Propagator for the Harmonic Oscillator
We will now evaluate the propagator for the harmonic oscillator. The Lagrangian for this system is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\tfrac{1}{2}m\dot{x}^2-\tfrac{1}{2}m\omega^2x^2.}
Before we begin, let us prove that the propagator will separate into two factors; one of these comes entirely from the classical motion of the system, and the other comes entirely from quantum fluctuations about said trajectory. To this end, let us write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=x_c+y,\!} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_c\!} is the classical trajectory and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\!} is the fluctuation, which will be a new integration variable for the path integral. If we take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_i\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_f\!} to be the initial and final times, respectively, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t_i)=y(t_f)=0.\!} Substituting this into the action, we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\int_{t_i}^{t_f} dt\,[\tfrac{1}{2}m(\dot{x}_c+\dot{y})^2-\tfrac{1}{2}m\omega^2(x_c+y)^2].}
We now expand out the squares, obtaining
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{x}_c^2-\tfrac{1}{2}m\omega^2x_c^2)+\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{y}^2-\tfrac{1}{2}m\omega^2y^2)+\int_{t_i}^{t_f} dt\,(m\dot{x}_c\dot{y}-m\omega^2x_cy).}
If we integrate by parts in the third term, we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{x}_c^2-\tfrac{1}{2}m\omega^2x_c^2)+\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{y}^2-\tfrac{1}{2}m\omega^2y^2)-\int_{t_i}^{t_f} dt\,m(\ddot{x}_c+\omega^2x_c)y.}
We know, however, that the classical motion obeys the equation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ddot{x}_c+\omega^2x_c=0.\!} As a result, the third term is zero, and the action separates into two contributions, one coming entirely from the classical motion, and the other coming entirely from quantum fluctuations. Denoting these two contributions as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_c(x_c)=\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{x}_c^2-\tfrac{1}{2}m\omega^2x_c^2)}
and
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_q(y)=\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{y}^2-\tfrac{1}{2}m\omega^2y^2),\!}
the propagator may now be written as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K(x_f,t_f;x_i,t_i)=e^{iS_c/\hbar}\int D[y(t)]\,e^{iS_q/\hbar}.}
We will now evaluate each of these contributions.
Contribution from Classical Path
We will begin by evaluating the "classical" contribution to the propagator. This is essentially just a problem of classical mechanics; we begin by solving for the classical motion of the particle. The equation of motion is, as stated earlier,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ddot{x}_c+\omega^2x_c=0.\!}
We impose the boundary conditions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t_i)=x_i\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t_f)=x_f.\!} The solution of the equation of motion that satisfies these boundary conditions is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_c(t)=x_i\frac{\sin{\omega(t_f-t)}}{\sin{\omega(t_f-t_i)}}+x_f\frac{\sin{\omega(t-t_i)}}{\sin{\omega(t_f-t_i)}},}
and the corresponding velocity is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_c(t)=-\omega x_i\frac{\cos{\omega(t_f-t)}}{\sin{\omega(t_f-t_i)}}+\omega x_f\frac{\cos{\omega(t-t_i)}}{\sin{\omega(t_f-t_i)}}.}
If we now substitute these expressions into the Lagrangian and simplify, we obtain
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\frac{m\omega^2}{2\sin^2{\omega(t_f-t_i)}}\{(x_i^2+x_f^2)\cos[2\omega(t-t_i)]-2x_ix_f\cos{\omega(t_i+t_f-2t)}\}.}
If we now substitute this into the action, we finally obtain
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_c=\frac{m\omega}{2\sin{\omega(t_f-t_i)}}[(x_i^2+x_f^2)\cos{\omega(t_f-t_i)}-2x_ix_f].}
Contribution From Fluctuations
We now turn our attention to the "quantum" contribution to the propagator. It is given by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_q=\int D[y(t')]\,\exp\left [\frac{i}{\hbar}\int_{0}^{\delta t}(\tfrac{1}{2}m\dot{y}^2-\tfrac{1}{2}m\omega^2y^2)\,dt'\right ].}
