Test 2: Morse Oscillator

As a next test of our numerical procedure we calculate the energies and eigenfunctions of a Morse oscillator. This problem is also analytically solvable so that we can directly compare with known exact results.

The Morse potential is defined as:

$$ V(r)=D_{\text{e}}\cdot \left(1-e^{-a\cdot (r-r_{\text{e}})}\right)^{2} $$

The analytically known eigenenergies are given by:

$$ E_{n} = h \nu_{0} (n+1/2) - \frac{ \left[ h \nu_{0} (n+1/2) \right]^2} {4D_{e}} $$

Here, \( \nu_ {0} \) is a frequency that can be calculated from the potential parameters as:

$$ \nu_ {0} = \frac{a}{2\pi} \sqrt{2D_{e}/m} $$

The eigenfunctions have the following analytic form:

$$ \Psi_{n}(z)=N_{n}z^{\lambda -n-\frac{1}{2}} e^{-{\frac {1}{2}}z}L_{n}^{(2\lambda -2n-1)}(z)$$

where

$$z=2\lambda e^{-\left(x-x_{e}\right)}{\text{; }}N_{n}=\left[{\frac {n!\left(2\lambda -2n-1\right)}{\Gamma (2\lambda -n)}}\right]^{\frac {1}{2}} $$

and

$$\lambda ={\frac {\sqrt {2mD_{e}}}{a\hbar }}$$