## 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

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# More on two spin systems.

READING: Covering section 26.5 of the text [1].

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.

**Rotation operator**

Recall the definitions of the raising or lowering operators

or

We have

and

So

unless .

is a matrix.

Combining rotations

If

(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 .

Recall that

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

We find

Using

In fact, this is correct. Proof “by construction” to follow.

denote also by

but will often omit the portion.

With

Look at

we must have

So must be a superposition of states with . Choosing is called the Conbon Shotley convention.

We now move down column.

So

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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