Lecture 14
Angular momentum operator algebra
In this lecture we present the theory of angular momentum operator algebra in quantum mechanics.
14.1 Basic relations
ˆ ˆ ˆ Consider the three Hermitian angular momentum operators Jx, Jy and Jz, which satisfy the commutation relations ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Jx, Jy = i~ Jz, Jz, Jx = i~ Jy, Jy, Jz = i~ Jx. (14.1)
The operator
ˆ2 ˆ2 ˆ2 ˆ2 J = Jx + Jy + Jz , (14.2) ˆ ˆ ˆ is also Hermitian and it commutes with Jx, Jy and Jz:
ˆ2 ˆ ˆ2 ˆ ˆ2 ˆ J , Jx = J , Jy = J , Jz = 0. (14.3)
These relations are not difficult to prove using the operator identity
A,ˆ BˆCˆ = A,ˆ BˆCˆ + BˆA,ˆ Cˆ. (14.4)
1 2 LECTURE 14. ANGULAR MOMENTUM OPERATOR ALGEBRA
For example,
ˆ2 ˆ ˆ2 ˆ2 ˆ2 ˆ J , Jz = Jx + Jy + Jz , Jz ˆ2 ˆ ˆ2 ˆ = Jx , Jz + Jy , Jz , (14.5)
ˆ because by definition Jz commutes with itself and, using (14.4),
ˆ2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ Jz , Jz = − Jz, Jz Jz − Jz Jz, Jz = 0. (14.6)
The remaining two commutators in the last row of (14.5) can be calculated using again (14.4):
ˆ2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ Jx , Jz = Jx, Jz Jx + Jx Jx, Jz ˆ ˆ ˆ ˆ = − i~ JyJx + JxJy , (14.7) and
ˆ2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ Jy , Jz = Jy, Jz Jy + Jy Jy, Jz ˆ ˆ ˆ ˆ = i~ JxJy + JyJx . (14.8)
ˆ2 ˆ ˆ2 ˆ The sum of Jx , Jz and Jy , Jz is therefore zero and from (14.5) it follows ˆ2 ˆ that J , Jz = 0. ˆ ˆ Rather then working with the Hermitian operators Jx and Jy, it is more convenient to work with the non-Hermitian linear combinations,
ˆ ˆ ˆ J+ = Jx + iJy, (14.9a) ˆ ˆ ˆ J− = Jx − iJy, (14.9b)
ˆ † ˆ where, by definition, (J−) = J+. For reasons that will become clear later, ˆ ˆ J+ and J− are called ladder operators. Using (14.1) and (14.3), it is straight- 14.1. BASIC RELATIONS 3 forward to show that
ˆ ˆ ˆ Jz, J+ = ~ J+, (14.10a) ˆ ˆ ˆ Jz, J− = − ~ J−, (14.10b) ˆ ˆ ˆ J+, J− = 2~ Jz, (14.10c) ˆ2 ˆ ˆ2 ˆ J , J+ = J , J− = 0. (14.10d)
ˆ ˆ ˆ ˆ ˆ ˆ The operators J+ and J− often appear in the products J+J− and J−J+, which are equal to
ˆ ˆ ˆ ˆ ˆ ˆ J+J− = Jx + iJy Jx − iJy
ˆ2 ˆ2 ˆ ˆ = Jx + Jy − i Jx, Jy ˆ2 ˆ2 ˆ = Jx + Jy + ~ Jz, (14.11) and
ˆ ˆ ˆ ˆ ˆ ˆ J−J+ = Jx − iJy Jx + iJy
ˆ2 ˆ2 ˆ ˆ = Jx + Jy + i Jx, Jy ˆ2 ˆ2 ˆ = Jx + Jy − ~ Jz, (14.12)
ˆ2 ˆ2 ˆ2 ˆ2 respectively. After noticing that (14.2) implies Jx + Jy = J − Jz , we can ˆ ˆ ˆ ˆ straightforwardly rewrite J+J− and J−J+ as
ˆ ˆ ˆ2 ˆ2 ˆ J+J− = J − Jz + ~ Jz, (14.13a) ˆ ˆ ˆ2 ˆ2 ˆ J−J+ = J − Jz − ~ Jz. (14.13b)
Adding equations (14.13) side-by-side and rearranging the terms, we obtain
1 Jˆ2 = Jˆ Jˆ + Jˆ Jˆ + Jˆ2. (14.14) 2 + − − + z 4 LECTURE 14. ANGULAR MOMENTUM OPERATOR ALGEBRA
Two additional useful relations are:
ˆ ˆ n ˆ n Jz, J+ = n~ J+ , (14.15a) ˆ ˆ n ˆ n Jz, J− = − n~ J− , (14.15b) where n = 0, 1, 2,..., is an integer number. We demonstrate, for example, ˆ 0 ˆ the first relation (14.15a). The proof is by iteration. If n = 0, then (J+) = I and (14.15a) becomes a trivial identity. If n = 1, we simply recover (14.10a). If n > 1, we first rewrite
ˆ ˆ n ˆ ˆ ˆ n−1 Jz(J+) = JzJ+(J+) , (14.16)
ˆ ˆ ˆ ˆ ˆ and then use (14.10a) to write JzJ+ = J+Jz + ~J+. This permits us to find the following recursion relation:
ˆ ˆ n ˆ ˆ ˆ ˆ n−1 Jz J+ = J+Jz + ~J+ J+ ˆ n ˆ h ˆ ˆ n−1i = ~ J+ + J+ Jz J+ . (14.17)
If we rewrite this equation replacing n with n − 1, we easily find
ˆ ˆ n−1 ˆ n−1 ˆ h ˆ ˆ n−2i Jz J+ = ~ J+ + J+ Jz J+ . (14.18)
Substituting this expression into (14.17), we obtain
ˆ ˆ n ˆ n ˆ n ˆ n−1 ˆ h ˆ ˆ n−2io Jz J+ = ~ J+ + J+ ~ J+ + J+ Jz J+