Welcome back everyone. I'm Charles Clark.
This is Exploring Quantum Physics. This part we're going to start studying
the quantum theory of angular momentum. And in particular we'll begin by
introducing and practicing a few basic techniques that are generally useful in
computing the commutation relations associated with operators in quantum
mechanics. They're particularly useful for
understanding level of detail, the theory of angular momentum.
So, in the next few parts of the lecture, we're going to, derive a number of
foundational properties of the angular momentum operator and then apply to
calculations. First we're going to start by reviewing
some standard procedures, the, these may be familiar to some of you but I think a
review here is appropriate. And, of interest to those who haven't
seen them before. They're really quite elementary ideas,
but very powerful in application. And the, first thing we discuss is the,
Einstein's summation con, convention was first introduced by Albert Einstein in
the context of, general relativity. But it's broadly useful and we will, see
how it applies too. Now in the discussion we're going to be,
continue to use vector notation. But, the symbols that we are, really are
using, are all operators so, in correspondence to classical mechanics,
the angle of momentum vector has three separate compliments, one along each
axis. so we'll write it symbolically as a
vector but just keep in mind that the copy, the components of that vector are
in fact operators given by this familiar expression.
Now, there are three useful tools that we're going to use hereafter.
And, in fact, you'll find them used quite widely throughout physics and I think
you'll see the advantage of that in a moment.
So, the first is a notational convention. Probably familiar to some of you, and
that is to replace the standard XYZ Cartesian coordinates.
By index at x1, x2, x3, which are just, I mean it doesn't matter what you choose.
There's just but it's, a convenience instead of having x, y, and z to have an
index set, because in running a, running over sums over all coordinates it just is
convenient to introduce them, by indices rather than different letters.
Then corespondingly we define an, appropriate set of unit vectors, which
are just related the old x, y, and z unit vectors call them e1, e2, and e3.
Okay, now the second aspect is the Einstein summation convention.
So in his work on general relativity, Einstein was evaluating many multiple
sums over coordinates and he made a observation.
And let's describe his observations, in terms of the familiar example, which is
the scalar product of two vectors, a.b. And in our, index notation we would write
that as a sum of I equal 1 to 3 of ai times bi.
Now Einstein noticed the following thing and that is; When you have a summation
sign and the sum is over indices that are repeated in the sum, you may just as well
eliminate the, the summation sign and write the expression thusly.
So this save space, especially if you're doing multiple sums.
So the message now is when indices are repeated in the expression, then the sum
is implied. And we'll, go through that, let's see if
you can apply it now to a simple problem. Okay, I hope that elementary construction
was clear to you. It's just a way of writing a vector and
just summing over all the cartesian coordinates of the projection on to that
coordinate times the unit vector on that coordinate, and you just use the Einstein
summation. Okay the third tool that we'll be using
is the Levi-Civita symbol. initially introduced by Tullio
Levi-Civita and it's a function with three arguments indices i, j and k.
I, J and K are, are, are taken from the values one, two and three.
This is simply used for vector manipulation and has the following
properties so when I, J, K is 123, 231 or 312 in that order the value of epsilon
ijk is one. Those are the so called cyclic
permutations, so you can see you can get from here to here just by wrapping.
Wrapping, it's like having the the arguments on a ring let's say, and now by
rotating the ring, you just take 1 to 2, 2 to 3, and 3 to 1.
So this, cyclic for cyclic permutations of the arguments from the natural order
the symbol takes the value plus 1 for odd permutations, which are just the
permutations you get by transposing two of these and then executing cyclic
permutation those. The symbol is minus 1 and otherwise it's
0, and this is important in the sense that it, vanishes when, any two, any two
of these indices are the same. So let's, try a simple example of
applying this to describe the conventional vector cross product.
