Morphing Quantum Spheres into Atoms: The Spherical Harmonics and Associated Legendre Functions

As a step along the path to creating a quantum mechanical model of an atom, we considered the solution to the problem of a single particle moving on the surface of a sphere. We saw that the Hamiltonian for the motion could be written simply in terms of the angular momentum operator, L². The eigenfunctions of this operator are well known and called the spherical harmonics or Y_l,m. We noted that the solutions depended on two quantum numbers, l and m_l.

We wrote down the Hamiltonian for a one-electron atom (the archetype would be the hydrogen atom) and discussed the form of the potential energy (Coloumbic or electrostatic attraction). We noted that we could simplify matters by assuming that nuclear motion was very slow compared to the motion of the electrons and therefore could be (to a first approximation) ignored.

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