21.3. HILBERTSPACE 335
[J−]jm,j′m′ =√
(j′+m′)(j′−m′+1) ̄hδjj′δm,m′− 1[J+]jm,j′m′ =√
(j′−m′)(j′+m′+1) ̄hδjj′δm,m′+1 (21.190)Intheformof∞×∞matricestheseare
J^2 = ̄h^2
0........
.^340......
. 0 34......
... 2 0 0...
... 0 2 0...
... 0 0 2...
.........
.........
Jz = ̄h
0........
.^120......
. 0 −^12......
... 1 0 0...
... 0 1 0...
... 0 0 1...
.........
.........
J+ = ̄h
0........
. 0 1......
. 0 0......
... 0
√
2 0...
... 0 0
√
2...
... 0 0 0...
.........
.........
J− = ̄h
0........
. 0 0......
. 1 0......
... 0 0 0...
...
√
2 0 0...
... 0
√
2 0...
.........
.........
(21.191)
WecanalsowritetheoperatorsJxandJyasmatrices,usingtheidentities
Jx =1
2
(J++J−)
Jy = −i
2