A Modern Approach to Quantum Mechanics · Problem 8.3
Translational and Rotational Symmetry in the Two-Body Problem — John S. Townsend
Physics — Quantum Mechanics · harmonic-oscillator-path-integral-propagator
Problem
Determine, up to an overall multiplicative function of time, the transition amplitude, or propagator, for the harmonic oscillator using path integrals. See Feynman and Hibbs, Path Integrals and Quantum Mechanics, Sections 3.5 and 3.6.Intuitive explanation
The propagator tells you the quantum amplitude for a particle to travel from one position to another in a given time—it's the heart of the path integral formulation. Rather than summing over classical paths, you'll sum over all paths with a phase weight , where is the classical action. For the harmonic oscillator, the key insight is that the action is quadratic in position, which means the path integral becomes a Gaussian integral you can evaluate exactly by completing the square. This gives you a closed form for the propagator that encodes all the quantum dynamics.
Intuitive explanation
The propagator tells you the quantum amplitude for a particle to travel from one position to another in a given time—it's the heart of the path integral formulation. Rather than summing over classical paths, you'll sum over all paths with a phase weight , where is the classical action. For the harmonic oscillator, the key insight is that the action is quadratic in position, which means the path integral becomes a Gaussian integral you can evaluate exactly by completing the square. This gives you a closed form for the propagator that encodes all the quantum dynamics.