Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 60 / 3 / a

Solution

Take the drift propagator with the correct initial normalization,

U_{0} (t, t_{0}) = e^{- i (t - t_{0}) H_{0}}

U = U_{0} U_{I}

. Differentiate this product and substitute the Schrödinger equation:

i \dot{U}_{0} U_{I} + i U_{0} \dot{U}_{I} = (H_{0} + \sum_{m} f_{m} H_{m}) U_{0} U_{I} .

Since

i \dot{U}_{0} = H_{0} U_{0}

, the drift terms cancel. Left multiplication by

U_{0}^{†}

gives

i \dot{U}_{I} = m \sum f_{m} (t) H_{m} (t) U_{I}, H_{m} = U_{0}^{†} H_{m} U_{0}, U_{I} (t_{0}, t_{0}) = I .

The printed

e^{- i t H_{0}}

is this expression with time origin

t_{0} = 0

. Its differential identity remains valid at arbitrary

t_{0}

, but its interaction-picture initial operator would then be

e^{i t_{0} H_{0}}

rather than the identity. Using

t - t_{0}

resolves that normalization without changing the derivation.

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