Question

Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators:

[L, x] = ihy, [L, y] = -ihₓ, [L, z] = 0 [4.122]
[L, Px] = ih py, [L, py] = −ihpₓ, [L2, P2] = 0.

Answer 1

The commutation relations involving angular momentum and position/momentum operators are derived using the canonical commutation relations, resulting in six commutation relations involving the angular momentum operator L and the position and momentum operators x, y, and z.

The commutation relation is given by:

[A,B] = AB - BA

Using the commutation relations for position and momentum: [x, p_x] = iħ, [y, p_y] = iħ, and [z, p_z] = iħ, we can determine the commutation relations involving the angular momentum operators:

[L, x] = iħy

[L, y] = -iħx

[L, z] = 0

[L, p_x] = iħp_y

[L, p_y] = -iħp_x

[L^2, p^2] = 0