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the closer it is approaching the classical theory. The classical trajectory is recovered as 0. In general, some of the paths may indicate movement at greater than light speed, or in violation of energymomentum conservation law. Those paths would be related to the virtual particles; they need only to obey the uncertainty principle. Their contribution to the transition amplitude is usually very small. Figure 01c1 shows the different paths in 3D perspective dimensions, each one of which contributes a value to the action S (defined by Eq.(1a))  
Figure 01c1 Action Principle 
as indicated by an arrow labelled S[q(t)]. The quantum mechanical transition amplitude for going from the fixed end points q(t_{1}) to q(t_{2}) via all possible paths, is known as the Feynman Path Integral : 
The blue curve shows the variation of S as a function of "a" (representing the different paths), while the red curve (not to scale) is the accumulated sum of _{} starting from a = 50. The insert displays a much wider range for _{} from a = 400 to a = +400. These diagrams demonstrate that the main contribution to _{} comes mainly from paths near the classical one (around the stationary point of S). The action S assumes a pattern of rapid oscillation as its value getting larger, the amplitudes tend to cancel out each other leaving a steady value for _{} at the asymptotic limit.  
Figure 01c2 Path Integral, Real [view large image] 
Figure 01c3 Path Integral, Img [view large image] 
The sum can be evaluated mathematically by breaking up every path into an infinite number of intermediate points: q_{1}, q_{2}, ..., q_{n}, ...q_{N}, and then takes the sum of each point, e.g., q_{n} in all the paths from  to + (Figure 01c4). For the free field case where L = (m/2)(dq/dt)^{2}, the functional integrations can be performed analytically. After applying the Gaussian integral to all the q_{n} points, the final result is (see footnote for details) : _{} = [m/i h(t_{2}  t_{1})]^{1/2} exp{[i m(q_{2}  q_{1})^{2}] / [2 (t_{2}  t_{1})]}  (1i) If we vary both the end point q(t_{2}) and t_{2}, it can be shown that the transition amplitude _{} in Eq.(1i) satisfies the timedependent Schrodinger's equation in one dimension for a nonrelativistic free particle : 

Figure 01c4 Sum of Paths [view large image] 

 (1k) 
An alternate way to perform the path integral is to sum over all the paths at different time t and then put them all together as shown in Figure 01c5, where the time coordinate is divided into N parts of equal interval . Thus T = N, and dt = 0. The transition amplitude _{}, where X = q_{N}, can be written as : For a particle in free field L = (m/2)(dq/dt)^{2} it can be evaluated analytically by applying the Gaussian integral _{} :  
Figure 01c5 Path Integral 
The integration is carried out by repeatedly applying the Gaussian integral (N1) times : 
The real part of the transition amplitude _{} is plotted in Figure 01c6 and 01c7 with T and X holding constant respectively. Numerical computation is performed by a home computer with Basic programming. The electron mass is assumed in the calculation with (m/h)^{1/2} ~ 0.4 and (m/2) ~ 0.5. For the case of constant T, X is evaluated within the range 3 to +3 in step of 0.001 (in unit of cm); while for the case of constant X, T is evaluated from 0.5 to 5 in step of 0.0001 (in unit of sec). Fiigure 01c6 is similar to Figure 01c2; while Figure 01c7 indicates that the transition occurs preferably at T ~ 0. This is related to the relative probability P = _{}*_{} = (m/hT) for the particle to arrive at point X. Classically, the velocity v ~ X/T as T 0 signifies the transition to occur instantly.  
Figure 01c6 X vs Transition Amplitude [view large image] 
Figure 01c7 T vs Transition Amplitude [view large image] 
BTW, in applying the Gaussian integral to the integration, a Wick Rotation has been performed, e.g., T i T, the operation is reverted afterward. 