CHAPTER 1. ANGULAR MOMENTUM
2
one easily establishes the following commutation relations for the Cartesian components of the quantum mechanical angular momentum operator: Lx Ly − Ly Lx = i¯hLz , Ly Lz − Lz Ly = i¯hLx ,
Lz Lx − Lx Lz = i¯hLy . (1.3) Since the components of L do not commute with each other, it is not possible to find simultaneous eigenstates of any two of these three operators. The operator L2 = L2x + L2y + L2z , however, commutes with each component of L. It is, therefore, possible to find a simultaneous eigenstate of L2 and any one component of L. It is conventional to seek eigenstates of L2 and Lz .
1.1.1
Quantum Mechanics of Angular Momentum
Many of the important quantum mechanical properties of the angular momentum operator are consequences of the commutation relations (1.3) alone. To study these properties, we introduce three abstract operators Jx , Jy , and Jz satisfying the commutation relations, Jx Jy − Jy Jx = iJz ,
Jy Jz − Jz Jy = iJx ,
Jz Jx − Jx Jz = iJy .
(1.4)
The unit of angular momentum in Eq.(1.4) is chosen to be ¯h, so the factor of ¯h on the right-hand side of Eq.(1.3) does not appear in Eq.(1.4). The sum of the squares of the three operators J 2 = Jx2 + Jy2 + Jz2 can be shown to commute with each of the three components. In particular, [J 2 , Jz ] = 0 .
(1.5)
The operators J+ = Jx + iJy and J− = Jx − iJy also commute with the angular momentum squared: (1.6) [J 2 , J± ] = 0 . Moreover, J+ and J− satisfy the following commutation relations with Jz : [Jz , J± ] = ±J± .
(1.7)
2
One can express J in terms of J+ , J− and Jz through the relations J2
= J+ J− + Jz2 − Jz ,
(1.8)
2
= J− J+ + Jz2 + Jz .
(1.9)
J
We introduce simultaneous eigenstates |λ, m of the two commuting operators J 2 and Jz : J 2 |λ, m
= λ |λ, m ,
(1.10)
Jz |λ, m
= m |λ, m ,
(1.11)
and we note that the states J± |λ, m are also eigenstates of J 2 with eigenvalue λ. Moreover, with the aid of Eq.(1.7), one can establish that J+ |λ, m and J− |λ, m are eigenstates of Jz with eigenvalues m ± 1, respectively: Jz J+ |λ, m Jz J− |λ, m
= =
(m + 1) J+ |λ, m , (m − 1) J− |λ, m .
(1.12) (1.13)