Introduction to Quantum Mechanics · Problem 11.36

Quantum Dynamics — David J. Griffiths

Physics — Quantum Mechanics · driven-harmonic-oscillator

Problem

The driven harmonic oscillator. Suppose the one-dimensional harmonic oscillator (mass mm, frequency ω\omega) is subjected to a driving force of the form F(t)=mω2f(t)F(t) = m\omega^2 f(t), where f(t)f(t) is some specified function. (I have factored out mω2m\omega^2 for notational convenience; f(t)f(t) has the dimensions of length.) The Hamiltonian is H(t)=−ℏ22m∂2∂x2+12mω2x2−mω2xf(t).(11.140)H(t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{1}{2}m\omega^2 x^2 - m\omega^2 x f(t). \tag{11.140} Assume that the force was first turned on at time t=0t = 0: f(t)=0f(t) = 0 for t≤0t \leq 0. This system can be solved exactly, both in classical mechanics and in quantum mechanics. (a) Determine the classical position of the oscillator, assuming it started from rest at the origin (xc(0)=x˙c(0)=0x_c(0) = \dot{x}_c(0) = 0). *Answer:* xc(t)=ω∫0tf(t′)sin⁡ ⁣[ω(t−t′)]dt′.(11.141)x_c(t) = \omega\int_0^t f(t')\sin\!\left[\omega(t-t')\right]dt'. \tag{11.141} (b) Show that the solution to the (time-dependent) Schrödinger equation for this oscillator, assuming it started out in the nnth state of the undriven oscillator (Ψ(x,0)=ψn(x)\Psi(x,0) = \psi_n(x) where ψn(x)\psi_n(x) is given by Equation 2.62), can be written as Ψ(x,t)=ψn(x−xc) eiℏ[−(n+12)ℏωt+mx˙c(x−xc2)+mω22∫0tf(t′)xc(t′)dt′].(11.142)\Psi(x,t) = \psi_n(x-x_c)\,e^{\frac{i}{\hbar}\left[-(n+\frac{1}{2})\hbar\omega t + m\dot{x}_c(x-\frac{x_c}{2}) + \frac{m\omega^2}{2}\int_0^t f(t')x_c(t')dt'\right]}. \tag{11.142} (c) Show that the eigenfunctions and eigenvalues of H(t)H(t) are ψn(x,t)=ψn(x−f);En(t)=(n+12)ℏω−12mω2f2.(11.143)\psi_n(x,t) = \psi_n(x-f); \quad E_n(t) = \left(n+\frac{1}{2}\right)\hbar\omega - \frac{1}{2}m\omega^2 f^2. \tag{11.143} (d) Show that in the adiabatic approximation the classical position (Equation 11.141) reduces to xc(t)≈f(t)x_c(t) \approx f(t). State the precise criterion for adiabaticity, in this context, as a constraint on the time derivative of ff. *Hint:* Write sin⁡[ω(t−t′)]\sin[\omega(t-t')] as (1/ω)(d/dt′)cos⁡[ω(t−t′)](1/\omega)(d/dt')\cos[\omega(t-t')] and use integration by parts. (e) Confirm the adiabatic theorem for this example, by using the results in (c) and (d) to show that Ψ(x,t)≈ψn(x,t) eiθn(t)eiγn(t).(11.144)\Psi(x,t) \approx \psi_n(x,t)\,e^{i\theta_n(t)}e^{i\gamma_n(t)}. \tag{11.144} Check that the dynamic phase has the correct form (Equation 11.92). Is the geometric phase what you would expect?

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