Path Integrals

Quantum Mechanics · Part 9

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Table of Contents

  1. 1.The Third Formulation of Quantum Mechanics
  2. 2.The Quantum Propagator
  3. 3.Time-Slicing Derivation
  4. 4.The Path Integral Formula
  5. 5.The Functional Measure
  6. 6.The Classical Limit
  7. 7.Recovering the Schrodinger Equation
  8. 8.Path Integral View of the Double Slit
  9. 9.Boundary Conditions and the Action Principle

Key Equations

$$K(x_b, t_b;\, x_a, t_a) = \langle x_b | \, e^{-i\hat{H}(t_b - t_a)/\hbar} \, | x_a \rangle$$
$$K(x_b, t_b;\, x_a, t_a) = \int \prod_{j=1}^{N-1} dx_j \;\prod_{k=0}^{N-1} \langle x_{k+1} | \, e^{-i\hat{H}\varepsilon/\hbar} \, | x_k \rangle$$
$$\mathcal{D}x(t) = \lim_{N\to\infty} \left(\frac{m}{2\pi i\hbar\varepsilon}\right)^{N/2} \prod_{j=1}^{N-1} dx_j$$
$$\psi(x, t+\varepsilon) = \left(\frac{m}{2\pi i\hbar\varepsilon}\right)^{1/2}\!\int\! dy\;\exp\!\left[\frac{im(x-y)^2}{2\hbar\varepsilon} - \frac{i\varepsilon}{\hbar}V(y)\right]\psi(y, t)$$

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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