PHY456H1F: Quantum Mechanics II. Lecture 17 (Taught by Prof J.E. Sipe). Two spin systems and angular momentum.
Posted by peeterjoot on November 10, 2011
Peeter’s lecture notes from class. May not be entirely coherent.
More on two spin systems.
READING: Covering section 26.5 of the text .
where is a triplet state for and the “singlet” state with . We want to consider the angular momentum of the entire system
Why bother? Often it is true that
so, in that case, the eigenstates of the total angular momentum are also energy eigenstates, so considering the angular momentum problem can help in finding these energy eigenstates.
Recall the definitions of the raising or lowering operators
is a matrix.
(something that may be hard to compute but possible), then
For fixed , the matrices form a representation of the rotation group. The representations are irreducible. (This won’t be proven).
It may be that there may be big blocks of zeros in some of the matrices, but they cannot be simplified any further?
Back to the two particle system
If we use
If a and a are picked then
is also a representation of the rotation group, but these sort of matrices can be simplified a lot. This basis of dimensionality is reducible.
A lot of this is motivation, and we still want a representation of .
Might guess that, for
Suppose that this is right. Then
Check for dimensions.
Q: What was this ?
It was just made up. We are creating a shorthand to say that we have a number of different basis states for each of the groupings. I Need an example!
Check for dimensions in general
In fact, this is correct. Proof “by construction” to follow.
denote also by
but will often omit the portion.
we must have
So must be a superposition of states with . Choosing is called the Conbon Shotley convention.
We now move down column.
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.