Hi everybody. Up to this point in the course Charles
and I have presented already a lot of material.
Which included both material that can be found in traditional quantum mechanical
text books and also some non standard exposure to quantum physics for example
super conductivity path integrals some elements of quantum localisation, etc,
etc. And I have to say, on my side, the
preference has been for the latter. I want to tell you more, about things
that we cannot find in traditional textbooks.
But today is certainly going to, going to be an exception, because today I'm
going to present a very traditional quantum mechanical problem that no
reasonable quantum mechanical course can avoid, and this is the problem of a
harmonic oscilllator. So the reason I cannot avoid this problem
is two-fold. So first of all quantum harmonic
oscillator, the problem of quantum harmonic oscillator is relevant to a
variety of various experimental systems, ranging from cold atoms who are atoms
that are trapped in a quadratic potential, to let's say molecules who's
low energy oscillations are described by harmonic oscillator, to quantization of
light, to quantization of oscillations increase those so-called
[UNKNOWN].
We're going to actually discuss them soon.
And many, many other things. And second, so the problem, on the
theoretical side, the solution to the problem of harmonic oscillator, is very
illuminating and useful, in that it gives you very useful tools, so-called creation
and anti-elation operators that are useful in a variety of context across
various fields of physics. And you're going to see this thing very
often in other courses if you pursue. physics states.
But before solving the quantum harmonic oscillator problem, let me remind you the
story behind the classical harmonic oscillator.
And here I have an example of the classical system which exhibits these
small oscillations and near an equilibrium position.
And these oscillations exactly are the harmonic oscillator motion.
So, on the, mathematical side, what happens here, of course, is that, the,
mass here experiences, the so-called Hook's force, which is proportional to
the displacement x of the, spring relative to its equilibrium position.
So let's say here if this is my x axis, and let's say we have an equilibrium
position somewhere here so this is my zero.
So this is x, and the force is proportional to the displacement, and in
this case at this moment of time it's this Hook's force.
X in this direction, in the negative x direction.
Now, I can write this force, of course, as a minus gradient in this case it's
just one-dimensional gradient, or just the derivative with respect to x, of some
potential. And to produce a linear force, of course,
I need to differentiate a quadratic potential.
So in this case is kx square over 2 and this is exactly the potential energy of
the harmonic oscillator that eventually is going to appear in our Schrodinger
Equation in the quantum version of this problem.
Now the classical side though, what I have to do is I have to solve the second
Newton law presented here. So, and the second Newton law, of course,
is just mass times acceleration, the second derivative of x is equal to all
the forces. In this case, there is just one force, in
this example, it's if we ignore gravity, which is not going to modify too much.
but in this case, it's going to be just one force which is the force due to the
Hook's law. And if I write this equation, if I divide
this equation by m and instead of writing the second derivative, I will write it in
the compound form x with two dots on top So I can write it as x two dots, plus
omega squared x, is equal to 0. Where omega, is equal to the square root
of k over m. And the solution to this, differential
equation is the sines, or cosines to some arbitrary phases, and it's up to us,
which, solution to choose. So there will be, some amplitude here,
amplitude here, so we can choose a sine or we can choose a cosine.
And as long as we have two free parameters which will determine the
initial conditions, the initial position and the initial velocity, these are all
general solutions. And solve these classical problems so let
me choose for example the sin solution and present it here.
So this is this solution to this harmonic oscillator.
Now to move further let me choose a particular intial condition which I can
do which corresponds to x of t equals zero being zero.
So let's say the initial moment of time the position of our harmonic oscillator
was exactly at the equilibrium point just for simplicity and in this case this is
my solution. Now if I want to find the velocity of the
oscillator as a function of time, so all I have to do is differentiate the
position, and in this case it's going to be just x naught.
Some amplitude of oscillations times omega times cosign of omega t.
So what I can do now, I can ride the energy of the system which is not a
function of time as we shall see in accordance with the conversation law.
Which would be a sum of the kinetic energy and m v squared over 2, and the
potential energy v of x, which is equal to one half k x squared.
k is the coefficient in the Hooke's law, or expressing it through the mass and
the, and frequency, I can write it as so, one half, m omega squared, x squared.
And, this is by the way, the form in which typically appears in the
literature, in particular in quantum mechanical literature.
