Phase Shifts

Quantum Mechanics · Part 8

217 KB10 sections4 key equationsLaTeX typeset
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Table of Contents

  1. 1.Physical Meaning
  2. 2.Levinson's Theorem
  3. 3.Sign of Phase Shift
  4. 4.Energy Dependence
  5. 5.Example: Hard Sphere
  6. 6.Example: Square Well
  7. 7.Scattering Length and Effective Range
  8. 8.Time Delay
  9. 9.Phase Shift Analysis
  10. 10.Connection to Bound States

Key Equations

$$\delta_\ell(0) - \delta_\ell(\infty) = n_\ell\pi$$
$$\delta_\ell = -ka + \arctan\left[\frac{j_\ell(ka)}{n_\ell(ka)}\right]$$
$$\delta_0 \approx -ka, \quad \delta_1 \approx -\frac{(ka)^3}{3}, \quad \delta_2 \approx -\frac{(ka)^5}{15}$$
$$\tan\delta_0 = \frac{\tan(k'a) - ka}{1 + ka\tan(k'a)}$$

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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