Now, the way I phrased that previous example within, in the video quiz was to
indicate the in factors, there's no fundamental significance to the use of
any symbol for an index. Very often we have, a large number of,
of, index symbols that need to be kept independent so we just keep throwing them
in. So here is, important contraction
identity for the Levi-Civita symbol. And again I'm writing this using Einstien
summation convention. So this, this identity here implicitly
refers to a summation over the repeated index I.
So, it says it when you have a product of two Levi-Civita symbols, and you're
selling over one index, then you get a contraction.
And it's so if, again it has this, expresses the symmetry of the system, so
this is, the, the value of this is delta js, delta kt minus delta jt, delta ks.
where the delta functions the Kronecker delta which is 1 if the arguments agree
and 0 otherwise. Now, you can actually work this out, it's
not a deep identity you can just look at the several cases.
And it might be helpful for you to pause here and just try a few cases for
yourself and persuade, persuade yourself that this is true because it, it is
something that's broadly useful. And it's a good idea for you to develop
some intuition about how and when it can be used correctly.
Okay, we're going to conclude this part with two examples of the use of this
formalism from, with applied to the standard problems of vector calculus.
So, let's first look at this, this product, A.B cross C, a tripe vector
product. Okay, so how do we write this in the
Einstein summation convention notation? Well this is just that, because we're
now. We're, we're, we're going to go down to
specific indexing of the, the various components and see what we get.
So, this is, this is a scalar process. A sub i times the i'th component of the
vector B cross C. Well, from the definition of the cross
product that you, uh,uh, look, you, you reviewed in the, in the M video quiz, the
i'th component of E cross C is epsilon ijkBjCk.
So now, we can just bring the A inside and we have, you see we now have
something that can be evaluated because we have implied summations over all three
indicies. So this, this expresses this A.B cross C
in a highly symmetric form, and now by, just.
The, these identities here and here are trivial, they just involve cycling, doing
cycling permutations on the indices here, and rearranging the order of a, b and c.
Now here we're talking about classical vectors, so there's no issue computation.
And so, what you can see is, that this original, expression A.B cross C, is the
same as B.C cross A, and the same as C.A cross B.
That's a, that's a an identity that I think is made quite transparent in this,
in this particular notation. let's take another example with vector
identity A cross B cross C. So this will give us, show us how
contraction identity works and give us use of multiple Levi-Civita symbols.
So, we start just by expanding in compliments, and this is the, the whole
idea of this whole approach, is to take an expression, expand it out in the
components, and then run contractions over all indices that are possible.
So, we start by saying, okay, here's the cross product of, of, we put in the
vector, A, epsilon ijk, epsilon iaj, then B cross C, the K complement of that.
Now, what's the K complement of B cross C?
Well we just throw in some more indices. That's epsilon KBSCT minus BTCS epsilon
KSTBSCT. So this is now, expressed with the
cross-product, and, let's see. Well, I've real, I've run a cyclic,
permutation of the index ijk to get that. That's the same, the same thing.
I'll, just pass this epsilon through to get a highly symmetric looking product
here. So, we have the product of two
Levi-Civita symbols with a repeat, one repeated in index k.
So then we take this identity here, and perform the contraction.
Now the notation's somewhat different, the, the, the specific symbols used for
the indices, differ. But I think you can see that the whole
point is, you take the common, the repeated index K and then the contraction
is delta is delta jt minus delta js, delta it, that in other words, plus 1 for
the same order of n to z minus 1 to the different order.
So, now that gives us this expression. And let's see what do we have?
Well we have a contraction on, we have a contraction on both indices so this gives
us, for example, the delta is, the delta jt gives us eibi.
So, that's that one there and then AjCj. And then, correspondingly, are it gives
us, eiCi and js gives us Aj Bj. So, now we see that this here is the
vector B and this here is the .product A.C and similarly.
So we get this, we get this spectral identity out in a fairly transparent way.
Okay, so I think it might be a good idea for you to review this and make sure that
you're you're comfortable with the innervations.
And we'll next, in the next part we'll start applying them to the angular
momentum operator.