Welcome back everyone, I'm Charles Clark, 
this is Exploring Quantum Physics. 
today this lecture we're continuing the 
study of angular momentum and now getting 
to a really important part. 
Which is to see how the angular momentum 
appears in any prov of motion independent 
of whatever the, type of external 
potential is, as an effective potential, 
so as how of the angular momentum, and is 
a fundamental way into the equations of 
motion, so a particle moving at a 
potential. 
So in particular, here we're going to 
show how the angular momentum operator. 
Contributes to the kinetic energy 
operator, and will do this based on just 
the basic commutation relations and the 
identities we've done, preformed so far. 
I consider this to be a fundamental and 
important derivation because it shows 
something deep about the effects of 
angular momentum in quantum mechanical 
systems And there's a part which I think 
you really need to work through yourself 
in order to understand it. 
I'll, I'll, I'll outline the steps, but 
it's really up to you to implement them. 
And I can only say that if you complete 
this derivation and are happy with it. 
you'll have done something that's a 
pretty fundamental calculation in quantum 
physics, and I think that the results of 
it will give you an insight into a lot of 
other physical problems, as well. 
So, let's proceed. 
Now, just to review where we are from 
this definition [UNKNOWN] minimum 
operator. 
And this identity alone everything else 
follows but here are some important 
weight stations which simplify life a lot 
in thinking about how to proceed on more 
complicated calculations then that, these 
show that the the of the commutator of 
the angle incremental with either 
position or the momentum operator leads 
to a common, effectively sort of 
rotation, in the space indeed the same 
the same result as the change for the 
angle incremental operate itself. 
And then we have this important fact 
that. 
The commutative any compliment angle 
momentum with the square root angle 
momentum vanishes. 
I just emphasize that in some sense if 
you just have these first 2 pieces, you 
can do all the rest. 
But you'll find it useful to have A 
recollection of these, these staged 
identities,the staged in order of 
complexity. 
The key result of this part is to see how 
the square of the angular momentum Enters 
in to the kinetic energy operator in a 
very fundamental way. 
So there's some algebra associated with, 
with with deriving this result. 
So we start, as before, by. 
Expanding everything into the simplest 
possible expansion, that is, we'll just 
write out in all components, the the 
value of L squared just by expanding 
this. 
So in fact, the Einstein notation give us 
one rather a simple way of writing out of 
this, fairly dense in terms of indices. 
But so here I've actually I've taken the 
H bar squared and divided L squared by 
it. 
Just for further simplicity just so we 
don't keep getting factors of the H bar 
around. 
And so this is now R cross grad is just 
xjdk, where dk refers to d dxk. 
just a simplified notation here. 
And so here are these again, now look at 
here, here's the repeated sum Because 
we've got the ith, we're taking a, a 
scalar product. 
So the first index is repeated, and then 
we have the others in a fairly general 
way. 
this gets moved over into simplified 
form, and applying this contraction 
identity. 
you see we get to this. 
And in fact, in just a few steps we have 
what looks like it might be pretty much 
as far as one can go. 
Okay, but, let's Keep moving because 
there, there's still is an issue with 
this expression in the sense that we have 
the let's say here djxk. 
And potentially when j and, when j and k 
are the same that doesn't commute. 
We want to, we want to basically move all 
the parts over so there's. 
As little ambiguity about the, presence 
of non-moving bodies. 
In other words, so that we have to 
evaluate the thing in the appropriate, in 
the simplest possible order. 
Which means, for example, getting, 
something like xk xk dj dj. 
If we could, that would be something like 
x squared times, or something like r 
squared, d squared, dr squared if you see 
what I mean. 
Okay, so this, I think, is something that 
You now have to start looking at and you 
can go and simplify using these 
expressions. 
This this general rule for exchanging the 
order of operators /g. 
And let me suggest that you just Just 
pause for a while and try your hand at 
this. 
so the next slide is going to show the 
central resolved and then we'll work out 
some implications of that. 
Well, here we have. 
The a the the the answer, this is the the 
the answer can be expressed in this form, 
so this is equal to minus l squared over 
h squared. 
Well now, you see what can be done 
because this here, the Dell squared is a 
Kinetic Energy operator. 
So from that we can write. 
The Hamiltonian for particle three 
dimensional potential has formed. 
This is, this is the familiar form that 
you've seen earlier many times, earlier. 
And now, we just substitute the result 
that we obtained, and that we see that we 
get we get equational motion That has 
derivatives in the root, in the radial 
coordinates. 
So r squared is equal to x squared plus y 
squared plus z squared. 
And that the angle minimum operator 
basically appears in that expression, if 
you, if you, if you extract this out, 
it's something that has to do with the 
derivatives here. 
Which is the kinetic energy operator and 
the angular momentum is also part of the 
kinetic energy operator but it looks like 
a potential of the type l squared. 
Keep in mind this is still an operate, we 
haven't said anything more about it 
there. 
L squared over 2 mr squared so it's so we 
can actually incorporate, we can think 
about a combined potential that consists 
both of the externally applied potential 
and the angular momentum. 
So a couple of notes about this. 
first of all we didn't obtain this by 
transforming to spherical coordinates in 
any explicit way, but by just taking 
considering the properties of the angular 
momentum operator, we see that it 
motivates a description of the problem in 
terms of the spherical coordinate r and 
then the, the angles of the source of 
theta a and b. 
So basically the way that the angular 
momentum enters directly in the 
Hamiltonian, does, indicate the, an 
appropriate choice of perspective from 
the spherical coordinate system. 
So the second point I'd like to make is 
that there are many other equations in 
physics that a involve a Laplacian 
operator dell squared. 
Electromagnetism, acoustics, motion in 
fluids optics and so on. 
So any time that you have An equation 
with a Laplacian, you have something like 
quantum angular momentum. 
in fact, there's, the, expansion of the 
electromagnetic field in, in states of 
definite or angular momentum actually 
resembles the way that we describe the 
short, the the angular momentum of the 
Schrodinger equation. 
So, your better understanding of angular 
momentum and quantum mechanics, I think, 
it will give you, as it did me, a renewed 
appreciation of many classical problems 
of physics. 
third point I'd like to say it this way. 
Angular momentum is more powerful than 
gravity. 
It's more powerful than electricity. 
By this, I mean the the angular, the 
effect of potential angular momentum, V 
effective, goes as one over r squared, 
whereas for gravity. 
The effective potential goes as 1 over r, 
and the same for electricity. 
So this is a highly repulsive potential. 
It to, it has a greater, approach to 
infinity as r goes to 0 than does any of 
the, do any of the conventional 
potentials. 
In in physics so that, that it shows how 
the, that small r, the angular momentum 
will always dominate over any other type 
of the pairwise potential in the problem, 
as it turns out. 
finally we want to generalize this 
approach to N dimensional systems. 
So instead of thinking about, of a 
particle moving in three dimensions, we 
have to think of a particle moving in, 
in, well hundreds of dimensions. 
Now you may ask, well how relevant to the 
world in which we live. 
And the answer is that in some cases to 
discuss collective particle motions, for 
example lets say you have a motion of two 
particles. 
Well there are six dimensions in which 
two particles move and in some cases if 
you want to identify collective 
coordinates it makes sense to start 
thinking of the, the two particle has 
something that's being described by a 
six, six dimension and to see about the 
most general possible description that 
one can get in a six dimension. 
And this leads to a generalized angular 
momentum. 
Which has to do with the behavior, the 
wave function. 
Under rotations in the space that 
includes all particles. 
So this is something that's been applied 
in nuclear and atomic physics, and it 
introduces many of the same concepts that 
we find in the quantum theory of the 
angular momentum.