Attention is drawn to the fact that in a number of recent books the relative phases of the given spherical harmonics are not such as will lead to the commonly accepted matrices for quantities such as Ix. Some historical discussion is added.
Fock studied the hydrogen atom problem in momentum space by projecting the space into a 4-dimensional hyperspherical space. He found that as a consequence of the symmetry of the problem in this space the eigenfunctions are the R4 spherical harmonics and that the eigenvalues are determined only by the principal quantum number n. In this paper we note that if his method is applied to the 2-dimensional Kepler problem in momentum space, the eigenfunctions are R3 spherical harmonics, Ylm, and the eigenvalues are determined only by the quantum number l. These facts enable one to give a visualizable geometrical discussion of the dynamical degeneracy.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
The recently established existence of spherical harmonic functions, Yℓm(θ,ϕ) for half-odd-integer values of ℓ and m, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles θs,ϕs, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators, S2,Sz, are represented by the usual differential operators in spherical polar coordinates (commonly used for L2,Lz), and their electron-spin eigenfunctions are sinθsexp(±ϕs/2). This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions, α,β, that ``integration over spin coordinates'' is a true integration (over the angles θs,ϕs). In addition they facilitate construction of many electron wavefunctions in which the electron spins are neither parallel nor antiparallel, but inclined at an intermediate angle. In particular this may have application to the description of EPR correlation experiments.