334 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA
oftheelectron.Inthatcasethe|θ,φ>basisisinadequate,andtheuseoftheangular
momentumbasis|jm>becomesindispensible.
Theeigenstatesofangularmomentumarewrittenasunitcolumnvectors
φ 00 =
1 0 0......
φ (^12) , 12 =
0 1 0......
φ (^12) ,− 12 =
0 0 1 0.....
φ 11 =
0 0 0 1 0 0...
φ 10 =
0 0 0 0 1 0...
φ 00 =
0 0 0 0 0 1 0..
(21.188)
Usingeq. (21.186)above,andalsotheangularmomementumraisingandlowering
operators
J−|jm> =√
(j+m)(j−m+1) ̄h|j,m− 1 >J+|jm> =√
(j−m)(j+m+1) ̄h|j,m+ 1 > (21.189)wegetthematrixelements
[
J^2
]
jm,j′m′ = j(j+1) ̄h(^2) δ
jj′δmm′
[Jz]jm,j′m′ = m ̄hδjj′δmm′