CRC Press eBooks

Generalization to 3D

Steven M. Pascal

Published Jul 12, 2023• 0 citations

Abstract

With some understanding of the TISE in one dimension behind us, it is now time to graduate to the three-dimensional TISE. This will be required if we wish to understand electrons and atoms, which reside in three-dimensional space. The best coordinate system to use to study atoms is the spherical coordinate system (r, θ, ϕ), so we will only use this system. For a potential that is radial (that is, a central potential, such as an electron drawn to the center of the atom by the nucleus), the solutions ψ(r, θ, ϕ) can be split into separate functions of r, θ, and ϕ, similar to how Ψ(x,t) was separated into ψ(x) times f(t). Thus, the 3D TISE can be split into three separate TISE, one for each dimension r, θ, and ϕ. Since we are considering only radial potentials V(r) with no θ and ϕ dependence, the θ and ϕ TISE do not change. We provide the solutions for these two equations in this chapter, and needn't solve them again for other radial potentials. The θ and ϕ solutions, multiplied by each other, are called the spherical harmonics.

DOI: 10.1201/9781003356172-18

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