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| {{Quantum Mechanics A}} | | {{Quantum Mechanics A}} |
| Although our heuristic analysis yielded an exact free-particle propagator, we will now repeat the calculation without any approximation to illustrate the partial integration.
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| Consider <math>U(x_{N}, t_{N}; x_{0}, t_{0})</math>. The peculiar labeling of the end points will be justified later. Our problem is to perform the path integral
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| <math>\int_{_{X0}}^{_{XN}}e\tfrac{tS\left [ x(t) \right ]}{h}D\left [ x(t) \right ]</math>
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| <br/>Where<br/> <br/>
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| <math>\int_{_{X0}}^{_{XN}}D\left [ x(t) \right ]</math>
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| is a symbolic way of saying "integrate over all paths connecting <math>{x_{0}}</math> and <math>{x_{N}}</math> (in the interval <math>{t_{0}}</math> and <math>{t_{N}}</math>).
| | We will now evaluate the kernel <math>K(x_f,t_f;x_i,t_i)\!</math> for a free particle. In this case, the action is just |
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| ." Now, a path <math>{x_{t}}</math> is fully specified by an infinity of numbers <math>{x(t_{0})}</math>,..., <math>{x(t)}</math>, ...,<math>{x(t_{N})}</math>, namely, the values of the function <math>{x(t)}</math> at every point <math>{t}</math> is the interval <math>{t_{0}}</math> to <math>{t_{N}}</math>.To sum over all paths, we must integrate over all possible values of these infinite variables, except of course <math>{x(t_{0})}</math> and <math>{x(t_{N})}</math>, which will be kept fixed at <math>{x_{0}}</math> and <math>{x_{N}}</math>, respectively. To tackle this problem,we trade the function <math>{x_{t}}</math> for a discrete approximation which agrees with <math>{x_{t}}</math> at the <math>{N+1}</math> points.agrees with x{t) at the N + 1 points <math>t_{n}=t_{0}+n\varepsilon</math>, n = 0,.. . , N, where <math>\varepsilon =\frac{t_{n}-t_{0}}{N}</math>. In this approximation each path is specified by N+ 1 numbers <math>x(t_{0}),x(t_{1}),...,x(t_{N})</math>. The gaps in the discrete function are interpolated by straight lines. We hope that if we take the limit <math>N\to \infty</math> 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
| | <math>S=\int_{t_0}^{t_N}dt\,\frac{1}{2}m\dot{x}^2.</math> |
| <math>
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| S=\int_{t_{0}}^{t_N}\mathcal{L}(t)dt\int_{t_{0}}^{t_N}\frac{1}{2}m\dot{x}^2dt</math> | |
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| <math>S\int_{t_{0}}^{N-1}\frac{m}{2}[\frac{x_{i+1}-x_{i}}{\varepsilon }^2]\varepsilon</math> | | Note that we renamed <math>t_i\!</math> to <math>t_0\!</math> and <math>t_f\!</math> to <math>t_N;\!</math> the reason for this will become clear shortly. Let us now discretize the path that the particle takes, so that the intermediate positions are <math>x_1,\,x_2,\,\ldots,\,x_{N-1}.</math> We discretize the time axis similarly, with a spacing <math>\delta t\!</math> between two subsequent times. The action may then be written as |
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| where <math>x_{i}=x(t_{i})</math>. We wish to calculate
| | <math>S=\tfrac{1}{2}m\sum_{i=0}^{N-1}\frac{(x_{i+1}-x_{i})^2}{\delta t}.</math> |
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| <math>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}</math> | | The kernel now becomes |
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| | <math>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}</math> |
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| It is implicit in the above that <math>x(t_{0})</math> and <math>x(t_{N})</math> 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 <math>N\to \infty</math> is taken. | | It is implicit in the above that <math>x(t_{0})</math> and <math>x(t_{N})</math> 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 <math>N\to \infty</math> is taken. |
We will now evaluate the kernel 
for a free particle. In this case, the action is just
Note that we renamed 
to 
and 
to 
the reason for this will become clear shortly. Let us now discretize the path that the particle takes, so that the intermediate positions are 
We discretize the time axis similarly, with a spacing 
between two subsequent times. The action may then be written as
The kernel now becomes
It is implicit in the above that 
and 
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 
is taken.
Let us first switch to the variables
We then want
where
Although the multiple integral looks formidable, it is not. Let us begin by doing the
integration. Considering just the part of the integrand that involves , we get
Consider next the integration over yr. Bringing in the part of the integrand involving
and combining it with the result above we compute next
