Quantum Mechanics: Concepts and Applications · Problem 3.13

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · ehrenfest-theorem-charged-oscillator

Problem

Consider a charged oscillator, of positive charge qq and mass mm, which is subject to an oscillating electric field E0cos⁡ωtE_0 \cos \omega t; the particle's Hamiltonian is H^=P2/(2m)+kX^2/2+qE0X^cos⁡ωt\hat{H} = P^2/(2m) + k\hat{X}^2/2 + qE_0\hat{X}\cos\omega t. (a) Calculate d⟨X^⟩/dtd\langle \hat{X} \rangle/dt, d⟨P^⟩/dtd\langle \hat{P} \rangle/dt, d⟨H^⟩/dtd\langle \hat{H} \rangle/dt. (b) Solve the equation for d⟨X^⟩/dtd\langle \hat{X} \rangle/dt and obtain ⟨X^⟩(t)\langle \hat{X} \rangle(t) such that ⟨X^⟩(0)=x0\langle \hat{X} \rangle(0) = x_0.
Intuitive explanation

The key here is using the Heisenberg equations of motion to connect time derivatives of expectation values to commutators with the Hamiltonian. For part (a), you'll apply d⟨A^⟩dt=iℏ⟨[H^,A^]⟩+⟨∂A^∂t⟩\frac{d\langle \hat{A} \rangle}{dt} = \frac{i}{\hbar}\langle[\hat{H}, \hat{A}]\rangle + \langle \frac{\partial \hat{A}}{\partial t}\rangle to each observable; the commutators [H^,X^][\hat{H}, \hat{X}] and [H^,P^][\hat{H}, \hat{P}] will give you classical-looking equations. For part (b), recognize that the equation for d⟨X^⟩/dtd\langle \hat{X} \rangle/dt is a driven harmonic oscillator equation — the time-dependent force term qE0cos⁡ωtqE_0\cos\omega t acts like an external drive, so you can solve it using standard ODE methods (homogeneous plus particular solution).

Intuitive explanation

The key here is using the Heisenberg equations of motion to connect time derivatives of expectation values to commutators with the Hamiltonian. For part (a), you'll apply d⟨A^⟩dt=iℏ⟨[H^,A^]⟩+⟨∂A^∂t⟩\frac{d\langle \hat{A} \rangle}{dt} = \frac{i}{\hbar}\langle[\hat{H}, \hat{A}]\rangle + \langle \frac{\partial \hat{A}}{\partial t}\rangle to each observable; the commutators [H^,X^][\hat{H}, \hat{X}] and [H^,P^][\hat{H}, \hat{P}] will give you classical-looking equations. For part (b), recognize that the equation for d⟨X^⟩/dtd\langle \hat{X} \rangle/dt is a driven harmonic oscillator equation — the time-dependent force term qE0cos⁡ωtqE_0\cos\omega t acts like an external drive, so you can solve it using standard ODE methods (homogeneous plus particular solution).

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