Parabolic Potential Well
An example of an extremely important class of one-dimensional bound state in quantum mechanics is the simple harmonic oscillator whose potential can be written as
where K is the force constant of the oscillator.
The Hamiltonian operator is given by
The Schrödinger equation that gives the possible energies of the oscillator is
This equation can be simplified by choosing a new measure length and a new measure of energy, each of which is dimensionless. , where . With these substitutions, Equation E.3 becomes
In looking for bounded solutions, one can notice that as η approaches infinity and becomes too small compared to ζ2, the resulting differential equation can be easily solved to yield
This expression for the asymptotic dependence ...