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.