Spectroscopic Transitions for the Simple Harmonic Oscillator

This Mathcad worksheet determines whether an SHO spectroscopic transition is allowed assuming that

the Bohr frequency condition is satisfied. It requires only the quantum numbers of the initial and final states.

The v = 0 to v = 1 transition is allowed because the position distribution function, Y^{*}Y, exhibits oscillating dipole character.

Initial state:

Final state:

Set plot parameters:

Construct time-dependent superposition of the initial and final states:

Calculate and plot Y^{*}Y:

In this contour plot the horizontal axis is the spatial axis. Time is graphed on the veritcal axis. If Y^{*}Y asymmetric in time the transition is allowed.

The v = 0 to v = 2 transition is not allowed because the position distribution function, Y^{*}Y, does not exhibit oscillating dipole character.

Initial state:

Final state:

Set plot parameters:

Construct time-dependent superposition of the initial and final states:

Calculate and plot Y^{*}Y:

In this contour plot the horizontal axis is the spatial axis. Time is graphed on the veritcal axis. If Y^{*}Y asymmetric in time the transition is allowed.