Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 12, Exercise 12.5.3.
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For 3-d angular momentum, we’ve seen that the components and can be written in terms of raising and lowering operators
In the basis of eigenvectors of and (that is, the states ) the raising and lowering operators have the following effects:
We can use these relations to construct the matrix elements of and in this basis. We can also use these relations to work out expectation values and uncertainties for the angular momentum components in this basis.
First, since diagonals of both the and matrices have only zero elements,
To work out and , we can write these operators in terms of the raising and lowering operators:
We can then use the fact that the basis states are orthonormal, so that
The required squares are
The diagonal matrix elements and will get non-zero contributions only from those terms that leave and unchanged when operating on . This means that only the terms that contain an equal number of and terms will contribute. We therefore have
From 10 we see that the only terms that contribute to are the same as the corresponding terms in , so the result is the same:
We can check that and satisfy the uncertainty principle, as derived by Shankar. That is, we want to verify that
On the LHS
On the RHS
Using the same technique as that above for deriving we have
We therefore need to verify that
for all allowed values of . We know that , so
Thus the inequality is indeed satisfied.
In the case we have
so the inequality saturates (becomes an equality) in that case.