# What numerical techniques are used to find eigenfunctions and eigenvalues of a differential operator?

Of course numerical techniques won't give an expression for eigenfunction, but I am looking for a vector of values of the eigenfunction in a given range. Also, do Scilab or Octave have any built-in functions for finding them?

For instance, I was trying to solve the Schrodinger equation for the hydrogen atom and I ended up with the following eigenvalue equation,

$$-\bigg(\frac12\frac{d^2}{dr^2}+\frac1{r}\bigg)\psi=E\psi$$

with the initial condition that

$\psi(0.0001)=0.0001$ and $\psi'(0.0001)=1$

I am starting a bit away from origin because, I have $r$ in the denominator. I just want to find the eigenvalues and plot it's eigenfunctions.

• The exact solutions of the radial Schroedinger equation are listing in many text on quantum mechanics. I can recommend "Quantum Physics" by Gasiorowicz and "Physics of Atoms and Molecules", by Bransden and Joachain. – Carl Christian Nov 6 '17 at 18:41

## 1 Answer

One technique is called the "wag-the-dog" method, discussed in Griffith's Introduction to Quantum Mechanics, works as follows: Pick a number for the eigenvalue, and solve the equation numerically with your initial conditions, say using Runge-Kutta. If the number you pick for the eigenvalue is in fact not an eigenvalue, then $\psi$ will blow up as $r\to \infty$ to either $+\infty$ or $-\infty$. Pick another value for the eigenvalue, and try again. Once you have two "eigenvalues", one where $\psi$ diverges to $+\infty$ and another which diverges to $-\infty$, then you can start bisecting, and recover the eigenvalue to high precision. Once you have the eigenvalue to very high precision, the wavefunction recovered numerically will be very good until very large $r$.

Note that this method is not especially sophisticated, and that any initial condition might only give a subset of all the eigenvalues. But it's not a bad first method of attack against this type of problem.