 Solution

ID: 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.

 New to topics? Read the docs here!