The Quantum Harmonic Oscillator with Time-Dependent Boundary Condition in the Causal Interpretation

Exact nontrivial analytic solutions of the trajectory equation in the Bohm approach can be found only for a very limited number of cases. Fortunately, for a single nonstationary state, an analytical solution exists for the trajectories in both the classical and quantum treatments. In this special case, the classical and the quantum trajectories are the same (up to a constant factor), because the quantum potential has a space-independent amplitude. The wavefunction obeys the Schrödinger equation with the potential term . The auxiliary function is defined by , from which the analytical wavefunction is given by , with , where is the mass and , and so on. The function is the time-dependent length of the moving walls, , , , with the initial width , the amplitude , and the frequency . In the quantum case, the analytical solution for the quantum trajectories is derived from the phase of the wavefunction in the eikonal form: . Therefore, the equation for the quantum trajectory with is given by .

The Euler–Lagrange equations imply the classical (or Newtonian) equation of motion of the particles: , whose solution is .

Reference

[1] A. J. Makowski and P. Peplowski, "On the Behaviour of Quantum Systems with Time-Dependent Boundary Conditions," Physics Letters A 163(3), 1992 pp. 143–151.