The Free-Particle Propagator

From PhyWiki
Revision as of 13:14, 19 June 2011 by StephanieReynolds (talk | contribs) (New page: Although our heuristic analysis yielded an exact free-particle propagator, we will now repeat the calculation without any approximation to illustrate the partial integration. Consider <mat...)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

Although our heuristic analysis yielded an exact free-particle propagator, we will now repeat the calculation without any approximation to illustrate the partial integration. Consider 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 U(x_{N}, t_{N}; x_{0}, t_{0})} . The peculiar labeling of the end points will be justified later. Our problem is to perform 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 \int_{_{X0}}^{_{XN}}e\tfrac{tS\left [ x(t) \right ]}{h}D\left [ x(t) \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 \int_{_{X0}}^{_{XN}}D\left [ x(t) \right ]}

is a symbolic way of saying "integrate over all paths connecting 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}}} 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_{N}}} (in the interval 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}}} 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_{N}}} ).

." Now, a path 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}}} is fully specified by an infinity of numbers 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_{0})}} ,..., 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)}} , ...,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_{N})}} , namely, the values of the function 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)}} at every 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 {t}} is the interval 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}}} 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_{N}}} .To sum over all paths, we must integrate over all possible values of these infinite variables, except of course 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_{0})}} 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_{N})}} , which will be kept fixed at 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}}} 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_{N}}} , respectively. To tackle this problem,we trade the function 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}}} for a discrete approximation which agrees with 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}}} at 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 {N+1}} points.agrees with x{t) at the N + 1 points 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_{n}=t_{0}+n\varepsilon} , n = 0,.. . , N, 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 \varepsilon =\frac{t_{n}-t_{0}}{N}} . In this approximation each path is specified by N+ 1 numbers 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_{0}),x(t_{1}),...,x(t_{N})} . The gaps in the discrete function are interpolated by straight lines. We hope that if we take 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 N\to \infty} at the end we will get a result that is insensitive to these approximations.t Now that the paths have been discretized, we must also do the same approximations.paths discretized, we must also do the same to the action integral. We replace the continuous path definition 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_{0}}^{t_N}\mathcal{L}(t)dt\int_{t_{0}}^{t_N}\frac{1}{2}m\dot{x}^2dt}

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_{0}}^{N-1}\frac{m}{2}[\frac{x_{i+1}-x_{i}}{\varepsilon }^2]\varepsilon}

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_{i}=x(t_{i})} . We wish to calculate

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 U(x_{N},t_{N};x_{0},t_{0})=\int_{x_{0}}^{x_N} exp\frac{i S[x(t)]}{\hbar}D[x(t)]=lim A\int_{-\infty }^{\infty } \int_{-\infty }^{\infty } ...\int_{-\infty }^{\infty } exp(\frac{i}{\hbar}\frac{m}{2}\sum_{i=0}^{N-1}\frac{(x_{i+1}-x_{{i}})^2}{\varepsilon })dx_{1}...dx_{N-1}}

It is implicit in the above 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 x(t_{0})} 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_{N})} have the values we have chosen at the outset. The factor A in the front is to be chosen at the end such that we get the correct scale for U when 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 N\to \infty} is taken. Let us first switch to the variables

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_{i}=(\frac{m}{2\hbar\varepsilon })^{1/2}x_{i}}

We then want

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 lim {A}'\int_{-\infty }^{\infty } \int_{-\infty }^{\infty } ...\int_{-\infty }^{\infty } exp(-\sum_{i=0}^{N-1}\frac{(y_{i+1}-y_{{i}})^2}{i } dy_{1}...dy_{N-1}}

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 {A}'=(\frac{2\hbar\varepsilon }{m })^{\frac{N-1}{2}}}

Although the multiple integral looks formidable, it is not. Let us begin by doing 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 y_{1}} integration. Considering just the part of the integrand that involves 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_{1}} , 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 \int_{-\infty }^{\infty }exp\left (\frac{-1}{i}(y_{2}-y_{1})^2+(y_{1}-y_{0})^2\right )dy_{1}=(\frac{i\pi }{2})^{1/2}e^\frac{-(y_{2}-y_{0})^2}{2i}}

Consider next the integration over yr. Bringing in the part of the integrand involving 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_{2}} and combining it with the result above we compute next


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 (\frac{i\pi}{2})^{1/2} \int_{-\infty }^{\infty }e^\frac{-(y_{3}-y_{2})^2}<math>{i}e^\frac{-(y_{2}-y_{0})^2}{2i}dy_{2}=}

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 (\frac{(i\pi )^{2}}{3})^{\frac{1}{2}}e^\frac{-(y_{3}-y_{0})^2}{3i}}

Retrieved from "http://wiki.physics.fsu.edu/index.php?title=The_Free-Particle_Propagator&oldid=12473"