Degenerate
Quantum Mechanics · Part 6
219 KB8 sections4 key equationsLaTeX typeset
Table of Contents
- 1.Why Non-Degenerate Theory Fails
- 2.The Degenerate Subspace
- 3.Finding the "Good" Basis
- 4.The Secular Equation
- 5.Splitting of Degeneracy
- 6.Two-Fold Degeneracy in Detail
- 7.Symmetry and the Good Basis
- 8.Accidental vs Essential Degeneracy
Key Equations
$$|n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle}{E_n^{(0)} - E_m^{(0)}} |m^{(0)}\rangle$$
$$|\phi\rangle = \sum_{\alpha=1}^{g} c_\alpha |n^{(0)},\alpha\rangle \quad \Rightarrow \quad \hat{H}_0|\phi\rangle = E_n^{(0)}|\phi\rangle$$
$$\sum_{\beta=1}^{g} W_{\alpha\beta}\, c_\beta^{(i)} = E^{(1),i}\, c_\alpha^{(i)}, \quad i = 1, 2, \ldots, g$$
$$W = \begin{pmatrix} W_{11} & W_{12} \\ W_{21} & W_{22} \end{pmatrix} = \begin{pmatrix} W_{11} & W_{12} \\ W_{12}^* & W_{22} \end{pmatrix}$$
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 ...