# Particle in a box under harmonic driving

Is the particle in a box under harmonic driving electric field solvable analytically?

Here is the Schrodinger equation:

$$i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t)$$

where the potential $V(x)$ is

$$V(x)= \begin{array}{cc} \Big\{ & \begin{array}{cc} 0 & 0\leq x\leq L \\ \infty & \text{otherwise} \\ \end{array} \\ \end{array}$$

The driving force $F(t)$ is

$$F(t)=F_0*\cos(\omega_0*t)$$

and $F_0$ and $\omega_0$ are constants.

• If the driving force amplitude $F0$ is small compared to the eigenenergies of your system, then time-dependent perturbation theory would give you the solutions that you are looking for. Essentially, you solve the unperturbed problem of particle in a box and then apply corrections to it due to the drive. See here for details: tcm.phy.cam.ac.uk/~bds10/aqp/handout_dep.pdf You may also want to learn about the rotating wave approximation and Fermi's golden rule: en.wikipedia.org/wiki/Rotating_wave_approximation en.wikipedia.org/wiki/Fermi%27s_golden_rule – eqb Jun 9 '14 at 2:36
• Thanks, but I'm interested in exactly the non-perturbative limit, where the Rabi oscillation can happen. – xslittlegrass Jun 9 '14 at 3:54
• In that case, first write the solutions of the particle in a box, then pick the two states that you are interested in (usually the ground and first excited states), and then expand the solutions to the driven Hamiltonian in terms of the eigenstates of the undriven one that you chose. That's exactly how people solve for Rabi oscillations. – eqb Jun 9 '14 at 4:12