In this video I'm going to talk about the, perhaps the simplest, kind of,
problem in time-dependent quantum mechanics.
That is the problem of a sudden perturbation, so, which happens when a
potential and a quantum problem changes very fast as compared to all other
relevant time scales and the problem. So in another term for such a change,
sudden change in the potential is a Quantum Quench.
So, and this may happen, for instance if we have a potential which changes.
Very rapidly from one form to a different form.
So, let's say here for example I'm showing a harmonic oscillator, but don't
show and for a negative time, we have a particle, let's say, just sitting here on
a stationary state described by the usual Gaussian wave function with is the,
Eigenfunction of the grounds, the grounds that wave function the harmonic
oscillator. And let's imagine that at t equals zero
the frequency of the harmonic oscillator potential changes rapidly.
So it increases rapidly, such that it becomes a completely different potential
now. So what kind of questions can we ask?
So well the main before asking questions let's see what's going on here.
So the mean sort of inside in this problem is that while the potential
changes very rapidly, the wave function does not have the time to change.
So the wave function right before the quench is exactly equal to the wave
function right after the quench. In the approximation of an infinitely
fast or an infinitely sudden you know, quench of the potential.
Let's say in this case from one harmonic oscillator frequency to a different
harmonic oscillator frequency. So but they another important insight is
that the quantum state that used to be a good stable stationary ground state.
It's not longer, an eignestate of our potential, but what is it?
So this wave function, the old wave function, is just some wave function for
the, from the point of view of the new Hamiltonian.
So for instance let me call this guy here psi naught of 1, which refers to the
ground state of the old potential that we had for negative for t is smaller than
zero. And now after we quench it, so this sign
naught 1 is no longer an Eigenstate of the of the new Hamiltonian.
Let's say we have this Hamiltonian h 2 when it acts on sine naught 1.
It's no longer equal to an energy of anything times sine naught 1.
It's no longer proportional to sine naught one so this state is no longer an
eigen state. What it is though, it's, as always we can
use the eigen states of the new Hamiltonian so we can use these eigen
states as a base. So we can, what we can do, we can expand
our wave function which was the eigenstate of the old Hamiltonian in
terms of the well some coefficients here, let me call it c sub n, and n in this
case goes from zero to infinity. In this particular case of Harmonic
oscillator. So and this can be expanded in terms of
the wave functions of the new harmonic oscillator Hamiltonian.
But in general, this coefficients are all nonzero.
So what basically what happens after we do this quench, is that the wave
function, which used to be nice, pure. State with a well defined energy, well
defined frequency, now becomes essentially a bunch of de, incoherent
harmonics with different frequencies. It's something if you, if I were trying
to suggest a possible classical analogue of what happens here.
Imagine you have a guitar string and you, you generated a pure frequency of you
know, pure sound with the initial frequency of the string, and then you
suddenly change the length of the string, well just you know pinch it at a certain
distance. So then you're going to generate all
kinds of new sounds all kinds of harmonic, it's no longer going to be a
pure sound. A similar thing is happening here but
instead of a sound, instead of the strings we're having.
You know, quantum states which become now a linear combination of the new
eigenstates. So an, an interesting illustration
perhaps of a physically relevant phenomenon that can be discussed in this
context here. So let's say again let me go back to,
this original potential. So let's say we have many particles
sitting on the ground state, in the ground state.
So this basically means that there is a [UNKNOWN] of sorts, which we gain a
produced [UNKNOWN]. In many example labratories world wide,
so this is now. done very frequently and the recumulous.
So, and let me say I change the frequency of the trap a, a as I just showed you.
So and what happens is that this particles that used to be in the ground
state of the old potential did now get redistributed in essential all possible
states of the new potential. And after time goes by, so there is some
relaxation going on and the system relaxes to new state, but now the
temperature of the state is going to be large because the temperature is actually
essentially the measure of the typical energy of a particle emitted by the
system. So essentially what I'm saying here is
that any change in the potential in this case it's a harmonic oscillator, but it's
a completely general statement. Any sudden change of the potential always
hits it up. So, it doesn't matter whether we increase
the frequency, if we decrease the frequency.
If we, let's say had the lowest energy state in the beginning with the lowest
possible temperature, let's say even zero, absolute zero in this model, you
could do the quench. So the particle are going to be
redistributed among all possible energies.
And the adventure that I'm going to relax, to find a temperature.
Which is always higher than the original one.
So this is an interesting phenomena so we can understand without any calculations,
just by this discussion. Now another thing that we just mentioned
in the end so here in the context of this of this expansion of the regional wave
function in terms of the new eigenfunctions of the harmonic oscillator
or well any other potential for this matter.
So this coefficient determine the probability of us finding the particle in
the single particle case or the density of particles let's say in the mini
particle case in this specific new eigenstate.
