Question

Derive the matrix representation of the lowing operator ? ^- for a two-spin system.

Answer 1

Spin 1/2

In Quantum Mechanics, the angular momentum operator L = r×p = L_xxˆ+L_yyˆ+L_zzˆ satisfies

L²|jm| = hj(j + 1)|jm| (1)

L_z |jmi = hm |jmi (2)

It is useful to define the rising and lowering operators L_± ≡ L_x ± iL_y,

which have the following property

L ± |jm| = hp j(j + 1) − m(m ± 1)|jm ± 1i| (3)

And L_x and L_y are obtained from

L_x = (L₊ + L₋)/2

L_y = (L₊ − L₋)/2i (4)

Now, Spin 1/2

If j = 1/2, the spin-space is spanned by two states: { | 1/2 1/2 |, | 1/2 -1/2| }.

The properties Eq.(2) and Eq.(3) for this particular case are

L_z |1/2 ± 1/2| = ±h/2 |1/2 ± 1/2| (5)

L₊ |1/2 1/2| = 0 (6)

L₊ |1/2 -1/2| = h|1/2 1/2 | (7) (8)

If we use the matrix representation (1 0)T ≡ |1/2 1/2| and (0 1)T ≡ |1/2 -1/2|, the operators are