The Classical Limit
It is intuitively clear that when quantum mechanics is applied to a macroscopic system it should reproduce the results of classical mechanics, very much the way that relativistic dynamics, when applied to slowly moving (v/c«.l) objects, reproduces Newtoni
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system it should reproduce the results of classical mechanics, very much the way that relativistic dynamics, when applied to slowly moving (v/c«.l) objects, reproduces Newtonian dynamics. In this chapter we examine how classical mechanics is regained from quantum mechanics in the appropriate domain. When we speak of regaining classical mechanics, we refer to the numerical aspects. Qualitatively we know that the deterministic world of classical mechanics does not exist. Once we have bitten the quantum apple, our loss of innocence is permanent. We commence by examining the time evolution of the expectation values. We find d dt
d
(Q)=dt('l'ir!i'l') = 'P,u.ru.A =
(
\
~)
JrA
1,."4
(6.9)
1
If we choose L1"" 10- 13 em. say, which is the size of a proton, AP ~I 0 14 gem/sec. For a particle of mass 1 g, this implies L1 V"" 10- 14 em/ sec, an uncertainty far below the experimentally detectable range. In the classical scale, such a state can be said to have well-defined values for X and P, namely, x 0 and p 0 , since the uncertainties (fluctuations) around these values are truly negligible. If we let such a state evolve with time, the mean values xo(t) and po{t) will follow Hamilton's equations, once again with negligible deviations. We establish this result as follows. Consider Eqs. (6.6) and (6.8) which govern the evolution of (X)= x 0 and (P) = p0 . These would reduce to Hamilton's equations if we could replace the mean values of the functions on the right-hand side by the functions of the mean values:
( 6.10) and
( 6.11)
If we consider some function of X and P, we will find in the same approximation ( 6.12) Thus we regain classical physics as a good approximation whenever it is a good approximation to replace the mean of the functions iJH/iJP, ··-aH/cX, and Q(X, P) by the functions of the mean. 'fhis in turn requires that the fluctuations about the mean have to be smalL (The result is exact £(there are no fluctuations.) Take as a concrete example Eqs. ( 6.10) and ( 6. 11). There is no approximation involved in the first equation since (aH/iJP) is just (P/m)=p 0 /m. In the second one, we need to approximate (iJH/iJX)=(dV/dX)=(V'(X)) by V'(X=x 0 ). To see when this is a good approximation, let us expand V' in a Taylor series around x 0 • Here it is convenient to work in the coordinate basis where V(X) = V(x). The series is
Let us now take the mean of both sides. The first term on the right-hand side, which alone we keep in our approximation, corresponds to the classical force at xo, and thus reproduces Newton's second law. The second vanishes in all cases, since the mean of x- x 0 does. The succeeding terms, which are corrections to the classical approximation, represent the fact that unlike the classical particle, which responds only to the force F=- V' at x 0 , the quantum particle responds to the force at neighboring points as well. (Note, incidentally, that these terms are zero if the potential is at the most quadratic in the variable x.) Each of these terms is a product of two factors, one of which measu
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