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.