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.