Suppose has an isolated

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch Slow change, sudden change and the Berry phase

Problem

Suppose H^(R⃗)\hat H(\vec R) has an isolated, non-degenerate eigenvalue En(R⃗)E_n(\vec R) throughout a region of parameter space, with instantaneous eigenstate ∣n(R⃗)⟩|n(\vec R)\rangle (normalised to one). At each point of parameter space ∣n(R⃗)⟩|n(\vec R)\rangle is fixed only up to a phase: any smooth real function ξ(R⃗)\xi(\vec R) gives an equally good choice ∣n′(R⃗)⟩=eiξ(R⃗)∣n(R⃗)⟩|n'(\vec R)\rangle = e^{i\xi(\vec R)}|n(\vec R)\rangle, still normalised, still an eigenstate with the same eigenvalue En(R⃗)E_n(\vec R). Write the Berry connection as A⃗n(R⃗)=i⟨n(R⃗)∣∇Rn(R⃗)⟩\vec A_n(\vec R) = i\langle n(\vec R)|\nabla_R n(\vec R)\rangle, so that the extra phase accumulated adiabatically by ∣n⟩|n\rangle along a path CC in parameter space, beyond the dynamical phase θn\theta_n, is γn(C)=∫CA⃗n(R⃗)⋅dR⃗\gamma_n(C) = \int_C \vec A_n(\vec R)\cdot d\vec R. ((a)) Derive how A⃗n\vec A_n changes when the phase convention is changed from ∣n⟩|n\rangle to ∣n′⟩|n'\rangle. ((b)) Show that γn(C)\gamma_n(C) computed with ∣n′⟩|n'\rangle equals γn(C)\gamma_n(C) computed with ∣n⟩|n\rangle when CC is a closed loop (R⃗a=R⃗b\vec R_a=\vec R_b), but that it differs by ξ(R⃗a)−ξ(R⃗b)\xi(\vec R_a)-\xi(\vec R_b) when CC runs between two distinct points R⃗a→R⃗b\vec R_a\to\vec R_b. ((c)) Show that the dynamical phase θn=−1ℏ∫En dt\theta_n=-\frac1\hbar\int E_n\,dt is unaffected by the choice of ξ\xi, on any path, and say in one line what this means for which phases in the adiabatic problem are directly measurable.

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