 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/ib/paper-3/20d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

i ℏ \partial_{t} Ψ = H Ψ

. The normalized eigenstates of the Hermitian Hamiltonian

H_{3}

with distinct eigenvalues are orthogonal, so they form an orthonormal basis of this two-state system. Substitute the expansion of

Ψ

and compare the two coefficients:

i ℏ \overset{a}{˙}_{0} = - a_{0}, i ℏ \overset{a}{˙}_{1} = a_{1} .

Hence

a_{0} (t) = a_{0} (0) e^{i t /ℏ}

and

a_{1} (t) = a_{1} (0) e^{- i t /ℏ}

. For the normalized test state

ϕ = α χ_{0} + β χ_{1}

, the Born rule gives

P_{ϕ} (t) = ∣ ⟨ ϕ, Ψ (t)⟩ ∣^{2} = ∣ \overset{α}{ˉ} a_{0} (t) + \overset{ˉ}{β} a_{1} (t) ∣^{2} .

The complex conjugates are essential; relative phases can affect this probability even though the energy-state populations remain constant under

H_{3}

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