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8: Central Potentials

Last updated

Apr 1, 2025
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- 7.E: Orbital Angular Momentum (Exercises)
- 8.1: Derivation of Radial Equation

Richard Fitzpatrick
University of Texas at Austin

( \newcommand{\kernel}{\mathrm{null}\,}\)

In this chapter, we shall investigate the interaction of a non-relativistic particle of mass $m$ and energy $E$ with various so-called central potentials, $V (r)$ , where $r = {(x^{2} + y^{2} + z^{2})}^{1 / 2}$ is the radial distance from the origin. It is, of course, most convenient to work in spherical coordinates— $r$ , $θ$ , $ϕ$ —during such an investigation. (See Section [s8.3].) Thus, we shall be searching for stationary wavefunctions, $ψ (r, θ, ϕ)$ , that satisfy the time-independent Schrödinger equation (see Section [sstat]) $\begin{matrix} (8.1) & H ψ = E ψ, \end{matrix}$ where the Hamiltonian takes the standard non-relativistic form $\begin{matrix} (8.2) & H = \frac{p^{2}}{2 m} + V (r) . \end{matrix}$

8.1: Derivation of Radial Equation
8.2: Infinite Spherical Potential Well
8.3: Hydrogen Atom
8.4: Rydberg Formula
8.E: Central Potentials (Exercises)

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