The harmonic oscillator is an extremely important physics problem. Many potentials look like a harmonic oscillator near their minimum. This is the first non-constant potential for which we will solve the Schrödinger Equation.
The harmonic oscillator Hamiltonian is given by
It is standard to remove the spring constant
from the Hamiltonian,
replacing it with the classical oscillator frequency.
To solve the Harmonic Oscillator equation, we will first change to dimensionless variables, then find the form of the solution for , then multiply that solution by a polynomial, derive a recursion relation between the coefficients of the polynomial, show that the polynomial series must terminate if the solutions are to be normalizable, derive the energy eigenvalues, then finally derive the functions that are solutions.
The energy eigenvalues are
The ground state wave function is.
The first excited state is an odd parity state, with a first order polynomial multiplying the same Gaussian.
The second excited state is even parity, with a second order polynomial multiplying the same Gaussian.
Note that is equal to the number of zeros of the wavefunction. This is a common trend. With more zeros, a wavefunction has more curvature and hence more kinetic energy.
The general solution can be written as
Jim Branson 2013-04-22