X_{j} = \frac{ℏ}{2 mω} (a_{j} + a_{j}^{*}), P_{j} = - i \frac{mω ℏ}{2} (a_{j} - a_{j}^{*}) .

L

gives

L = - \frac{i ℏ}{2} [(a_{1} + a_{1}^{*}) (a_{2} - a_{2}^{*}) - (a_{2} + a_{2}^{*}) (a_{1} - a_{1}^{*})] .

Operators belonging to distinct modes commute. Therefore the terms with two annihilation operators and those with two creation operators cancel, and

a_{1}^{*} a_{2} = a_{2} a_{1}^{*}

a_{2}^{*} a_{1} = a_{1} a_{2}^{*}

L = i ℏ (a_{1} a_{2}^{*} - a_{2} a_{1}^{*}) .

ii

Solution

From part (a),

[N, a_{j}] = - a_{j}

and

[N, a_{j}^{*}] = a_{j}^{*}

[N, a_{1} a_{2}^{*}] [N, a_{2} a_{1}^{*}] = [N, a_{1}] a_{2}^{*} + a_{1} [N, a_{2}^{*}] = - a_{1} a_{2}^{*} + a_{1} a_{2}^{*} = 0, = [N, a_{2}] a_{1}^{*} + a_{2} [N, a_{1}^{*}] = - a_{2} a_{1}^{*} + a_{2} a_{1}^{*} = 0.

Consequently

[N, L] = 0.

Each term in

L

transfers one excitation between the two modes, preserving the total number operator eigenvalue. This is an instance of the oscillator bilinear commutator; every number operator eigenspace is an invariant subspace of

L

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