A silica nanosphere of mass is levitated in an optical trap that acts 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

A silica nanosphere of mass m=1.0×10−17 kgm = 1.0\times10^{-17}\,\mathrm{kg} is levitated in an optical trap that acts as a one-dimensional harmonic potential V(x)=12mω2x2V(x) = \tfrac12 m\omega^2 x^2 with ω=2π×100 kHz\omega = 2\pi\times 100\,\mathrm{kHz}. The motion is one-dimensional and undamped, and the trap is harmonic to the precision needed here. (a) Starting from the Heisenberg equation of motion dA^Hdt=iℏ[H^,A^H]\frac{d\hat A_H}{dt}=\frac{i}{\hbar}[\hat H,\hat A_H], derive d⟨x^⟩dt=⟨p^⟩m\frac{d\langle\hat x\rangle}{dt}=\frac{\langle\hat p\rangle}{m} and d⟨p^⟩dt=−⟨V′(x^)⟩\frac{d\langle\hat p\rangle}{dt}=-\langle V'(\hat x)\rangle, and show that for this potential the second relation is exact for every state, with no correction term, so ⟨x^⟩\langle\hat x\rangle obeys ⟨x^⟩¨=−ω2⟨x^⟩\ddot{\langle\hat x\rangle}=-\omega^2\langle\hat x\rangle exactly. (b) At t=0t=0 the sphere is prepared in the coherent state ∣α0⟩|\alpha_0\rangle, the eigenstate of the lowering operator a^\hat a with eigenvalue α0\alpha_0 real and positive, for which ⟨x^⟩0=x0=5.0 nm\langle\hat x\rangle_0 = x_0 = 5.0\,\mathrm{nm} and ⟨p^⟩0=0\langle\hat p\rangle_0=0. Using ⟨α∣x^∣α⟩=2ℏ/mω Re α\langle\alpha|\hat x|\alpha\rangle=\sqrt{2\hbar/m\omega}\,\mathrm{Re}\,\alpha, find the numerical value of α0\alpha_0. (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 ∣α0⟩=e−α02/2∑nα0nn!∣n⟩|\alpha_0\rangle=e^{-\alpha_0^2/2}\sum_n\frac{\alpha_0^n}{\sqrt{n!}}|n\rangle and the energy eigenvalues 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, and use this to recompute ⟨x^⟩(t)\langle\hat x\rangle(t). Confirm it agrees with part (c), then evaluate ⟨x^⟩(t1)\langle\hat x\rangle(t_1) at t1=16×2πωt_1 = \tfrac16\times\tfrac{2\pi}{\omega}.

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