We discuss the general mathematical condition in one-dimensional time-independent quantum mechanics for an energy eigenfunction to have zero curvature over an extended region of space and still be valid. This condition and its solution are not often discussed as part of quantum mechanics texts at any level, yet have interesting consequences for experimental, pedagogical and theoretical investigations of systems with piecewise-constant potential energy functions. We present the solutions for several standard cases in which zero-curvature energy eigenfunctions are allowed as bound states, scattering states and threshold states. These states are of interest to supersymmetric quantum mechanics and the procedures we outline can be used to 'design' quantum wells for studies of wave packet dynamics.
All Science Journal Classification (ASJC) codes
- Physics and Astronomy(all)