A mechanical 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

A mechanical oscillator has mass m=1.0×10−25 kgm = 1.0\times10^{-25}\,\mathrm{kg} and angular frequency ω=1.0×104 rad s−1\omega = 1.0\times10^{4}\,\mathrm{rad\,s^{-1}}. At t=0t=0 it is prepared in the coherent state ∣α⟩|\alpha\rangle with α=2\alpha = 2 (real). Consider the time t=T/8t = T/8, where T=2π/ωT=2\pi/\omega is the classical period. ((a)) Using U^(t)=e−iH^t/ℏ\hat U(t) = e^{-i\hat Ht/\hbar} acting on the number-state expansion ∣α⟩=e−∣α∣2/2∑nαnn!∣n⟩|\alpha\rangle = e^{-|\alpha|^2/2}\sum_n\frac{\alpha^n}{\sqrt{n!}}|n\rangle, show that U^(t)∣α⟩=e−iωt/2∣αe−iωt⟩\hat U(t)|\alpha\rangle = e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle, and use ⟨β∣x^∣β⟩=2ℏ/mω Re β\langle\beta|\hat x|\beta\rangle = \sqrt{2\hbar/m\omega}\,\mathrm{Re}\,\beta to compute ⟨x^⟩\langle\hat x\rangle at t=T/8t=T/8. ((b)) Independently, use the Heisenberg-picture ladder operator a^H(t)=a^ e−iωt\hat a_H(t) = \hat a\,e^{-i\omega t} (from the Heisenberg equation of motion) to compute ⟨α∣x^H(t)∣α⟩\langle\alpha|\hat x_H(t)|\alpha\rangle at the same time, and compare with part (a).

Related theory

Browse the problem library