Partial Waves
Quantum Mechanics · Part 8
251 KB9 sections4 key equationsLaTeX typeset
Table of Contents
- 1.Why Partial Waves?
- 2.Plane Wave Expansion in Partial Waves
- 3.Effect of the Potential: Phase Shifts
- 4.Scattering Amplitude from Phase Shifts
- 5.Total Cross Section
- 6.The Radial Schrodinger Equation
- 7.Semiclassical Picture of Partial Waves
- 8.Truncation of the Partial Wave Sum
- 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 ...