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.