A spin- particle sits in a uniform magnetic field along the -axis

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Function spaces and operator theory · Function spaces and operator theory

The Structure of Quantum Mechanics, Ch Observables, spectra and the spectral theorem

Problem

A spin-12\tfrac12 particle sits in a uniform magnetic field along the xx-axis, giving it the Hamiltonian H^=ℏω2σ^x\hat H = \dfrac{\hbar\omega}{2}\hat\sigma_x, where σ^x=(0110)\hat\sigma_x=\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix} is the Pauli matrix written in the S^z\hat S_z eigenbasis {∣↑⟩,∣↓⟩}\{|\uparrow\rangle,|\downarrow\rangle\}, and ω=1.0×1011 rad s−1\omega=1.0\times10^{11}\,\text{rad s}^{-1} is a given constant. Take t=1.0×10−11 st=1.0\times10^{-11}\,\text{s}. (a) Find the eigenvalues of H^\hat H and the corresponding normalised eigenvectors (in the {∣↑⟩,∣↓⟩}\{|\uparrow\rangle,|\downarrow\rangle\} basis), and write the two spectral projectors P^+,P^−\hat P_+,\hat P_- as 2×22\times2 matrices. (b) Use the spectral theorem to write H^=∑aaP^a\hat H=\sum_a a\hat P_a and check by direct matrix multiplication that this reproduces ℏω2σ^x\dfrac{\hbar\omega}{2}\hat\sigma_x. (c) Construct U^(t)=e−iH^t/ℏ=∑ae−iat/ℏP^a\hat U(t)=e^{-i\hat Ht/\hbar}=\sum_a e^{-iat/\hbar}\hat P_a and give its matrix at the stated tt, to three significant figures. (d) Check your result against the identity e−iβσ^x=cos⁡β 1−isin⁡β σ^xe^{-i\beta\hat\sigma_x}=\cos\beta\,\mathbb1-i\sin\beta\,\hat\sigma_x for a real number β\beta.

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