Time in Quantum Mechanics
The treatment of time in quantum mechanics is still an important and challenging open question in the foundation of the quantum theory. This book describes the problems, and the attempts and achievements in defining, formalizing and measuring different ti
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4.1 Introduction It is ironic that experimentally time is the most accurately measured physical quantity, while in quantum mechanics one must struggle to provide a definition of so practical a concept as time of arrival. Historically, one of the first temporal quantities analyzed in quantum mechanics was lifetime, a property of an unstable state. The theory of this quantity is satisfactory in two ways. First, with only the smallest of white lies, one predicts exponential decay, and generally this is what one sees. Second, at the quantitative level, one finds good agreement with a simply derived formula, the Fermi–Dirac Golden rule, 2π ρ(E)|f |H|i|2 . (4.1) Γ = Equation (4.1) uses standard notation. Γ is the transition rate from an initial (unstable) state |i to a final state |f . The transition occurs by means of a Hamiltonian H. The density of (final) states is ρ, evaluated at the (common) energy of the states |i and |f . In terms of Γ , the lifetime is τL = 1/Γ . The lifetime τL is not a property of any one atom (or whatever), but rather of an ensemble of like atoms. For much of the twentieth century this was sufficient. One was taught not to inquire too closely about the time evolution of an individual member of an ensemble. An exception to this informed neglect arose as technology allowed experimentalists to focus on transitions in individual atoms [1]. Although one can recast these phenomena in ensemble terms, the ensemble is typically conditioned on the fact of the ultimate decay of the system studied. But a similar extension of naive ensemble interpretations was already present in studies of tunneling time. The barrier penetration phenomenon of quantum mechanics was sufficiently provocative in its denial of classical notions that one sought places where conventional ideas could be applied, e.g., trying to assign a time of passage through the barrier. This subject has a long history and a collection of recent views can be found in [2]. Again, in principle, for barrier penetration one deals with ensembles, but if one
L. S. Schulman: Jump Time and Passage Time: The Duration of a Quantum Transition, Lect. Notes Phys. 734, 107–128 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-73473-4 4
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measures passage time there would need to be conditioning on the fact of the transition, observations of individual transits, and a time interval measured for each. Our notation for tunneling time (without distinguishing among the many definitions) is τT . The tunneling-time concept allowed further probes of the Copenhagen view of quantum mechanics. A decaying particle, for example, a nucleus in the Gamow model of alpha decay, was said to undergo a quantum jump. The idea (I guess) was that you could measure the particle in its initial state or in its final state. But getting from one to the other was a “jump.” It took a measurement to distinguish one state from the other, putting the jump itself beyond the scope of quantum mechanics, or at least of ordinary unitary time evolution. How
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