# Discrete Eigenvalues and Bound States - The Harmonic Oscillator and the WKB Approximation

### From FSUPhysicsWiki

This chapter is devoted to a further discussion of systems with bound states; here we discuss two important problems. We first discuss the solution of the harmonic oscillator in quantum mechanics. This problem is of great interest because not only is it exactly solvable, but also because it emerges, at least as an approximation, in the study of systems of physical interest, such as molecular vibrations and phonons in solids. We also discuss the problem of a charged particle in a magnetic field, since it shares a number of features in common with the harmonic oscillator and many of the same techniques are applicable to it. Finally, we develop the WKB approximation, which is a semiclassical approximation to the solution of the Schrödinger equation, and show how to obtain approximate bound-state energies. We find that this approximation reduces to the Bohr-Sommerfeld quantization condition from the old quantum theory, thus showing that said theory is in fact a semiclassical approximation to the modern quantum theory that we have been studying.