Principles of Quantum Mechanics · Problem 21.9
Path Integrals—II — R. Shankar
Physics — Quantum Mechanics · path-integrals-statistical-mechanics
Problem
Use the imaginary-time path integral to compute the partition function of the harmonic oscillator and verify that Check the high-temperature () and low-temperature () limits.Intuitive explanation
The partition function counts all quantum states weighted by their Boltzmann factor, and the imaginary-time path integral reformulates this as a sum over all closed paths in Euclidean time. For the harmonic oscillator, the path integral evaluates to a Gaussian, whose determinant gives you the partition function directly—the key is recognizing that the functional determinant of the kinetic operator encodes the density of states. Once you have , the limits reveal physics: high temperature should recover the classical equipartition result (many states accessible), while low temperature should show only the ground state contributes.
Intuitive explanation
The partition function counts all quantum states weighted by their Boltzmann factor, and the imaginary-time path integral reformulates this as a sum over all closed paths in Euclidean time. For the harmonic oscillator, the path integral evaluates to a Gaussian, whose determinant gives you the partition function directly—the key is recognizing that the functional determinant of the kinetic operator encodes the density of states. Once you have , the limits reveal physics: high temperature should recover the classical equipartition result (many states accessible), while low temperature should show only the ground state contributes.