Section 13.9: Exploring Solutions to the Coulomb Problem

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The entire solution to the Coulomb problem can be represented as

ψ_nlm = A_nl R_nl(r) Y_l^m(θ,φ) ,                     (13.56)

where A_nl are the normalization constants, R_nl(r) are the radial wave functions, 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 wave function, first imagine the rotation of ψ_nlm about the z axis, this gives you the shape of the wave function.  Then, to get the phase of the wave function, project  the phase (color) from Φ_m(φ).

1. For a given value of n and l, how does the number of angular lobes in P_l^m (θ) change with m?
2. 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?
3. How do the non-zero l values affect the radial wave function?
4. How do the non-zero m values affect the radial wave function?
5. For n = 3, how many times does the radial wave function 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.

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