Propagator for the Harmonic Oscillator

The classical action $S$ can be evaluated as follows:

$S=\int _{0}^{t}(KE-PE)dt$

Where $KE\!$ is the kinetic engergy and $PE\!$ is the potential energy.

Equation of motion for harmonic oscillator: $x_{cl}(t')=A\cos(\omega t')+B\sin(\omega t')\!$
$A\!$ and $B\!$ are constants.

At $t'=0\!$ (starting point),$x_{cl}(0)=x_{0}\rightarrow A=x_{0}$.

At $t'=t\!$ (final point), $x_{cl}(t)=x\rightarrow B={\frac {x-x_{0}\cos(\omega t)}{\sin(\omega t)}}$ . </math>

is a symbolic way of saying "integrate over all paths connecting ${x_{0}}$ and ${x_{N}}$ (in the interval ${t_{0}}$ and ${t_{N}}$)." Now, a path ${x_{t}}$ is fully specified by an infinity of numbers ${x(t_{0})}$,..., ${x(t)}$, ...,${x(t_{N})}$, namely, the values of the function ${x(t)}$ at every point ${t}$ is the interval ${t_{0}}$ to ${t_{N}}$. To sum over all paths, we must integrate over all possible values of these infinite variables, except of course ${x(t_{0})}$ and ${x(t_{N})}$, which will be kept fixed at ${x_{0}}$ and ${x_{N}}$, respectively. To tackle this problem, we follow this idea that was used in section 1.10: we trade the function ${x_{t}}$ for a discrete approximation which agrees with ${x_{t}}$ at the ${N+1}$ points. Substitute:

$x_{cl}(t')=x_{0}\cos(\omega t')+{\frac {x-x_{0}\cos(\omega t)}{\sin(\omega t)}}\sin(\omega t')\Rightarrow {\frac {dx_{cl}(t')}{dt'}}=-\omega x_{0}\sin(\omega t')+\omega {\frac {x-x_{0}\cos(\omega t)}{\sin(\omega t)}}\cos(\omega t')$
$KE={\frac {1}{2}}m\left({\frac {dx_{cl}}{dt}}\right)^{2}={\frac {1}{2}}m\left[-\omega x_{0}\sin(\omega t')+\omega {\frac {x-x_{0}\cos(\omega t)}{\sin(\omega t)}}\cos(\omega t')\right]^{2}$
$PE={\frac {1}{2}}k(x_{cl}(t'))^{2}={\frac {1}{2}}k\left[x_{0}\cos(\omega t')+{\frac {x-x_{0}\cos(\omega t)}{\sin(\omega t)}}\sin(\omega t')\right]^{2}$
Substituting, integrating from time 0 to time $t\!$ and simplifying, we get:

$S=S(t,x,x_{0})={\frac {m\omega }{2\sin(\omega t)}}((x^{2}+x_{0}^{2})\cos(\omega t)-2xx_{0})$