The quantum harmonic oscillator problem and its solution go far beyond the simple
generalization of a particle and a spring problem in Classical Physics.
And they appear in a variety of contexts in quantum physics, as already mentioned.
For example electromagnetic radiation can be thought of as a collection of harmonic
oscillators, the photons. And much more generally various complex
structures in quantum mechanics. Oftentimes can be thought of as being
built of the small building blocks being quantum harmonic oscillation.
But in this particular case, the meaning of these individual oscillators is
different. And today, I'm going to show you the
appearance of these quantum harmonic oscillators.
And the context of a very interesting problem that sort of underlies solid
state physics or a condensed measure of physics.
And this is a problem of excitations in crystals, which are collections of many,
many ions that form a periodic structure. And these ions can oscillate, and these
oscillations, quantum oscillations, are called phonons.
And we will see how these phonons appear, and we will see also that they can be
described. In terms of the creation-annihilation
operators that we introduced in the first lecture this week.
To understand the concept of a collective mode, such as a phonon for example, in a
complex quantum system. We first can go back to classical physics
and remind ourselves. The notion of a normal mode, which is
equivalent to the quantum collective mode.
And these normal modes appear in complex classical systems.
So, here, I have a primitive classical system if you want this is a simpler
harmonic oscillator. And in this case, as we know, the
frequency of the oscillation is propor-, is equal to square root of the stiffness
of this spring divided by the mass of this of this object.
now if we make the system a little more complicated, and instead of just one
oscillator. Consider let's say two masses attached to
let's say three springs in this case. So, we have a system here with two
degrees of freedom. And this system will have two different
oscillation frequencies corresponding to different types of oscillatory motions.
So, for instance this type of motion when the two masses sort of oscillate in in
phase with one another. So, this let's see first type of motion
is described by the frequency which is exactly equal to omega.
But there is also a second type of motion that we can imagine this slightly more
complicated system. In which case let's say these guys
oscillate out of phase in this fashion, sort of.
And this oscillation, this oscillation frequency let me call it two in this
case, is equal to square root of times the basic frequency of these guys.
So, let's assume they all have the same stiffness.
And we can make things more and more complicated, and let's say in this case
have six six masses attached to springs. In which case we will have six different
degrees of freedom. And six different oscilation frequencies.
So, and this oscillation frequencies normal so called normal modes in
classical physics. They have an analog in quantum physics.
In particular I would say in this example, in this example of two harmonic
oscillators. We just by knowing the results of the
classical calculation. We can write down, as a matter of fact
the spectrum of the quantum problem, or the corresponding quantum problem.
So and the spectrum of the corresponding quantum problem will be.
simply the energy is going to be equal to h omega 1 and 1 plus 1 half h plus h
omega 2, and 2 plus 1 half. And its going to depends on two quantum
numbers, n1 and n2 corresponding to two harmonic oscillators.
But these are not going to be actually the harmonic oscillators built out of
let's say the coordinates and momentum of the two the two particles.
So these oscillators or the corresponding creation-annihilation operators are going
to be a certain non-trivial linear combination of these x1's and p1's, x2
and p2. which sort of describe the generalization
of these in-phase and out-of-phase motions to quantum physics.
And in general, whenever we have a classical system with some normal modes.
There is a corresponding quantum system with a spectrum associated with these
normal modes. A very interesting example of a complex
mini particle quantum system is a crystal.
We know that crystalline structures of various types appear in a variety of
solid state materials. And in this structure so what happens is
that ions or atoms of which this solicited material sort of built.
They arrange themselves into a unique order and pattern.
And so here, for instance, I'm showing a fragment of a so called simple cubic
lattice. And these blue balls in this picture,
correspond to atoms sitting on their equilibrium position on their lattice
size. But this picture is a bit deceptive,
actually it is probably better to say incorrect in that the atoms here, in
reality, are not static. So, they indeed have some preferential
positions, and some equilibrium positions.
But they move around all the time, so there, this position that they occupy,
corresponds to a minimum in some effective potential they fill.
