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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

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In this chapter, we shall investigate the interaction of a non-relativistic particle of mass $m$ and energy $E$ with various so-called , $V(r)$ , 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— $r$ , $\theta$ , $\phi$ —during such an investigation. (See Section [s8.3].) Thus, we shall be searching for stationary wavefunctions, $\psi(r,\theta,\phi)$ , that satisfy the time-independent Schrödinger equation (see Section [sstat])

$\label{e9.1} H\,\psi = E\,\psi,$ where the Hamiltonian takes the standard non-relativistic form

$\label{e9.2} H = \frac{p^{\,2}}{2\,m} + V(r).$

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