Quantized Morse Potential

In the classical position probability calculation, the energy value E=1 was arbitrary, and not consistent with quantized action.

Compute the first few quantum states of the Morse potential

where the momentum is in the range

.

Numerically invert the integral

(in units ħ²/2m=1) to obtain the first few quantum states (n=0, 1, 2, 3).