An oscillator has mass and angular frequency

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch Coherent states, Ehrenfest's theorem and the classical limit

Problem

An oscillator has mass m=1.0×10−26 kgm=1.0\times10^{-26}\,\mathrm{kg} and angular frequency ω=1.0×107 rad s−1\omega=1.0\times10^{7}\,\mathrm{rad\,s^{-1}}. It is prepared in a coherent state ∣α⟩|\alpha\rangle with ⟨x^⟩=9.19×10−8 m\langle\hat x\rangle=9.19\times10^{-8}\,\mathrm{m} and ⟨p^⟩=0\langle\hat p\rangle=0. ((a)) Using ⟨x^⟩=2ℏ/mω Re α\langle\hat x\rangle=\sqrt{2\hbar/m\omega}\,\mathrm{Re}\,\alpha and ⟨p^⟩=2ℏmω Im α\langle\hat p\rangle=\sqrt{2\hbar m\omega}\,\mathrm{Im}\,\alpha, find α\alpha. ((b)) Find the mean phonon number nˉ\bar n and the number spread Δn\Delta n. ((c)) A second coherent state of the same mode has β=2α\beta=2\alpha. Find the overlap probability ∣⟨β∣α⟩∣2|\langle\beta|\alpha\rangle|^2.

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