The Classical Limit
The aim of this book is to provide you with an introduction to quantum mechanics, starting from its axioms. It is the aim of this chapter to equip you with the necessary mathematical machinery.
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8.1. The Path Integral Recipe We have already seen that the quantum problem is fully solved once the propagator is known. Thus far our practice has been to first find the eigenvalues and eigenfunctions of H, and then express the propagator U(t) in terms of these. In the path integral approach one computes U(t) directly. For a single particle in one dimension, the procedure is the following. To find U(x, t; x', t'): (1) Draw all paths in the x-t plane connecting (x', t') and (x, t) (see Fig. 8.1). (2) Find the action S[x(t)] for each path x(t). (3) U(x,t;x',t')=A
I
eiS[x(I)J/~
(8.1.1)
all paths
where A is an overall normalization factor.
t The nineteen forties
that is, and in his twenties. An interesting account of how he was influenced by Dirac's work in the same direction may be found in his Nobel lectures. See, Nobel Lectures-Physics, Vol. III, Elsevier Publication, New York (1972).
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224 CHAPTER 8
•
g
(x,t)
+
:.(x!t')
Figure 8.1. Some of the paths that contribute to the propagator. The contribution from the path x(t) is Z=exp{iS[x(t)]/:li}.
8.2. Analysis of the Recipe Let us analyze the above recipe, postponing for a while the proof that it reproduces conventional quantum mechanics. The most surprising thing about it is the fact that every path, including the classical path, xc1 (t), gets the same weight, that is to say, a number of unit modulus. How are we going to regain classical mechanics in the appropriate limit if the classical path does not seem favored in any way? To understand this we must perform the sum in Eq. (8.1.1). Now, the correct way to sum over all the paths, that is to say, path integration, is quite complicated and we will discuss it later. For the present let us take the heuristic approach. Let us first pretend that the continuum of paths linking the end points is actually a discrete set. A few paths in the set are shown in Fig. 8.1. We have to add the contributions Za = e;srx.(t)J/Ii fwm each path Xa(t). This summation is done schematically in Fig. 8.2. Since each path has a different action, it contributes with a different phase, and the contributions from the paths essentially cancel each other, until we come near the classical path. Since S is stationary here, the Z's add constructively and produce a large sum. As we move away from xc1 (t), destructive interference sets in once again. It is clear from the figure that U(t) is dominated by the paths near xc1 (t). Thus the classical path is important, not because it contributes a lot by itself, but because in its vicinity the paths contribute coherently. How far must we deviate from Xc1 before destructive interference sets in? One may say crudely that coherence is lost once the phase differs from the stationary value S[xc1 (t)]j1i = Sc~/1i by about 1r. This in turn means that the action for the -;oherence paths must be within lire of Sc~. For a macroscopic particle this means a very tight constraint on its path, since Sc1 is typically ~ 1 erg sec~ 1027 1i, while for m electron there is quite a bit of latitude. Consider th
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