Central Potential

Quantum Mechanics · Part 4

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Table of Contents

  1. 1.Separation in Spherical Coordinates
  2. 2.Angular Equation
  3. 3.Radial Equation
  4. 4.Effective Potential
  5. 5.Reduced Radial Equation
  6. 6.Quantum Numbers
  7. 7.Behavior at Origin
  8. 8.Asymptotic Behavior
  9. 9.Conservation of Angular Momentum

Key Equations

$$\psi(r,\theta,\phi) = R(r)Y(\theta,\phi)$$
$$-\frac{\hbar^2}{2m}\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) + \left[V(r) + \frac{\hbar^2\ell(\ell+1)}{2mr^2}\right]R = ER$$
$$-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + V_{\text{eff}}(r)u = Eu$$
$$R(r) \sim r^\ell \quad \text{as } r \to 0$$

Equations are rendered with MathJax in the PDF with professional LaTeX typesetting.

Course Context

This PDF is part of the Quantum Mechanics course on CoursesHub.World. Master quantum mechanics from mathematical foundations to advanced topics. Covers Hilbert spaces, wave functions, angular momentum, perturbation theory, scattering, and path integrals with 450+ pages ...

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