The Schrodinger equation in 1-D is extended to 3-D in much the same way as the energy method in 1-D is extended to 2-D to deal with central-force motion. The kinetic-energy term associated with the angular variable(s) is written as a function of r and the angular momentum l, and is treated as a form of potential energy (the centrifugal potential energy). The resulting equation looks (somewhat in the case of the Schrodinger equation) like a 1-D problem with an "effective" potential energy. Although the program that finds energy levels and plots wavefunctions is itself very simple, its development involves a fair bit of detail (hint: skip to the program if you are not interested in detail).
The Bohr model of the hydrogen atom uses the classical techniques of central-force motion along with quantized angular momentum, and finds the energy levels successfully. It is a convenient starting point here because it provides the quantities we need to scale the Schrodinger equation into dimensionless form. The three equations needed for the model are Newton's law:
Energy conservation:
Quantization of the angular momentum:
where n has integral values 1, 2, 3.. The three equations can be solved for the radii and energies of the orbits:
The dimensionless group of constants represented by a is known as the fine-structure constant and is approximately equal to 1/137. We will use the first orbit radius (approximately 0.053 nm) and energy as the length and energy units in the dimensionless Schrodinger equation.
The Schrodinger equation looks fairly complicated when written in spherical polar coordinates. The solution is written as the product of a function R of r and a function Y (known as a spherical harmonic) of the angular variables. There are elegant ways of dealing with spherical harmonics in the quantum mechanical theory of angular momentum, and there are even more elegant ways in group theory where they are basis functions for the rotation group in 3-D. (There is nothing "quantum mechanical" about the spherical harmonics.) We will say nothing more about the angular variables here beyond noting that they contribute the angular momentum term to the radial Schrodinger equation:
where u is non-zero at the origin. When we make this substitution,
This is the equation we solve numerically using the Feynman algorithm. (The term in 1/r dictates the value of du/dr at the origin.) The program operates in two modes: (1) it finds the energy levels for which the wavefunction tends asymptotically to zero at large r, and (2) it plots the radial probability density (using Bohr atom energy values). The usual s, p, d, f notation is used for l = 0, 1, 2, 3. The horizontal axis is divided into four divisions, each equal to the radius of the nth Bohr orbit. Suitable vertical scale factors are input along with the n and l values for the selected state.