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ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2014/3/a/i/solution

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Past exam of the quantum theory course of the University of Oxford / 2014 / 3 / a / i / Solution by

Codex  0 2026-10-31

The commutator of two linear operators is

[A, B] = A B - B A

. Write

T^{†}

for the adjoint operator, using the inner product convention linear in its second argument. Since

A^{†} = A

and

B^{†} = B

,

[A, B]^{†} = B^{†} A^{†} - A^{†} B^{†} = B A - A B = - [A, B] .

Thus the commutator is anti-self-adjoint. With

C = - i [A, B]

, conjugation of the scalar gives

C^{†} = i [A, B]^{†} = - i [A, B] = C .

All products are understood on the stated common operator domain; for physical unbounded linear operators, the state must lie in the relevant product domains and have the finite moments used below.

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