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Section 13.9: Exploring Solutions to the Coulomb Problem

n = | l = | m =

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The entire solution to the Coulomb problem can be represented as

ψnlm = Anl Rnl(r) Ylm(θ,φ) ,

where Anl are the normalization constants, Rnl(r) are the radial energy eigenfunctions, and Ylm(θ,φ) are the spherical harmonics.

In this Exploration, Animation 1 depicts ψnlm in the zx plane only. In Animation 2, Φm(φ), Plm (θ), R(r), and ψnlm are visualized in one of four panels. The entire solution, ψnlm, is visualized in the zx plane only. To generate the spherical energy eigenfunction, first imagine the rotation of ψnlm 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 Φm(φ).

  1. For a given value of n and l, how does the number of angular lobes in Plm (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 Φm(φ) change with m?
  3. How do the non-zero l values affect the radial energy eigenfunction?
  4. How do the non-zero m values affect the radial energy eigenfunction?
  5. 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.

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