An elementary derivation of the oscillator propagator

Josef Fischer

L. Moriconi

The harmonic oscillator propagator is found straightforwardly from the free particle propagator within the imaginary-time Feynman path integral formalism. The derivation is simple, and requires only elementary mathematical manipulations and no clever use of Hermite polynomials, annihilation and creation operators, cumbersome determinant evaluations, or involved algebra.

Marco Moriconi

We provide an elementary derivation of the one-dimensional quantum harmonic oscillator propagator, using a mix of approaches, such as path integrals, canonical operators, and ladder operators. This way we take the best of each world, and find the propagator with as few tears as possible.

T. NITTA

Xi-Jun Ren

Jiushu Shao

Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.