 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-61/1/d/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Differentiate

U_{I} = U_{0}^{†} U

. The free propagator satisfies

\dot{U}_{0}^{†} = (i /ℏ) U_{0}^{†} H_{0}

i ℏ \dot{U}_{I} = - U_{0}^{†} H_{0} U + U_{0}^{†} (H_{0} + f H_{1}) U = f U_{0}^{†} H_{1} U_{0} U_{I} .

Thus the interaction picture removes the drift and gives

i ℏ \dot{U}_{I} = f (t) H_{I} (t) U_{I}, H_{I} = U_{0}^{†} H_{1} U_{0}

, with

U_{I} (0) = I

. The transformed control Quantum Hamiltonian is time-dependent even though the original control matrix is constant.

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