Partial Waves

Quantum Mechanics · Part 8

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Table of Contents

  1. 1.Why Partial Waves?
  2. 2.Plane Wave Expansion in Partial Waves
  3. 3.Effect of the Potential: Phase Shifts
  4. 4.Scattering Amplitude from Phase Shifts
  5. 5.Total Cross Section
  6. 6.The Radial Schrodinger Equation
  7. 7.Semiclassical Picture of Partial Waves
  8. 8.Truncation of the Partial Wave Sum
  9. 9.Interference Between Partial Waves

Key Equations

$$\boxed{e^{ikz} = e^{ikr\cos\theta} = \sum_{\ell=0}^{\infty}(2\ell+1)\,i^\ell\,j_\ell(kr)\,P_\ell(\cos\theta)}$$
$$u_\ell(r) \xrightarrow{r\to\infty} A_\ell\sin\!\left(kr - \frac{\ell\pi}{2} + \delta_\ell\right)$$
$$\boxed{f_\ell = \frac{e^{2i\delta_\ell} - 1}{2ik} = \frac{1}{k}\,e^{i\delta_\ell}\sin\delta_\ell}$$
$$\boxed{\sigma_{\text{tot}} = \frac{4\pi}{k^2}\sum_{\ell=0}^{\infty}(2\ell+1)\sin^2\delta_\ell}$$

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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