The stretching vibration of a carbon monoxide (CO) molecule is modelled as a…

PHYSICS · Subjective · 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

The stretching vibration of a carbon monoxide (CO) molecule is modelled as a one-dimensional harmonic oscillator in the reduced-mass coordinate xx, the departure of the bond length from equilibrium, with reduced mass μ=1.14×10−26 kg\mu=1.14\times10^{-26}\,\mathrm{kg} and angular frequency ω=4.1×1014 rad s−1\omega=4.1\times10^{14}\,\mathrm{rad\,s^{-1}}. Treat the potential as exactly harmonic, V(x)=12μω2x2V(x)=\tfrac12\mu\omega^2x^2, and the motion of the relative coordinate is treated as one-dimensional and undamped. (a) Starting from the Heisenberg equation of motion, derive d⟨x^⟩dt=⟨p^⟩μ\frac{d\langle\hat x\rangle}{dt}=\frac{\langle\hat p\rangle}{\mu} and d⟨p^⟩dt=−⟨V′(x^)⟩\frac{d\langle\hat p\rangle}{dt}=-\langle V'(\hat x)\rangle, and show the second relation carries no correction term for this potential, for any state, giving ⟨x^⟩¨=−ω2⟨x^⟩\ddot{\langle\hat x\rangle}=-\omega^2\langle\hat x\rangle exactly. (b) At t=0t=0 the vibrational state is the coherent state ∣α0⟩|\alpha_0\rangle, eigenstate of a^\hat a with α0\alpha_0 real and positive, for which ⟨x^⟩0=x0=0.010 nm\langle\hat x\rangle_0=x_0=0.010\,\mathrm{nm} and ⟨p^⟩0=0\langle\hat p\rangle_0=0. Using ⟨α∣x^∣α⟩=2ℏ/μω Re α\langle\alpha|\hat x|\alpha\rangle=\sqrt{2\hbar/\mu\omega}\,\mathrm{Re}\,\alpha, find α0\alpha_0 numerically. (c) Solve the equation of part (a) with these initial conditions for ⟨x^⟩(t)\langle\hat x\rangle(t). (d) Starting instead from the number-state expansion of ∣α0⟩|\alpha_0\rangle and En=(n+12)ℏωE_n=(n+\tfrac12)\hbar\omega, show directly that U^(t)∣α0⟩=e−iωt/2∣α0e−iωt⟩\hat U(t)|\alpha_0\rangle=e^{-i\omega t/2}|\alpha_0e^{-i\omega t}\rangle, recompute ⟨x^⟩(t)\langle\hat x\rangle(t), confirm it matches (c), and evaluate ⟨x^⟩(t1)\langle\hat x\rangle(t_1) at t1=16×2πωt_1=\tfrac16\times\tfrac{2\pi}{\omega}. (e) The real bond follows the Morse potential of the chapter's four-systems worked example, with range 1/a≈43 pm1/a\approx43\ \mathrm{pm}, whose restoring force near equilibrium is −V′(x)=−μω2x(1−32ax+⋯)-V'(x)=-\mu\omega^2x\big(1-\tfrac32ax+\cdots\big). By what fraction does the true force at x0x_0 differ from the harmonic one, and can the answer to (d) be trusted to two significant figures?

Related theory

Browse the problem library