But we can oscillate near the minimum of this potential and these oscillations are
basically just grabbed locally as a essentially a harmonic oscillator motion.
So, better illustration of what a crystal is that I put together well to simply sit
here for a two dimensional case. So, it's presented here.
So, instead of thinking about atoms which are sort of tied together in this
crystalline structure by some rigid objects.
We should think about atoms being tied together by some elastic springs with
some stiffness. Which is fine at and so when we look at
this system so my analogy to what I had discussed, what I had just discussed in
the previous slide. We can think about this system as a many
oscillator system with a moderate degrees of freedom.
So, in reality in this material so they're not two, or three or six atoms,
like in the previous slide, but they're min, billions and billions of atoms.
And therefore, there are many frequencies, eigen-frequencies of normal
modes that I classical normal modes. That appear and they correspond to
various kinds of complicated oscillations in this system.
And in the limit of an infinite number, basically a so called thermodynamic limit
when we consider essentially an infinite crystal.
So, these create normal modes, they merge into continuum curves which essential
corresponded to elastic waves running through this system.
And mathematically these waves are described by familiar expressions.
So, if we think let's say about the displacement of an atom with regard to
its equilibrium position, some delta r as a function of time, for a particular
atom. Let's say nth atom, so it's proportional
to the familiar exponential of key dot r, so where this atom is located, minus
omega t. But this omega is itself a function of
key. And so to find this so called dispersion,
phonon dispersion is the main question that appears in the context of this
problem. And this is what is going to be of
interest to us. Of particular interest is the question of
what are the low energy modes of this system.
So, what it actually means is the following, suppose we have a completely
static crystal. So, while as static is a guess, in
quantum mechanics there is always zero point motion, but we can even consider
classical system. Let's say we just built physically built
this model by attaching a bunch of masses to the springs, and attach, and arranging
them. In the, in equilibrium pattern and let's
say we touch this system a little bit, we apply very weak perturbation.
So the question I am asking about the law energy modes is.
Have we got inside any oscillation in this crystal ?
Or this weak perturbation is not going to have any effect?
Or, in other words, do we have to overcome any threshold, or gap as
condensed measure people call it. In order to excite the minimal
perturbation in the crystal. And as we will see in a later, in, in
today's lecture, so the answer will be that, it doesn't matter how weak the
perturbation is. There is always a very low energy mode
which corresponds to long wave length. Or small key wave running through the
samples, through this crystal. and this type of an excitation is called
an acoustic mode or it corresponds to sound.
Sound modes running through through this crystal.
And it turns out that this appearance of this acoustic sound modes.
Has a very profound explanation which goes as far as relativistic quantum
physics actually where. There is a theorem called Goldstone
theorem which guarantees the appearance of certain low energy modes.
And even though I'm talking about classical system, let me nevertheless try
to connect it to this theorem. But I will present sort of a poor man's
version of this Goldstone theorem which is relevant to our problem.
And then hopefully, we will be able to understand how they are connected.
So this going to be a little bit of detail work.
So, the poor man's version of the Goldstone theorem that I'm presenting is
related to the notion of symmetry in quantum physics that we already
discussed. So, suppose we have a quantum system
which is invariant or symmetric with respect to certain class of
transformations. What it means is that there exists
transformations, let me call them u. And this operator commutes with the
Hamiltonian which describes this system. So, this is what it means or in other
words U, H, U minus 1 is equal to H itself.
And let me consider this situation, when there is not just one or two symmetry
operations that exist. But as a matter a fact we have a certain,
this whole family of such operations let's see labeled by some parameter.
Let me call it alpha. So, an example of such a symmetry for
instance, could be symmetry with respect to translation.
So, let's say if we, if our Hamiltonian is symmetric with respect with
translating everything, every single particle in space, let's say by certain
distance. So, it could be a distance of 1 meter or
1 centimeter, 1 kilometer. So, there is a whole continuum of
distances and directions we can translate our system.
In which case is alpha, it's actually a three dimensional vector of translations.
And while another example could be symmetry with respect to rotations.
