General series solution for finite square-well energy levels for use in wave-packet studies

We develop a series solution for the bound-state energy levels of the quantum-mechanical one-dimensional finite square-well potential. We show that this general solution is useful for local approximations of the energy spectrum ~which target a particular energy range of the potential well for high accuracy!, for global approximations of the energy spectrum ~which provide analytic expressions of reasonable accuracy for the entire range of bound states!, and for numerical methods. This solution also provides an analytic description of dynamical phenomena; with it, we compute the time scales of classical motion, revivals, and super-revivals for wave-packet states excited in the well. © 2000 American Association of Physics Teachers.