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At each fixed time,

∣ ψ (x, t) ∣^{2}

is a bell-shaped Gaussian centred at

x = h (t)

, with width determined by

Re F (t) > 0

. Completing the square as in part (d) makes the density symmetric about

h (t)

. Therefore its position expectation value is

⟨ x ⟩_{ψ} = h (t) .

The sketch is a Gaussian peak whose centre oscillates between the two turning positions found in part (e), while its width varies periodically through

Re F (t)

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 324 / 1 / b / i / Solution Created 2026-09-28 Updated 2026-10-03

Since

P ∣ ψ ⟩ = λ_{P} ∣ ψ ⟩

P^{2} = I

, and

P

anticommutes with

Q

⟨ ψ ∣ Q ∣ ψ ⟩ = ⟨ ψ ∣ PQP ∣ ψ ⟩ = - ⟨ ψ ∣ Q ∣ ψ ⟩,

so the expectation value of

Q

is zero. The two spectral projectors of the Hermitian Pauli operator

Q

are

(I \pm Q) /2

, and hence the Born rule gives

Pr (λ_{Q} = \pm 1) = ⟨ ψ \frac{I \pm Q}{2} ψ ⟩ = \frac{1}{2} .

(2)

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 4D / a / Solution Created 2026-09-24 Updated 2026-09-25

Write the expectation value as

⟨ O ⟩_{ψ} = ⟨ ψ ∣ O ∣ ψ ⟩ .

Differentiating and retaining the possible explicit time dependence of the observable gives

\frac{d}{d t} ⟨ O ⟩_{ψ} = ⟨ \dot{ψ} ∣ O ∣ ψ ⟩ + ⟨ ψ \frac{\partial O}{\partial t} ψ ⟩ + ⟨ ψ ∣ O ∣ \dot{ψ} ⟩ .

(2)

The Schrodinger equation and its adjoint are

i ℏ∣ \dot{ψ} ⟩ = H ∣ ψ ⟩, - i ℏ ⟨ \dot{ψ} ∣ = ⟨ ψ ∣ H,

(3)

where the Hamiltonian operator is Hermitian. Substitution yields

\frac{d}{d t} ⟨ O ⟩_{ψ} = \frac{i}{ℏ} ⟨ ψ ∣ H O ∣ ψ ⟩ - \frac{i}{ℏ} ⟨ ψ ∣ O H ∣ ψ ⟩ + ⟨ \frac{\partial O}{\partial t} ⟩_{ψ} = \frac{i}{ℏ} ⟨[H, O] ⟩_{ψ} + ⟨ \frac{\partial O}{\partial t} ⟩_{ψ} .

(4)

This is the proof of Ehrenfest theorem from the Schrodinger equation.

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 35B / a / Solution Created 2026-09-24 Updated 2026-09-25

A density operator satisfies

ρ = ρ^{†}, ρ ⪰ 0, tr ρ = 1.

N \times N

Hermitian matrix has

N^{2}

real parameters, and the trace condition removes one, leaving

N^{2} - 1

(2)

real free parameters.

A_{H} (t) = U (t)^{†} A_{S} U (t)

, equality of expectation values requires

tr (ρ_{H} U^{†} A_{S} U) = tr (U ρ_{H} U^{†} A_{S}) .

(3)

Therefore

ρ_{S} (t) = U (t) ρ_{H} U (t)^{†} .

(4)

Pauli correlation tensor 2026-10-07

For a two-qubit density operator, the real three-by-three matrix

T

encodes the spin-product expectation values: unit measurement axes

a, b

have

E (a, b) = a^{T} T b

. The tensor does not include local Bloch-vector data and alone need not specify the complete density operator. The CHSH optimum from the correlation tensor depends on the two largest eigenvalues of

T^{T} T