Numerical Method

Y''(x) = -2µ ( E-V(x) ) Y

Schrödinger's wave equation indicates that the second derivative or curvature of wavefunction is proportional to the value of the wavefunction. The constant of proportionality is negative when E > V(x), causing the wavefunction to curve downward when the wavefunction is positive and to curve upward when the wavefunction is negative. The solution Y is forced to contiually turn, oscillating about zero like a wave. Hence our solution is called a wavefunction and this second order differential equation which forces our solution to bend continually is called a wave equation.

The extent of each turn will depend on the difference between the eigenvalue and the potential [E-V(x)]. We can use this simple observation along with Euler's method of integration to find a solution to Schrödinger's Equation:

Euler's Method

If we know Y and Y'= dY/dx at some point x we can approximate Y and Y' at a neighboring point using Euler's numerical integration

Y' (x+Dx) = Y'(x) + Y''(x) Dx
and
Y (x+Dx) = Y(x) + Y'(x) Dx

To start this method we need a guess for Y(0) and Y'(0). Then Euler's equations will give the value of the function at Dx. Letting x = x + Dx and repeatedly using Euler's method will give the wavefunction at each x over a finite interval L. The error at each step is on the order of Dx² , so fairly accurate results can be obtained if the step size Dx is kept small.

Shooting Method

The harmonic potential is used as an example at the top of this page. In this case the boundary condition that the wavefunction must satisfy is Y(L)=0. If as in the first graph the nnumerical solution is less than zero at the boundary, we can repeat the whole calculation with a smaller E to give the function less curvature until Y(L)=0. Seeking the appropriate value of E is not unlike shooting a gun at a long-distance target. When we hit the mark we have the right solution. Hence, this numerical procedure for solving a boundary value problem is called a shooting method.

Sliding the eigen value scroll bar allows you to shoot the wavefunction at its target boundary value. When you have a solution that is close, then press the automatic button, and the program will try to find the closest eigen value that makes the wavefunction zero at the end point. It looks for this solution by finding a value of the energy E₊ that makes the wavefunction positive and E_- that makes the wavefunction negative. The average of these two values is used for the next guess. Depending on the sign of the resultant wavefunction E₊ or E_- is replaced with the average energy and the process is repeated until a stable energy is found. Unfortunately the automatic method can be effected by numerical errors. If numerical error dominates the solution at the end point you may have to try the automatic integration several times or change the range of integration to exclude the region where the errors grow exponentially.

This numerical method is conceptually simple but robust enough to handle complicated potential functions. Some potentials require special handling. The hydrogen atom potential -1/r poses a problem for very small r. To make the solution more numerically stable the related function P(r) = r Y(r) is integrated.