A spin-1/2 particle sits in a uniform magnetic field along a fixed unit direction

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Time evolution as a unitary group, and the three pictures

Problem

A spin-1/2 particle sits in a uniform magnetic field along a fixed unit direction n^=(sin⁡θcos⁡φ,sin⁡θsin⁡φ,cos⁡θ)\hat n=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta), with time-independent Hamiltonian H^=12ℏω n^⋅σ^\hat H=\tfrac12\hbar\omega\,\hat n\cdot\hat\sigma, where σ^=(σ^x,σ^y,σ^z)\hat\sigma=(\hat\sigma_x,\hat\sigma_y,\hat\sigma_z) are the Pauli matrices and ω\omega is a constant. (a) Starting from A^H(t)=U^(t)†A^U^(t)\hat A_H(t)=\hat U(t)^\dagger\hat A\hat U(t) with U^(t)=e−iH^t/ℏ\hat U(t)=e^{-i\hat Ht/\hbar}, derive the Heisenberg equation dA^H/dt=iℏ[H^,A^H]d\hat A_H/dt=\tfrac{i}{\hbar}[\hat H,\hat A_H]. (b) Using the Pauli identity [n^⋅σ^,σ^i]=−2i(n^×σ^)i[\hat n\cdot\hat\sigma,\hat\sigma_i]=-2i(\hat n\times\hat\sigma)_i for each Cartesian component ii, show that the Heisenberg-picture spin operator obeys dσ^Hdt=ω n^×σ^H(t).\frac{d\hat\sigma_H}{dt}=\omega\,\hat n\times\hat\sigma_H(t). (c) Hence show that the component of σ^H(t)\hat\sigma_H(t) along n^\hat n is constant in time while the components perpendicular to n^\hat n rotate about n^\hat n at angular frequency ω\omega, and identify this as Larmor precession of the operator itself.

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