Introduction to Quantum Mechanics · Problem 11.39

Quantum Dynamics — David J. Griffiths

Physics — Quantum Mechanics · numerical-driven-harmonic-oscillator

Problem

We can use the technique of Problem 11.38 to investigate time evolution when the Hamiltonian does depend on time, as long as we choose Δt\Delta t small enough. Evaluating H\mathsf{H} at the midpoint of each time step we simply replace Equation 11.154 with [1+12iΔtℏH ⁣(t+Δt2)]Ψ(t+Δt)=[1−12iΔtℏH ⁣(t+Δt2)]Ψ(t).(11.155)\left[1 + \frac{1}{2}i\frac{\Delta t}{\hbar}\mathsf{H}\!\left(t + \frac{\Delta t}{2}\right)\right]\boldsymbol{\Psi}(t+\Delta t) = \left[1 - \frac{1}{2}i\frac{\Delta t}{\hbar}\mathsf{H}\!\left(t + \frac{\Delta t}{2}\right)\right]\boldsymbol{\Psi}(t). \tag{11.155} Consider the driven harmonic oscillator of Problem 11.36 with f(t)=Asin⁡(Ωt),(11.156)f(t) = A\sin(\Omega t), \tag{11.156} where AA is a constant with the units of length and Ω\Omega is the driving frequency. In the following we will set m=ω=ℏ=A=1m = \omega = \hbar = A = 1 and look at the effect of varying Ω\Omega. Use the same parameters for the spatial discretization as in Problem 11.38, but set Nt=1000N_t = 1000. For a particle that starts off in the ground state at t=0t = 0, create a movie showing the numerical and exact solutions as well as the instantaneous ground state from t=0t = 0 to t=2π/Ωt = 2\pi/\Omega for (a) Ω=ω/5\Omega = \omega/5. In line with the adiabatic theorem, you should see that the numerical solution is close (up to a phase) to the instantaneous ground state. (b) Ω=5ω\Omega = 5\omega. In line with what you've learned about sudden perturbations, you should see that the numerical solution is barely affected by the driving force. (c) Ω=6ω/5\Omega = 6\omega/5.

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