Two independent situations.

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

Two independent situations. ((a)) A particle of mass mm sits in the ground state of a harmonic oscillator of angular frequency ω1\omega_1. At t=0t=0 the spring constant is suddenly quadrupled, so the new angular frequency is ω2=2ω1\omega_2 = 2\omega_1, in a time much shorter than 1/ω11/\omega_1. Using the sudden approximation, and the standard overlap of two oscillator ground states of frequencies ω\omega and ω′\omega', ∣⟨0ω′∣0ω⟩∣2=2ωω′ω+ω′,|\langle 0_{\omega'}|0_\omega\rangle|^2 = \frac{2\sqrt{\omega\omega'}}{\omega+\omega'}, find the probability that the particle is found in the ground state of the new oscillator immediately afterwards. ((b)) A spin-one-half magnetic moment is carried once around a cone in field space, the field direction making a constant half-angle β=45∘\beta = 45^\circ with the cone's axis. Using Ω(C)=2π(1−cos⁡β)\Omega(C) = 2\pi(1-\cos\beta) for the solid angle subtended by the circuit and γ±=∓12Ω(C)\gamma_\pm = \mp\tfrac12\Omega(C), find the geometric (Berry) phase acquired by the state that starts (and ends) as the eigenstate with spin pointing opposite to the local field direction (n^⋅S⃗^=−ℏ/2\hat n\cdot\hat{\vec S}=-\hbar/2), the ∣−⟩|-\rangle state of the text.

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