Feynman Path Integral Evaluation of the Propagator: Difference between revisions
Jump to navigation
Jump to search
(New page: The propagator for 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...) |
No edit summary |
||
| Line 1: | Line 1: | ||
The propagator for harmonic oscillator can be evaluated as follows: | 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> | <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. | where <math>y(t')</math> is the deviation of possible trajectories about the classical trajectory. | ||
Revision as of 12:36, 6 July 2011
The propagator for harmonic oscillator can be evaluated as follows:
![{\displaystyle <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'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00d098a20301891a5c311fef1138c165dc18583c)
where is the deviation of possible trajectories about the classical trajectory.
