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Physical Review Letters

How fast can a quantum state change with time?

Peter Pfeifer

Published May 31, 1993• 200 citations
Publisher

Abstract

Lower and upper bounds are derived for the decay and transitions of quantum states, evolving under a time-dependent Hamiltonian, in terms of the energy uncertainty of the initial and final state. The bounds are simultaneously a rigorous version of Fermi's golden rule and of the time-energy uncertainty relation. They are sharp, refer to short times, and are compared with recent long-time results for time-independent Hamiltonians. Illustrations for tunneling systems, laser-driven processes, and neutron interferometry in time-dependent magnetic fields are given.

DOI
10.1103/physrevlett.70.3365

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Lower and upper bounds are derived for the decay and transitions of quantum states, evolving under a time-dependent Hamiltonian, in terms of the energy uncertainty of the initial and final state. The bounds are simultaneously a rigorous version of Fermi's golden rule and of the time-energy uncertainty relation. They are sharp, refer to short times, and are compared with recent long-time results for time-independent Hamiltonians. Illustrations for tunneling systems, laser-driven processes, and neutron interferometry in time-dependent magnetic fields are given.
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We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at initial time t=0t=0t=0. This expression is a function of the time ttt at which a measurement is performed to determine if the state has decayed and, if yes, in which one of the infinitely many possible final states. For large times the energy spread is, as expected, given by the decay width $\ensuremath{\Gamma}$ of the initial unstable state. However, if the measurement of the final state is performed at a time ttt comparable to (or smaller than) the mean lifetime of the state $1/\ensuremath{\Gamma}$, then the uncertainty in the energy of the final state is much larger than the decay width $\ensuremath{\Gamma}$. Namely, for short times an uncertainty of type 1/t1/t1/t dominates, while at long times the usual spread $\ensuremath{\Gamma}$ is recovered. Then we turn to a generic two-body decay process and describe the energy uncertainty of each of the two outgoing particles. We apply these formulas to the two-body decays of neutral and charged pions and to the spontaneous emission process of an excited atom. As the last step, we study a case in which the nonexponential decay is realized and show that for short times eventual asymmetric terms are enhanced in the spectrum.
Often one needs to calculate the evolution time of a state under a Hamiltonian with no explicit time dependence when only numerical methods are available. In cases such as this, the usual application of Fermi's golden rule and first-order perturbation theory is inadequate as well as being computationally inconvenient. Instead, what one needs are conditions under which the evolution time may be obtained from the easily calculated energy uncertainty. This work derives some general conditions for obtaining the evolution time from the energy uncertainty.
A quantum mechanical wave of a finite size moves like a classical particle and shows a unique decay probability. Because the wave function evolves according to the Schrödinger equation, it preserves the total energy but not the kinetic energy in the intermediate-time region of a decay process where those of the parent and daughters overlap. The decay rate computed with Fermi's golden rule requires corrections that vary with the distance between the initial and final states, and the energy distribution of the daughter is distorted from that of plane waves. The corrections have universal properties in relativistically invariant systems and reveal macroscopic quantum phenomena for light particles. The implications for precision experiments in beta decays and various radiative transitions are presented.
A quantum mechanical wave of a finite size moves like a classical particle and shows a unique decay probability. Because the wave function evolves according to the Schrodinger equation, it preserves the total energy but not the kinetic energy in the intermediate-time region of a decay process where those of the parent and daughters overlap. The decay rate computed with Fermi's golden rule requires corrections that vary with the distance between the initial and final states, and the energy distribution of the daughter is distorted from that of plane waves. The corrections have universal properties in relativistically invariant systems and reveal macroscopic quantum phenomena for light particles. The implications for precision experiments in beta decays and various radiative transitions are presented.