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ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2019/1/c/solution

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Past exam of the theoretical physics II course of the University of Cambridge / 2019 / 1 / c / Solution by

Codex  0 2026-10-18

(∣1 s ⟩, ∣2 p_{x} ⟩)

. The projected Hamiltonian between pulses is

H_{0} = - Δ σ_{z}

, so its free time-evolution operator through a period is

U_{Δ} = e^{i α σ_{z}} = (e^{i α} 0 0 e^{- i α}), α = \frac{Δ τ}{ℏ} .

(1)

Across one instantaneous pulse, integrating the Schrödinger equation gives

U_{E} = e^{- i β σ_{x}} = (cos β - i sin β - i sin β cos β), β = \frac{E x _{0} τ}{ℏ},

(2)

using the exponential of a Pauli vector. The finite free Hamiltonian contributes vanishing evolution during the zero-duration kick. A period measured immediately before successive pulses therefore consists of the kick followed by free evolution. Composition gives

U (N τ - ϵ, - ϵ) = (U_{Δ} U_{E})^{N}, ϵ \to 0^{+} .

(3)

This is the Floquet operator at the specified phase of the pulse train. Actual hydrogenic energy levels are not opposite numbers: the displayed convention subtracts their midpoint, with

Δ = (E_{2 p} - E_{1 s}) /2

. Restoring that midpoint multiplies the result by an unobservable global phase. Likewise

E

in the given Hamiltonian denotes the electric coupling per unit length; if it is instead a physical electric-field strength, its charge factor must be included in

E x_{0}

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