 Solution

ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2015/2/b/ii/solution

The squared norm is obtained by moving the adjoint operator to the other side of the inner product:

∥ a_{-} ψ ∥^{2} = ⟨ ψ, a_{+} a_{-} ψ ⟩ = 2 m ⟨ ψ, (H - \frac{1}{2} ℏ ω) ψ ⟩ .

∥ a_{-} ψ ∥^{2} = 2 m (E - \frac{1}{2} ℏ ω) ∥ ψ ∥^{2} .

Since a squared norm is nonnegative, every nonzero energy eigenstate has

E \geq ℏ ω /2

. Moreover

a_{-} ψ = 0

exactly when

E = ℏ ω /2

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