Note that we changed variables in the time integral to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t'=t-t_i;\!} this will simplify the subsequent calculations. To further simplify our notation, we introduce the quantity, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta t=t_f-t_i.\!}
Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t'=0)=y(t'=\delta t)=0,\!} we may expand Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t')\!} in a Fourier sine series:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t')=\sum_{n=1}^{\infty} a_n \sin\left(\frac{n\pi t'}{\delta t}\right)}
We now re-express the path integral in terms of the coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n\!} of this series. One may verify, with the aid of the fact that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^{\delta t}dt'\,\sin\left (\frac{m\pi t'}{\delta t}\right )\cos\left (\frac{n\pi t'}{\delta t}\right )=0}
for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\!} and
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^{\delta t}dt'\,\sin\left (\frac{m\pi t'}{\delta t}\right )\sin\left (\frac{n\pi t'}{\delta t}\right )=\int_0^{\delta t}dt'\,\cos\left (\frac{m\pi t'}{\delta t}\right )\cos\left (\frac{n\pi t'}{\delta t}\right )=0}
if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\neq n,\!} that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_q=C\int_{-\infty}^{\infty} da_1\,\int_{-\infty}^{\infty} da_2\,\ldots\,\exp{\left[\sum_{n=1}^{\infty}\frac{im}{2\hbar}\left(\left(\frac{n\pi}{\delta t}\right)^2-\omega^2\right)a^2_n\right]},}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\!} is a constant that is independent of the frequency that comes from the Jacobian of the transformation. The important point is that it does not depend on the frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega\!} . The integrals over the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n\!} are just Gaussians; evaluating them gives us
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_q=C'\prod_{n=1}^{\infty}\left[\left(\frac{n\pi}{\delta t}\right)^2-\omega^2\right]^{-\frac{1}{2}}= C'\prod_{n=1}^{\infty}\left(\frac{\delta t}{n\pi}\right) \prod_{n=1}^{\infty}\left[1-\left(\frac{\omega\,\delta t}{n\pi}\right)^2\right]^{-\frac{1}{2}}, }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C'\!} is a new constant directly related to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\!} that is also independent of the frequency of motion. Since the first product in this expression is also independent of the frequency of motion, we will absorb it into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C'\!} , thus defining yet another constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C''.\!} The remaining, frequency-dependent, product then evaluates to
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_q=C''\sqrt{\frac{\omega\,\delta t}{\sin(\omega\,\delta t)}}.}
In the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega\rightarrow 0} , we should recover the propagator for a free particle that propagates back to its initial position; using this fact, we find that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C''=\sqrt{\frac{m}{2\pi i \hbar\,\delta t}}.}
We have thus determined the full "quantum" contribution to the propagator,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_q=\sqrt{\frac{m}{2\pi i \hbar\,\delta t}}\sqrt{\frac{\omega\,\delta t}{\sin(\omega\,\delta t)}}=\sqrt{\frac{m\omega}{2\pi i \hbar \sin(\omega\,\delta t)}}, }
and therefore the full propagator,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K(x_f,t_f;x_i,t_i)=\sqrt{\frac{m\omega}{2\pi i \hbar \sin{\omega(t_f-t_i)}}}\exp\left \{\frac{im\omega}{2\hbar\sin{\omega(t_f-t_i)}}[(x_i^2+x_f^2)\cos{\omega(t_f-t_i)}-2x_ix_f]\right \}. }
Reference
Our evaluation of the "quantum" contribution to the propagator uses the method presented here: File:FeynmanHibbs H O Amplitude.pdf
For a more detailed evaluation of this problem, please see Barone, F. A.; Boschi-Filho, H.; Farina, C. 2002. "Three methods for calculating the Feynman propagator". American Association of Physics Teachers, 2003. Am. J. Phys. 71 (5), May 2003. pp 483-491.