Spherical Harmonics

Quantum Mechanics · Part 4

234 KB10 sections4 key equationsLaTeX typeset
Download PDFRead Online (Free)

Table of Contents

  1. 1.Definition & Eigenvalue Equations
  2. 2.Explicit Form
  3. 3.First Few Spherical Harmonics
  4. 4.Key Properties
  5. 5.Real Forms (for Visualization)
  6. 6.Addition Theorem
  7. 7.Applications Across Physics
  8. 8.📝 Chapter Summary
  9. 9.1. Orthonormality
  10. 10.2. Completeness

Key Equations

$$\hat{L}^2 Y_\ell^m = \hbar^2\ell(\ell+1) Y_\ell^m$$
$$Y_2^0 = \sqrt{\frac{5}{16\pi}}(3\cos^2\theta - 1) \quad \text{(d}_{z^2}\text{)}$$
$$= \sqrt{\frac{3}{16\pi^2}}(2\pi)\left[\frac{\sin^2\theta}{2}\right]_0^\pi = 0 \quad \checkmark$$
$$Y_{\ell,m}^{\text{cos}} = \frac{1}{\sqrt{2}}\left(Y_\ell^m + (-1)^m Y_\ell^{-m}\right) \propto P_\ell^m(\cos\theta)\cos(m\phi)$$

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 ...

← Scattering 3d
Addition →

Get Instant Access to 461+ PDF Study Guides

Professional LaTeX-typeset PDFs with complete derivations, worked examples, and beautiful equation rendering. Download any PDF, anytime. Cancel anytime.

$5
per month
All courses included
Save 17%
$50
per year
Best value
Subscribe Now
Secure payment via StripeCancel anytimeInstant access