The solution to the Schrodinger equation with this type of potential may be found by power series expansion and given in terms of Hermite polynomials, but a more elegant method makes use of ladder operators.
The Fock states of a single mode describe this system, and the operators used to describe second quantization were originally motivated by this system. Some systems modelled after this one are more accurately described by a model named for Morse. Inverse square forces and forces that cause these systems are the only ones that can result in closed orbits according to Bertrand's theorem. The solution to the Schrodinger equation with this type of potential may be found by power series expansion and given in terms of Hermite polynomials, but a more elegant method makes use of ladder operators. A small angle approximation allows one to model a pendulum as one of these. It is defined as having restoring force proportional to displacement. For 10 points, name this system involving sinusoidal oscillations. 
The Fock states of a single mode describe this system, and the operators used to describe second quantization were originally motivated by this system.
quantum harmonic oscillator [accept simple harmonic oscillator; prompt on "oscillator"; prompt on "Hooke's Law" or "linear force"]