Details

Excited states of the harmonic oscillator can be computed by applying the raising operator to the ground state wavefunction . Applying the raising operator times gives an unnormalized . The wavefunction can be normalized by dividing by . One can also define a lowering operator . The position and momentum operator can be expressed in terms of and as and . The variable is dimensionless and is related to the physical distance by where is the mass of the oscillator and is the angular frequency.