Chapter 3: Path Integral#

Units#

Path integral#: \(K(x_f,t_f;x_i,t_i) = \int \mathcal{D}[x]\, e^{iS[x]/\hbar}\). Sum over all paths weighted by the action phase.
Propagator#: Transition amplitude kernel satisfying the composition property and the Schrödinger equation.
Free-particle propagator#: \(K_{\text{free}} = \left(\frac{m}{2\pi i\hbar T}\right)^{d/2} \exp\!\left(\frac{im(\Delta x)^2}{2\hbar T}\right)\).
Stationary phase#: Dominant contribution from classical paths where \(\delta S = 0\). Quantum corrections are fluctuations around these paths.
WKB approximation#: Semiclassical ansatz \(\psi = A(x)e^{iS(x)/\hbar}\); valid when the potential varies slowly on the scale of the de Broglie wavelength.
Bohr-Sommerfeld quantization#: \(\oint p\,dx = 2\pi\hbar(n+\tfrac{1}{2})\).
Wick rotation#: \(t \to -i\tau\): connects quantum mechanics (\(e^{iS/\hbar}\)) to statistical mechanics (\(e^{-S_E/\hbar}\)).
Partition function#: \(Z(\beta) = \mathrm{Tr}(e^{-\beta H}) = \int \mathcal{D}[x]\, e^{-S_E/\hbar}\) over closed paths with period \(\beta = 1/k_BT\).