Chapter 11
Path integrals and other formulations
A third equivalent formulation: sum over histories, with the classical limit falling out.
- Sections
- 4
- Finalised
- 4/4
- Simulations
- 0
- Estimated time
- 2 hours
- Read first
- Chapter 03
- 11.1Feynman's path-integral ideaTear every barrier out of the double slit: a particle going from A to B takes every possible path, each contributing a phase e^{iS/ħ}. The classical world is just what survives near the stationary-phase point.
- 11.2The propagatorActually compute the "sum over all paths": the full derivation of the free-particle propagator. Its phase turns out to be exactly the classical action, and evolving a Gaussian wave packet with it reproduces wave-packet spreading to the letter.
- 11.3From the path integral to the Schrödinger equationLook at propagation over just one short interval ε and expand to first order — the Schrödinger equation grows out of the path integral step by step. From here on, the two formulations are officially equivalent.
- 11.4Instantons and tunnellingRotate time onto the imaginary axis and the potential flips upside down — under the barrier a perfectly respectable classical path appears: the instanton. Double-well level splitting and the ammonia maser are both settled by its Euclidean action.