Harmonic Oscillator: Integration Over Fluctuations: Difference between revisions
m (moved Harmonic fluctuations to Harmonic Fluctuations: Correct capitalization of title) |
|
(No difference)
|
Revision as of 10:49, 16 August 2013
Now, let's evaluate the path integral:
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=A(t)=\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'}}
Note that the integrand is taken over all possible trajectory starting at point 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_0} at time 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'=0} , ending at point 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} at time .
Expanding this integral,
where .
Expanding the path trajectory in Fourier series, we have
we may express in the form
where C is a constant independent of the frequency which comes from the Jacobian of the transformation. The important point is that it does not depend on the frequency . Thus, evaluating the integral of,
where C' is a constant directly related to C and still independent of the frequency of motion. Since the first product series in this final expression is also independent of the frequency of motion, we can absorb it into our constant C' to have a new constant, C. Simplifying further,
In the limit , we already know that
Thus,
and
Reference
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.