So, now, indeed if we plug in this result into the energy.
So, the energy is going to be equal to mv squared over 2.
So, we're going to have m over 2 x naught squared, omega squared, plus cosine
squared of omega t. From this velocity term and potential
energy plus m by two x squared, x not squared, omega squared from here, sine
squared of omega t, and then we use the fact that cosine squared of an arbitrary
angle plus and sine squared of the same angle is equal to one.
And so we see that indeed the energy is simply this.
So and the energy is conserved. Indeed in a, cor, in, in accordance with
the energy conservation law. But of course, even though the energy E
here, is not a function of time, the two terms in this equation, the kinetic and
the potential energy are functions of time.
So x is a function of time and the velocity is a function of time.
but they work in unison with one another in such a way that their sum, the sum, is
a constant. And if we plug this, so let's say the x.
X axis here is going to be m omega times the coordinate, and the y coordinate here
is going to be the momentum. So we see that this trajectory basically
in the face space describes the motion of a harmonic oscillator.
It's a motion on a circle. and the radius of the circle is
determined by the energy of our harmonic oscillator that we.
And this motion is a phase portrait of this classical system.
And, by the way, I will just mention in passing that there is a relation between
the phase portrait of the classical system and the corresponding spectrum of
quantum systems. But that's the only thing I'm going to
say here in this regard. So now let us move closer to the main
subject of today's lecture and make the oscillator quantum.
So the motivations to do so are many, as I said.
So, for example, we can think about instead of having a classical mass and a
spring, we can talk about, let's say a two-atom molecule and the oscillations of
this molecule relative to the equilibrium position of the atoms.
So at low energies these oscillations are going to be described by a quantum
harmonnic oscillator. So of course there are many more examples
as I mentioned in the beginnning of the lecture.
Let me just mention another one, so for instance Talking about the behavior of a
quantum particle in a rather arbitrary potential v of x so lets say somethign
like this. So if we are interested in low energy
behavior the particle somewhere near the minimum of the potential, we can always
Expand the potential view effects near the minimum, so it's going to be v of x
naught, so this point plus v prime of x naught x minus x naught.
And this is zero because this is the minimum, plus the second derivative over
2x minus x not squared. And model of this overall constant, which
is, just shifts the minimum of energy and this shift of the equilibrium position.
And this shift of the equilibrium position.
So the remaining part of this potential is really the quadratic potential that we
have in the harmonic oscillator problem. So what I'm saying here is that you can
always almost always approximate a reasonable potential, well near the
minimum of the well with the harmonic oscillator if you have a small function.
Of course if you have square quantum well we can not do that but in most cases you
have a smooth potential and in this case we can just replace it at low energies
with this harmonic oscillator. Okay, now let me raise this stuff and
move further so how to formally go from the classical harmonic oscillator in this
but does show the quantum harmonic oscillator.
So, to do so we have to pretty much do the following mathematical procedure.
We can just put a hat here. And that's it.
So that's how me made the oscillator quantum.
So well seriously what we have to do, we have to replace the energy of the
oscillator. So the energy we derived in the previous
side for the classical harmonic oscillator.
So now we replace the energy of the oscillator with the Hamiltonian which is
an operator the momentum which is just a variable in the classical description,
now we replace it with the operator that acts on the wave function.
And in principle we can put a hat also on top of the cord metal, though with the
action of the cord metal on a wave function in position space, is pretty
simple action. It's just a multiplication operator, but
this pretty much what what it means on the mathematical side to make, to go from
classical description with no operators involved to a quantum description with
operator involved. So this is the mapping, and the problem
that we want to solve now is to find solutions to the Eigenvalue problem.
So of course its a non its a time independent problem so there is no time
involved and what we want, we want to find a solution to this equation.
So when solving this Schrodinger equation and in particular when thinking about the
allowed energy levels. So what we should keep in mind about this
harmonic oscillator potential is one property which makes it very different
from what we have seen before. Namely that this potential grows.
The potential energy grows indefinitely to infinity as x become very large or
very negative. So there is no room here for plane
wave-like solutions propagating to infinity, and the only thing we may have
here, are discreet states, corresponding to particles localized in the vicinity of
this x equals 0. And to determine this these energy
levels, these harmonic oscillator spectrum Is exactly the main problem
we're going to be focusing on in the next couple of videos.