L = x p_{y} - y p_{x} = i (a_{y}^{†} a_{x} - a_{x}^{†} a_{y}) .

[H, x] = - i p_{x}, [H, p_{x}] = i x, [H, y] = - i p_{y}, [H, p_{y}] = i y .

Consequently

[H, L] = [H, x] p_{y} + x [H, p_{y}] - [H, y] p_{x} - y [H, p_{x}] = - i p_{x} p_{y} + i x y + i p_{y} p_{x} - i y x = 0.

Thus angular momentum preserves each energy eigenspace. More explicitly,

0 = ⟨ E ∣ [H, L] ∣ E^{'} ⟩ = (E - E^{'}) ⟨ E ∣ L ∣ E^{'} ⟩,

⟨ E ∣ L ∣ E^{'} ⟩ = 0 if E \neq = E^{'} .

Within a degenerate energy eigenspace, this argument places no further restriction on the matrix of

L

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