Variational
Quantum Mechanics · Part 6
222 KB6 sections4 key equationsLaTeX typeset
Table of Contents
- 1.The Variational Theorem
- 2.Proof from Completeness
- 3.The Variational Method in Practice
- 4.Example: Hydrogen Ground State with Gaussian Trial
- 5.Example: Harmonic Oscillator with Gaussian
- 6.Choosing Good Trial Functions
Key Equations
$$\boxed{\frac{\langle\psi|\hat{H}|\psi\rangle}{\langle\psi|\psi\rangle} \geq E_0}$$
$$|\psi(\alpha_1, \alpha_2, \ldots, \alpha_k)\rangle$$
$$\psi_{\text{trial}}(r) = A e^{-\alpha r^2}$$
$$\frac{dE}{d\alpha} = 0 \quad \Rightarrow \quad \alpha^* = \frac{8me^4}{9\pi^2\hbar^4(4\pi\epsilon_0)^2}$$
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 ...