Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 14D / a

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- Solution a

Solution

For a normalized state, write

A^{'} = A - ⟨ A ⟩_{ψ}, B^{'} = B - ⟨ B ⟩_{ψ} .

Then

Δ_{ψ} A = ∥ A^{'} ψ ∥

and

Δ_{ψ} B = ∥ B^{'} ψ ∥

. The Schwarz inequality gives

(Δ_{ψ} A) (Δ_{ψ} B) \geq ∣ (A^{'} ψ, B^{'} ψ) ∣ \geq ∣ Im (A^{'} ψ, B^{'} ψ) ∣.

Since

A^{'}

and

B^{'}

are Hermitian and their commutator equals

[A, B]

,

2 i Im (A^{'} ψ, B^{'} ψ) = (ψ, [A, B] ψ) .

Therefore

(Δ_{ψ} A) (Δ_{ψ} B) \geq \frac{1}{2} (ψ, [A, B] ψ),

which is the Robertson uncertainty principle.

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