A spin one-half is held in a field along the direction with the eigenstate convention…

PHYSICS · Subjective · 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

A spin one-half is held in a field along the direction n^(β,φ)\hat n(\beta,\varphi) with the eigenstate convention ∣+;β,φ⟩=cos⁡(β/2)∣↑⟩+sin⁡(β/2)eiφ∣↓⟩.|+;\beta,\varphi\rangle=\cos(\beta/2)|\uparrow\rangle+\sin(\beta/2)e^{i\varphi}|\downarrow\rangle. The field direction is held at fixed β=60∘\beta=60^\circ while φ\varphi is swept slowly from 00 to π\pi (halfway round the cone, not a full circuit).Someone argues: "The Berry connection for this state is Aφ=i⟨+∣∂φ+⟩=−sin⁡2(β/2)A_\varphi=i\langle+|\partial_\varphi+\rangle=-\sin^2(\beta/2). Integrating from φ=0\varphi=0 to φ=π\varphi=\pi gives γ=−∫0πsin⁡2(β/2) dφ=−πsin⁡2(30∘)=−π/4≈−0.785 rad.\gamma=-\int_0^\pi \sin^2(\beta/2)\,d\varphi=-\pi\sin^2(30^\circ)=-\pi/4\approx-0.785\ \mathrm{rad}. Since this number came out of a well-defined integral along the actual path taken, it must be a physically measurable phase difference between the state at φ=0\varphi=0 and the state at φ=π\varphi=\pi, detectable by interfering the spin with an unrotated reference spin."Find the error in this argument.

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