## Section 13.9: Exploring Solutions to the Coulomb Problem

Animation 1: Energy Eigenfunction Amplitude | Animation 2: Energy Eigenfunction Amplitude and Pieces

Please wait for the animation to completely load.

The entire solution to the Coulomb problem can be represented as

ψ_{nlm} = *A*_{nl }*R*_{nl}(*r*) *Y*_{l}^{m}(θ,φ) ,

where *A*_{nl} are the normalization constants, *R*_{nl}(*r*) are the radial energy eigenfunctions, and *Y*_{l}^{m}(θ,φ) are the spherical harmonics.

In this Exploration, Animation 1 depicts ψ_{nlm} in the *zx* plane only. In Animation 2, Φ_{m}(φ), *P*_{l}^{m} (θ), *R*(*r*), and ψ_{nlm} are visualized in one of four panels. The entire solution, ψ_{nlm}, is visualized in the zx plane only. To generate the spherical energy eigenfunction, first imagine the rotation of ψ_{nlm} about the z axis; this gives you the shape of the energy eigenfunction. Then, to get the phase of the energy eigenfunction, project the phase (color) from Φ_{m}(φ).

- For a given value of
*n*and*l*, how does the number of angular lobes in*P*_{l}^{m}(θ) change with*m*? - For a given value of
*n*and*l*, how does the number of wavelengths (from blue to blue is one wavelength) in Φ_{m}(φ) change with*m*? - How do the non-zero
*l*values affect the radial energy eigenfunction? - How do the non-zero
*m*values affect the radial energy eigenfunction? - For
*n*= 3, how many times does the radial energy eigenfunction cross zero (change signs) for each possible value of*l*? Try this for a few other values for the principal quantum number,*n*, and see if your conclusion holds.

*Note: Right click on any applet to make a copy of t**he image. The mouse coordinates may be observed by left-clicking within the graph.*