 Circular ladder operators of a two-dimensional oscillator

ID: circular-ladder-operators-of-a-two-dimensional-oscillator

a_{x}, a_{y}

a_{+}^{†} = \frac{a _{x}^{†} + i a _{y}^{†}}{2}, a_{-}^{†} = \frac{a _{x}^{†} - i a _{y}^{†}}{2}, N_{\pm} = a_{\pm}^{†} a_{\pm} .

a_{\pm}

[a_{\pm}, a_{\pm}^{†}] = 1

, with mixed commutators zero. Direct expansion gives

H = ℏ ω (N_{+} + N_{-} + 1), L_{z} = - i ℏ (a_{x}^{†} a_{y} - a_{y}^{†} a_{x}) = ℏ (N_{+} - N_{-}) .

Each circular creation adds one energy quantum and angular momentum

+ ℏ

- ℏ

. At total occupation

N

, the allowed angular momentum values are

N ℏ, (N - 2) ℏ, \dots, - N ℏ

, each once in this energy eigenspace.

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