Quantum-mechanical solution for the simple harmonic oscillator in a box

The quantum-mechanical problem of the simple harmonic oscillator confined into a box has been solved. We were able to solve the problem by casting the Schroedinger equation into the defining differential equation for parabolic cylinder functions. The probability amplitudes are parabolic cylinder functions. The energy levels are established by a numerical procedure as the roots of a polynomial and given in tabular form. A critical box length is found such that, when the box shrinks below it, the energy levels sharply increase over their unboxed values.