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}^{*}) .

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