 Solution

ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2014/3/a/iii/solution

Let

a = E_{ψ} (A)

, which is real. The dispersion, or quantum uncertainty, is the nonnegative standard deviation

Δ_{ψ} (A) = E_{ψ} (A^{2}) - E_{ψ} (A)^{2} = ∥ (A - a I) ψ ∥.

∥ (A - a I) ψ ∥^{2} = ⟨ ψ, (A - a I)^{2} ψ ⟩ = E_{ψ} (A^{2}) - 2 a E_{ψ} (A) + a^{2} = E_{ψ} (A^{2}) - a^{2} .

A squared norm is nonnegative, so the square root is defined. It vanishes precisely when

(A - a I) ψ = 0

, namely when

ψ

A

. Conversely, an eigenvector with eigenvalue

λ

has

a = λ

and hence zero dispersion. Thus zero dispersion is equivalent to being an eigenstate.

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