbations, which means fast changes in the potential and in the
third part of this lecture we're going to discuss slow.
Or, adiabatic pertubations, which lead to very interesting, so-called topological
quantum, phase that appears, in, in, in the solution to the Schrodinger Equation.
So, just, sort of to wrap up this, course introductory part, will you let me first,
eh, define what we actually mean by fast and slow, so as we discussed throughout
the course actually. well, if you are talking about the
physically quantity which has a non-trivial physical dimension be the
length or time or whatever. So you cannot really say that it's small
or large unless you compare it with a quantity which has the same physical
dimension. So when we have a time scale, let me call
it for instance tau of a certain time-dependent perturbation, let's say
this a typical time at which a potential changes so we in order for us to see to
define whether it's short time scale or long time scale, we have to get another
quantity to compare with. Which has the same physical dimension and
it turns out that in the case of quantum problems, if we have a spectrum, so lets
say this is our energy. Okay, so let's say these are different
energy, energies and there are some level spacing between the neighboring energy
levels. It's in the harmonic oscillator case it
would be H omega and the case of the potential well it would be something
else. And they will in general change with time
so, but let's say if we have a system which is quantum particle which is
sitting on this certain quantum level, and neighboring levels.
So the appropriate energy scale associated with it is delta, this level
spacing. And, and h over delta is another
timescale. So this is a timescale to compare with
when we are talking about fast or slow perturbations.
So if tau So if tau is much smaller than h over
delta, we will say that this is a fast or some perturbation.
If on the other hand if tau is much longer than h over level spacing, we're
going to be talking about slow or adiabatic perturbations.
And these are going to be the fast and slow perturbations again are going to be
the subjects of the next two parts of this lecture.
Now just the final thing I'm going to say here is that those of you who are
interested in a research on this sort of quantum dynamical systems and in sort of
exact non-perturbative methods of solving quantum problems.
So I gave a lecture, well it's a research level presentation but I think it shoudl
be understandable to most of you. So I gave a lecture on this subject at
the Cavalier Institute for Theoretical Physics a couple pf years ago and so if
you are interested, you may want to take a look at it.
Of course, there are a lot of other people working on this and they have a
number of amazing results. But I think there is another there is an
interesting development in this field which allows us to construct exact
solutions. But unfortunately, it is well beyond sort
of this school, this course and the level of this course, so I just put a link
here. So if you're interested take a look.
Otherwise we, we'll talk now about the sudden and slow perturbations and see how
quantum problems can be solved, in these cases, and what new phenomena appear
there. [NOISE]