So cn squared is exactly the probability of finding our particle in the state psi
sub n2, 2 again refers to the potential after the quench.
Well at, oops, at T greater than zero. Okay.
It's not always working this writing on a tablet.
Now, so now let me move actually to a similar but physically quite different
problem which also involves some perturbation, but it's a perturbation of
a completely different time. So the toy problem that we're going to
solve was actually a serious research problem back you know 70 years ago or so,
in the good old days of theoretical physics.
And it was, actually considered by. A famous Russian theorist mostly working
in nuclear physics Arkady Migdal in 1939. So this problem I'm going to discuss
actually appears also in the uh, [UNKNOWN] course on theoretical physics
for those of you who have access to this material.
and, Well, the problem, itself, is, quite,
straightforward. So they formulated the problem.
So let's say we have an atom. So, for the sake of simplicity, I'll be
talking about, the, a hydrogen atom. But, in principle.
We can consider any atom including heavy atoms for which it's the most
interesting. due to various applications in nuclear
physics. And lets imagine that a nucleus of this
atom which has isotomic electron or electrons in it's ground state.
Receives a sharp impulse, which gives it a velocity, v.
So we kick it with a with a radiation, or with a hammer, or we drop it from an
airplane, or whatever you want to think about.
So, and what we want to know is what's going to happen with the electron.
So basically here, the sharp, sudden perturbation.
Is that initially we have this atom at rest, and at certain moment of time,
let's sat t equals 0, this atom starts moving with a certain velocity, v.
And if it does so we want to see when the [INAUDIBLE] we want to see if the
electron gets excited. But how do we calculate the probability
for the electron to get excited you know that is to say that it would go to an
excited state with the higher quantum number or the atom would even get
ionized. That is, to say, that the electron just
leaves the atom and goes into the continuum.
And they just go two separate ways, the atom of the nucleus, and the electron.
So here naively it seems that the potential doesn't really change.
We don't quench the potential. but if we look that this problem a little
closer we see that well the atom when it receives this impulse or this kick.
So the atom starts moving so the, and the atom is the potential which binds the
electron to it in the first place. So therefore, the potential in which the
electron is supposed to be bound starts moving.
And so this is a sharp change. And so the original, but the wave
function of the electron does not have time to change.
So if let's say the electron was here, in the vicinity of the lowest orbit so it
was localized in the vicinity of the lowest orbit it will just try to stay
there. And the question we're asking is whether
the atom when it starts moving. with a certain velocity whether it will
carry the electron with it or whether it will leave it behind or maybe whether the
electron will sort of dissipate into higher energy states.
So and for this to solve this problem what turns out to be useful is to go into
another frame of reference. Which is moving together with the atom.
So essentially, we want just to move together with the atom and write the wave
function of the electron in this new frame of reference.
So this is really the key to solving the problem because in the frame of reference
moving together with atom, the the potential is exactly the same.
It was the [UNKNOWN] potential to begin with is the [UNKNOWN] potential here, and
in this frame of reference the atom, the nucleus itself is static, but the
electron now is moving and so essentially what we have to do, we have to write the
electron wave function after the quench, after, well, in this case it's not a
quench a, I'm sorry I shouldn't have said that, after this sort of jolt of the
atom. and this wave function will have two
pieces, so the first piece is going to be just, the standard, ground-state wave
function of the hydrogen atom, in the case of hydrogen atom.
And there will be a piece, a new piece, which is related to the velocity of the
electron moving, with velocity v, so here essentially it's a plane wave.
So a way to write the way function in the frame of reference moving with the
nucleus. is right as a product of a localized way
function in the lowest orbit times the plane wave.
And here again this mv is simply the momentum of the electron in this frame of
reference. And this is just the standard plain wave
expression. So now the final step is to calculate the
excitation probability of the electron. That is, to find the total probability of
an electron, of the electron going to, to some higher energy state, including the
continuum states. And well, the good thing is that we know
the wave function now. So in this moving frame of reference, let
me just write it again, so psi-naught of r is equal to 1 over the square root of
pi a-cubed, so this is Bohr's radius, e to the power minus r over the Bohr radius
and times this plain wave e to the minus i over h-bar mv dot r.
So this our wave function we just derived, sort of, which is by logical
reasoning. And if we want to know, let's say what
the probability of a transition into a higher energy state is, let's say, some
other state. I'm not going to specify what it is, but
some, let's say psi. Size of k, whatever k is.
So k could be a collection of indices involving the radial quantum numbers and
the angular quantum numbers or it can be a label of a continuum state.
Whatever this label, quantum label is. So the amplitude, the transition
amplitude is going to be, let me write it, a of zero to k will be equal to the
overlap integral of this synod. Let me write it star although it doesn't
really matter psi sub key of r integrated over the volume.
