Tunneling through a truncated harmonic oscillator potential barrier

The sum of a one‐dimensional, truncated harmonic oscillator potential and square well of the same range, defined in the positive half‐space, serves as a convenient and instructive example for which the Schrodinger equation may have both bound‐state and continuum solutions. A superposition of these solutions is used in a study of barrier penetration by a wave packet representing a particle with an initial position in the region of the potential well. The presence or absence of a bound state in the superposition is shown to be the key factor determining the evolution of the wave packet. If no bound state exists, the probability of the particle having a position within the potential well is a monotonically decreasing function of time. If the superposition includes a bound state, however, this probability oscillates slightly because of an interference between the bound‐state and continuum components of the wavefunction.