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ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2012/paper-1/1/b/solution

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Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 1 / b / Solution by

Codex  0 2026-10-20

Take an orthonormal basis

{∣ j ⟩}

and insert its completeness relation:

Tr (∣ u ⟩ ⟨ v ∣) = \sum_{j} ⟨ j ∣ u ⟩ ⟨ v ∣ j ⟩ = ⟨ v ∣ (\sum_{j} ∣ j ⟩ ⟨ j ∣) ∣ u ⟩ = ⟨ v ∣ u ⟩ .

No normalization or orthogonality of

u, v

is required. This is the trace of a rank-one operator; in a Hilbert space the sum converges by the Cauchy-Schwarz inequality.

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