7. Harmonic Oscillator
The harmonic oscillator is among the most important examples of explicitly solvable problems, whether in classical or quantum mechanics. It appears in every textbook in order to demonstrate some general principles by explicit calculations. In some respect, the classical and quantum properties of the harmonic oscillator are similar. For example, the classical observation that the oscillation frequency does not depend on the amplitude corresponds to the fact that all quantum states are periodic in time with the same period.
Like the particle in a box, the harmonic oscillator is a system that has only bound states. There is a discrete set of allowed energies and an associated basis of energy eigenfunctions. All energy eigenstates can be generated from the ground state by repeated application of a "creation operator". This approach to the solution of the eigenvalue problem is related to the "supersymmetry" of the harmonic oscillator.
It is rewarding to calculate the time-dependence of position and momentum observables which leads to the conclusion that their expectation values always follow the laws of classical mechanics. Moreover, all eigenstates of the harmonic oscillator are also eigenstates of the Fourier transformation, hence the motion in momentum space looks exactly like the motion in position space. If the initial state is a translated eigenstate, then the position and momentum distributions oscillate back and forth without changing their shape.
Among the more mathematical results obtained in this chapter is a proof of the completeness of the eigenfunctions and the calculation of the integral kernel of the unitary time-evolution operator ("Mehler kernel"). As a result, the Schr\"oödinger equation for an arbitrary initial function can be solved with an integration. Of particular interest is the behavior of Gaussian wave packets, because their motion is very similar to the motion of a classical particle. The most "particle-like" states are the "coherent states" -- Gaussian functions which optimize the uncertainy relation for all times. The corresponding initial state is a translated ground state and can also be defined as an eigenvector of the "annihilation operator" A.