Past exam of the quantum theory course of the University of Oxford / 2015 / 2 / b / i

Solution

From

H = a_{+} a_{-} / (2 m) + ℏ ω /2

[H, a_{+}] = \frac{1}{2 m} a_{+} [a_{-}, a_{+}] = ℏ ω a_{+}, [H, a_{-}] = \frac{1}{2 m} [a_{+}, a_{-}] a_{-} = - ℏ ω a_{-} .

Consequently, if

H ψ = E ψ

H (a_{\pm} ψ) = (E \pm ℏ ω) (a_{\pm} ψ) .

ℏ ω

, and the annihilation operator lowers it by

ℏ ω

, provided the resulting vector is nonzero. The zero vector is not an eigenstate; in particular the ground state will be annihilated by

a_{-}

. The displayed operator identity holds even in that case.

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