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.
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.
Abstract An elementary treatment of the quantum harmonic oscillator is proposed. No previous knowledge of linear differential equation theory or Fourier analysis are required, but rather only a few basics of elementary calculus. The pivotal role in our analysis is played by the sole particle localization constraint, which implies square integrability of stationary-state wavefunctions. The oscillator ground-state characterization is then achieved in a way that could be grasped, in principle, even by first-year undergraduates. A very elementary approach to build up and to characterize all higher-level energy eigenstates completes our analysis.