Quantum Gaussian wells and barriers

We apply the variational method to the ground-state and first-excited-state energies of a Gaussian well and show that the expansion of the approximate energy when the potential strength approaches zero does not agree with the exact result. We argue that a recent semiclassical approach to tunneling through a Gaussian barrier does not agree with an accurate calculation of the transmission probability. We point out the advantages of converting the Schrödinger equation into a dimensionless eigenvalue equation.