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 Incoming links: Quantum kicked rotor

Past exam of the theoretical physics II course of the University of Cambridge / 2016 / Paper 1 / 1 / a / Solution 2026-10-20

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Take periodic orbital wavefunctions on the ring, and write

H_{0} = - (ℏ^{2} /2 I) \partial_{θ}^{2}

. Between kicks the evolution over one period is

U_{T} = e^{- i H_{0} T /ℏ} = exp (\frac{i ℏ T}{2 I} \partial_{θ}^{2}) .

(1)

Integrating through an infinitesimal kick gives

U_{η} = exp [- \frac{i}{ℏ} (- ℏ η cos θ)] = e^{i η c o s θ} .

(2)

The free-motion contribution during the vanishing kick duration disappears in the ideal impulsive limit. The sign of the phase follows from the negative kick potential.

Starting just after a kick, the particle evolves freely first and is kicked at the end of the interval. Operators act right to left, so one step is $U_{η} U_{T}$ , and

U (0^{+} \to N T^{+}) = (U_{η} U_{T})^{N} .

(3)

The one-period product is the Floquet operator of this quantum kicked rotor. There is no additional kick at the initial endpoint

0^{+}

. Changing the stroboscopic convention to just before each kick would change the displayed order; it is not appropriate for these endpoints.

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