And the probability to go from this ground state to an excited state, w sub
k, is going to be equal to the absolute value squared of this transition
amplitude. So this is what it means.
So another way to connect it to what I. Talked about previously is to notice that
this wave function can expanded in the basis of all eigen states of our problem.
In this case actually it's both discreet state we describe atoms, I'm sorry
electron localized near the atom and also the continuum states.
Okay, and it seems that, well the problem is very complicated, so we have to sum
over all this probability. But, well fortunately we don't have to do
so, really. So this was just a, an unnecessary
comment. As a matter of fact, if we're interested
in calculating the total excitation probability, we, may want to notice that,
the total probability to find the electron in some quantum state is equal
to 1, so the electron will stay somewhere, right?
So it will be in some state. So therefore the probability to remain in
the ground state plus the excitation probability, which is essentially the sum
in this ungeneralized sense it also involves an integral of this w sub k.
This this must be equal to 1. And this excitation probability is
exactly what we're trying to find. So, from this equation we simply can
write that this excitation probability is equal to 1 minus the probability not to
get excited. The probability to remain.
In the grounds state now of the moving atom, of the moving nucleus.
Well I just realized actually that there is a slight problem with the notations
here. Because what we're actually looking for
is the transition amplitudes and the transition probability of, from an
electron. That is a moving electron relative to the
stationary nucleus. But we want to find transition's
amplitude into, stationary states, localized near this nucleus.
So therefore, you know? This, us, this case, for instance.
they do not have this, plane wave piece. So let me just.
sort of introduce tilde here which would refer to the fact that these guys are
moving together with that while this state psi-naught that we get in the
previous slide is actually moving with velocity v relative to that.
Now with this correction to notations and with this comment about the total
probability of. Staying in the new ground state, plus the
probability of excitation being equal to one, we can write down this excitation
probability. Let me just do that.
psi star is going to be equal to 1 minus the probability to stay in the ground
state, which means that I have to take this integral of psi-naught with
psi-naught-tilde. This is a ground state of the stationary
atom integrated over the volume and square the absolute value of it.
And so here I know everything there is to know, so for example, this part.
Here is, exactly, this psi naught tilde in my new inconvenient, perhaps,
notations. So we can just plug in this these
functions, and integrate over volume. And I will leave this calculation to you.
So let me just maybe write an intermediate step, and the final result.
So the intermediate step would be simply. Would simply involve writing down
explicitly this integral, so it's going to be 1 minus 1 over pi.
A Bohr radius cube from the normalization factor in the in the ground state wave
function times the exponential, well intregal of the exponential.
Minus 2r over the Bohr radius, minus this plane wave, i over h bar, p with r.
And so integrate it over the volume, and everything is squared.
And so this is really the integral that needs to be calculated.
Well I, I could, I could take another five minutes to, go over with you through
the calculation of the integral, but I think it's probably not the best use of
your time, and my time too, so let me just leave it, for those who are
interested, to do it on your own. And let me just write the final result.
So the final result will be this, 1 minus 1 divided by this ratio here, 1 over 4,
mv squared, n squared v squared a squared.
This is the Bohr radius divided by h bar squared, and everything to the fourth
power. So, I'm not necessarily saying that this
is the most important and illuminating result in physics, but it's a result,
something that you can actually measure and check, so what happens if we kick an
atom. So what is the probability of it being
excited and therefore it would mean protons and there all kinds of different
phenomenon associated with it. So, and well one can round the very
safety checks on this result. So let's say if the velocity is zero, so
we don't really keep the atom so it was stationary have one minus one.
So, the probability to excite the atom is exactly zero, as it should be, because we
don't really do anything. On the other hand, if the velocity here,
well, now in our altruistic case, goes to infinity.
So the, one or infinity is zero, so the probability to excite that atom is
exactly one. So we just leave the electron behind
that's what going to happen. The nucleus will just go away.
That's what's going to happen. In any case, so this is just an example
of that I wanted to show you. So it's actually two examples in this
part of, time dependent quantum mechanical problems.
And in both of these cases we dealt with very sharp, very sudden perturbation's.
To our Hamiltonian and they resulted in redistribution essentially of energies.
So the energy of the original state or is the, well is the original state of the
old Hamiltonian became a linear combination.
A rather complicated linear combination of Eigenstates of the new Hamiltonian.
And this is very typical for sharp perturbations or non adiabatic
perturbations. Now in the next video we're going to
study actually the opposite case of what's very slow perturbation or
adiabatic perturbation which in some sense is more interesting in that it
involves new mathematical structures. A new phenomena, so-called topological
Berry phase, which is currently actually a subject, the subject of intense
research in, in condensed matter physics.