The Free-Particle Propagator
We will now evaluate the kernel 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)\!} for a free particle. In this case, the action is just
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}dt\,\frac{1}{2}m\dot{x}^2.}
Note that we renamed 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\!} 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_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_f\!} 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;\!} the reason for this will become clear shortly. Let us now discretize the path that the particle takes, so that the intermediate positions are 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_1,\,x_2,\,\ldots,\,x_{N-1}.} We discretize the time axis similarly, with a spacing 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\!} between two subsequent times. The action may then 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 S=\tfrac{1}{2}m\sum_{i=0}^{N-1}\frac{(x_{i+1}-x_{i})^2}{\delta t}.}
The kernel now becomes
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_{N},t_{N};x_{0},t_{0})=\lim_{N\to\infty} 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}}