Quantum Recurrence Theorem

Abstract

A recurrence theorem is proved, which is the quantum analog of the recurrence theorem of Poincare\ Some statistical consequences of the theorem are stressed. I T is well known that in classical mechanics the following recurrence theorem holds, due to Poincare" (1890) 1 : "Any phase-space configuration (q,p) of a system enclosed in a finite volume will be repeated as accurately as one wishes after a finite (be it possibly very long) interval of time. ,, In this paper we shall show that a similar recurrence theorem holds in quantum theory; it can be formulated as follows: "Let us consider a system with discrete energy eigenvalues E n ; if \F(tf 0 ) is its state vector in the Schrodinger picture at the time to and e is any positive number, at least one time T will exist such that the norm \[^f(T)-^(to)\\ of the vector V(T)-y(t 0 ) is smaller than e." 2The proof of this theorem is simple and can be sketched in the following way: The equation of motion isi (d*(t)/dt) = H*(t);(1) the formal solution is 00 *(0 = £ r» exp(iv n -iEj)u(E»),(the r n 's being real positive numbers). From (2), |l^(r)-^(/o)|h2i:r n 2 (l-cosE n r); (r^T-t 0 ),and, if JV is suitably chosen,.n=JVConsequently, it is sufficient to prove that there is a value of r such thatBut this is actually the case according to a standard result of the theory of the almost-periodic functions. 3 1 For a modern formulation of this theorem see A.Furthermore it is easy to prove that this quantum recurrence theorem does not hold in general if the system has a continuous energy spectrum. The situation here is quite similar to the classical one: the quantum systems having a continuous energy spectrum correspond to classical systems not bounded to a finite volume. The analogy with the classical case is even deeper, since it is easy to prove (see Appendix) that also for the expectation values of the q's and p's a recurrence theorem holds, which in the classical limit goes over into the theorem of Poincare.The quantum recurrence theorem has statistical consequences rather similar to those of the Poincare's theorem in the classical case.Using Poincare's theorem, Zermelo (1896) was able to invalidate the unrestricted (nonstatistical) formulation of the Boltzmann ^-theorem and to conclude that the "Stosszahlansatz" is, strictly speaking, in contradiction with the dynamical laws, the effect of the "Stosszahlansatz" being that of averaging out the fluctuations. 4 The quantum analog to the "Stosszahlansatz" is the assumption about the number of transitions, 5 which is obtained by using the quantum-dynamical equations of motion and the conventional statistical postulate of equal a priori probabilities and random a priori phases.Analogously to the classical case, the quantum recurrence theorem shows that we cannot hope to obtain the assumption about the number of transitions without postulates of statistical nature.Our theorem shows furthermore that a similar conclusion is valid also for the probability transport equation.Finally we would like to emphasize that (contrary to a wid...

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