Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 6A / iii / Solution

Positivity of the squared norm of the supplied state gives, for every real

s

,

0 \leq ∥(p - i s x) ψ ∥^{2} = ⟨(p + i s x) (p - i s x)⟩ = ⟨ p^{2} ⟩ + s^{2} ⟨ x^{2} ⟩ + i s ⟨[x, p]⟩ = (Δ p)^{2} + s^{2} (Δ x)^{2} - s ℏ,

where the stated zero means were used in the last line. This quadratic in

s

is nonnegative for every real

s

, so its discriminant is nonpositive:

ℏ^{2} - 4 (Δ x)^{2} (Δ p)^{2} \leq 0.

This is the quadratic-norm proof of the Heisenberg uncertainty relation, and proves

Δ x Δ p \geq \frac{ℏ}{2} .

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