View source for Feynman Path Integral Evaluation of the Propagator

{{Quantum Mechanics A}}
The propagator for [[Harmonic oscillator spectrum and eigenstates|harmonic oscillator]] can be evaluated as follows: 

<math><x|\hat{U}(t,0)|x_0>=e^{\frac{i}{\hbar}S}\int_{y(0)=0}^{y(t)=0}D[y(t')]e^{\frac{i}{\hbar}\int_{0}^{t}(\frac{1}{2}my'^2-\frac{1}{2}ky^2)dt'}                                              </math>

where <math>y(t')</math> is the deviation of possible trajectories about the classical trajectory.