- Then, Schrödinger equation becomes
- with the change of variable,
, this equation becomes
where
is the angular frequency of the oscillator.
- This differential equation has an exact solution in terms of a quantum number
:
where
is a normalization constant.
- The function
is the physicists' Hermite polynomials of order
, defined by:
- The corresponding energy levels are
- Recursion formula:
with the first two:
and
.
Example py-file: The program to find the harmonic oscillator wavefunctions/probability densities
for up to 4 vibrational energy levels with the harmonic potential,
.
QHO.py
Figure 5.20:
Wavefunction representations for the first 5 bound eigenstates,
.
|
Figure 5.21:
Corresponding probability densities.
|