Let be a time-independent Hamiltonian with a complete

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Quantum dynamics and its formulations

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

Problem

Let H^\hat H be a time-independent Hamiltonian with a complete, orthonormal, discrete set of eigenstates {∣n⟩}\{|n\rangle\}, so H^∣n⟩=En∣n⟩\hat H|n\rangle=E_n|n\rangle with each EnE_n real, ⟨m∣n⟩=δmn\langle m|n\rangle=\delta_{mn} and ∑n∣n⟩⟨n∣=1^\sum_n|n\rangle\langle n|=\hat 1. The evolution operator built from this spectral decomposition is U^(t)=∑ne−iEnt/ℏ∣n⟩⟨n∣.\hat U(t)=\sum_n e^{-iE_nt/\hbar}|n\rangle\langle n|. ((a)) Show directly from this sum that U^(t)†U^(t)=1^\hat U(t)^\dagger\hat U(t)=\hat 1 for every real tt, so U^(t)\hat U(t) is unitary. ((b)) Show that U^(t2)U^(t1)=U^(t1+t2)\hat U(t_2)\hat U(t_1)=\hat U(t_1+t_2) for any real t1,t2t_1,t_2, and that U^(0)=1^\hat U(0)=\hat 1. ((c)) A spin-1/2 particle at rest in a uniform magnetic field along z^\hat z has H^=12ℏωσ^z\hat H=\tfrac12\hbar\omega\hat\sigma_z, with eigenstates ∣+⟩,∣−⟩|+\rangle,|-\rangle of eigenvalues +ℏω/2,−ℏω/2+\hbar\omega/2,-\hbar\omega/2. Using the spectral sum above, write U^(t)\hat U(t) explicitly as a 2×22\times2 matrix in the {∣+⟩,∣−⟩}\{|+\rangle,|-\rangle\} basis and verify by direct matrix multiplication that U^(t)†U^(t)\hat U(t)^\dagger\hat U(t) is the 2×22\times2 identity matrix.

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