Ladder Operators for the Harmonic Oscillator

The Hamiltonian for the linear harmonic oscillator can be written , in units with . The eigenstates are given by , , , where is a Hermite polynomial. An alternative reformulation of the problem can be based on the representation in terms of ladder operators and . The step-down or annihilation operator acts on the eigenfunctions according to , with . The step-up or creation operator satisfies

In this Demonstration, the eigenfunction is plotted in black. Also shown is either in red or in blue.