In which case alpha could represent a, an angle that we can choose, by which we can
rotate our system. But in principle, can be anything, so we
just basically postulate that the existence of some symmetry.
What is called symmetry group but I'm not going to go into this details.
with respect to H the Hamiltonian of the system is invariant.
But a very interesting fact is that, even though the model itself can be invariant
under a certain symmetry operations. The quantum states, the for instance
lowest energy state of the ground state, does not necessarily have to be
invariant. under the same symmetry U.
So, what it means is that the system will find itself in a state, let's say, let me
call it psi naught. Such that well, U acting on this psi
naught is not equal to psi naught. So, by translating let's say our system
in space, we do not necessarily represent the same state.
And in this case we call, we say that the symmetry is broken, spontaneously broken.
But the key statement, which is also the statement of this, of this Goldstone
theorem in this context, is the following.
If once again we have a situation where the Hamiltonian is a variant under a
certain continuum of symmetries. But some or all of these symmetries are
broken in this state in which the system finds itself.
In this state let's say psi naught, then, to each broken continuum symmetry.
There corresponds low energy mode, a sound-like mode if you want, that we just
discussed. and well, this statement, this
correspondence, is again the statement of this Goldstone theorem.
So, I realize that there has been, you know, a lot of words that I've said.
But perhaps the best way to understand the actual meaning of these words would
be to provide an example. Which also will explain why this
discussion is relevant to what we talked about before in the previous slides.
About hormones, crystals and all these things.
Now, let us look at a crystal structure once again.
So we have here a periodic arrangement of atoms.
But nobody puts this crystal structure by hand, so if we have a piece of some
solid, solid material, let's say a piece copper.
So, nobody builds it by hand like atom by atom.
We don't we don't have to do it so that nature does it itself.
So, this structure, they form spontaneous and there is some wave function if you
want psi naught. We describe the, describe, well very
complicated wave functions which describes these atom.
So, how do these atom's form in such a pattern?
So, are they you know, this, psi naught, is it a solution to a certain
Hamilitonian that describes those atoms. And these Hamilitonian , it has a well
kinetic energy of all the particles involved in the problem, all these atoms.
plus an energy describing its interactions.
And very importantly here, the interact, well, it depends only on the distance
between the atoms, so here n and m, they labeled various atoms.
Now, if you look at this Hamiltonian, so you, you see that, if we translate the
whole thing. If we, let's say change r to r plus some
constant distance in, let's go, let me go d, then the Hamiltonian's not going to
change. Since we have here only the distance
between particles, which is the position of fth particle minus the position of mth
particle. If we translate both of them by the same
distance, the distance between them is not going to change, right?
So, so therefore, in other words what I'm saying here is that here the Hamiltonian
isn't variant in translations in real space.
By an arbitrary it's very important, by an arbitrary distance.
So, but if you look at the crystalline arrangement here, so, it is no longer
invariant under an arbitrary translation. So, if I remove my particles let's say,
by a fraction of the slightest constant. I'm not going to reproduce the crystals.
So, the crystal is still symmetric, but it's it has a symmetry which is sort of
lower. It will, it has only discreet symmetry,
while the original symmetry of the problem was a continuum symmetry.
And there's a crucial difference and therefore here we have a broken,
spontaneously broken. If you want three continuum symmetries in
the x, y, z directions. And therefore there must exist three
modes or three or threefold degenerate mode or at least three low energy modes.
In this state and these modes are exactly the formats that we're going to find.
Of course in our solution and we'll, we'll see them automatically and they're
not going to be related to spontaneaous symmetry breaking itself.
But it's very interesting this connection is very interesting because you can sort
of predict the existance of such low energy modes.
Just by the fact that certain, certain types of structures form out of symmetric
systems. So, in the rest of the lectures, today
I'm not going to be alluding to these Goldstone theories.
So, this was just remark for those of you who are interested in advanced
understanding or advanced subjects. But I just wanted to make this remark
because I find it quite illuminating. Because you can see the appearance of
this is very powerful. The application of this is very powerful
theorem in a very simple example.