On the basis of the principles proposed by the first author in 1970 and 1979 the time-energy uncertainty relationship in quantum mechanics is constructed explicitly for transitions between adequately prescribed states of a simple system of a nonrelativistic particle moving in a constant uniform field of force. The implications of the uncertainties in time and energy contemplated for transitions are clarified by contrasting them with ordinary uncertainties of various dynamical variables evaluated in one single state of the system. Decay process and the mean lifetime related to it are discussed too in a particular case when a pair of states, initial and final, for a transition happen to be identical.
On the basis of the principles proposed by the first author in 1970 and 1979 the time-energy uncertainty relationship in quantum mechanics is constructed explicitly for transitions between adequately prescribed states of a simple system of a nonrelativistic particle moving in a constant uniform field of force. The implications of the uncertainties in time and energy contemplated for transitions are clarified by contrasting them with ordinary uncertainties of various dynamical variables evaluated in one single state of the system. Decay process and the mean lifetime related to it are discussed too in a particular case when a pair of states, initial and final, for a transition happen to be identical.
A precise form of the quantum-mechanical time-energy uncertainty relation is derived. For any given initial state (density operator), time-dependent Hamiltonian, and subspace of reference states, it gives upper and lower bounds for the probability of finding the system in a state in that subspace at a later or earlier time. The bounds involve only the initial data, the energy uncertainty in the initial state, and the energy uncertainty in the reference subspace. They describe how fast the state enters or leaves the reference subspace. They are exact if, but not only if, the initial state or the projection onto the reference subspace commutes with the Hamiltonian. The basic tool used in the proof is a simple inequality for expectation values of commutators, which generalizes the usual uncertainty relation. By introducing suitable comparison dynamics (trial propagators), the bounds can be made arbitrarily tight. They represent a time-dependent variational principle, in terms of trial propagators, which provides explicit error estimates and reproduces the exact time evolution when one varies over all trial propagators. As illustrations, we derive accurate lower bounds on the escape time of a particle out of a potential well modeling a quantum dot, and the total time before which a He+ ion moving in a uniform magnetic field loses its electron.
There are many controversies on energy—time uncertainty relation. Because of the confused role time plays in quantum mechanics, the references and meanings of energy—time uncertainty relation are different in different explanatory contexts. But what do time and energy really mean? What is the philosophical foundation of this relationship? The article bases on the three different meanings of time in quantum mechanics, discusses the references and meanings of energy—time uncertainty relation in various contexts, and aims to clarify the confusion on the references and the meanings of the uncertainty relation between energy and time.
A derivation from first principles is given of the energy-time uncertainty relation in quantum mechanics. A canonical transformation is made in classical mechanics to a new canonical momentum, which is energy E, and a new canonical coordinate T, which is called tempus, conjugate to the energy. Tempus T, the canonical coordinate conjugate to the energy, is conceptually different from the time t in which the system evolves. The Poisson bracket is a canonical invariant, so that energy and tempus satisfy the same Poisson bracket as do p and q. When the system is quantized, we find the energy-time uncertainty relation \ensuremath{\Delta}E\ensuremath{\Delta}T\ensuremath{\ge}\ensuremath{\Elzxh}/2. For a conservative system the average of the tempus operator T^ is the time t plus a constant. For a free particle and a particle acted on by a constant force, the tempus operators are constructed explicitly, and the energy-time uncertainty relation is explicitly verified.
Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form ∥1Re−iHtψ∥22≤Ct−3 have been derived, where ψ denotes the initial state, H the Hamiltonian, R a positive constant, and C is explicitly known. As intermediate step, bounds on the derivatives of the S-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with n=1,2,3, and the constants Cn,K explicitly known.
The time–energy uncertainty relation in nonrelativistic quantum mechanics has been intensely debated with regard to its formal derivation, validity, and physical meaning. Here, we analyze two formal relations proposed by Mandelstam and Tamm and by Margolus and Levitin and evaluate their validity using a minimal quantum toy model composed of a single qubit inside an external magnetic field. We show that the ℓ1 norm of energy coherence C is invariant with respect to the unitary evolution of the quantum state. Thus, the ℓ1 norm of energy coherence C of an initial quantum state is useful for the classification of the ability of quantum observables to change in time or the ability of the quantum state to evolve into an orthogonal state. In the single-qubit toy model, for quantum states with the submaximal ℓ1 norm of energy coherence, C<1, the Mandelstam–Tamm and Margolus–Levitin relations generate instances of infinite “time uncertainty” that is devoid of physical meaning. Only for quantum states with the maximal ℓ1 norm of energy coherence, C=1, the Mandelstam–Tamm and Margolus–Levitin relations avoid infinite “time uncertainty”, but they both reduce to a strict equality that expresses the Einstein–Planck relation between energy and frequency. The presented results elucidate the fact that the time in the Schrödinger equation is a scalar variable that commutes with the quantum Hamiltonian and is not subject to statistical variance.
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=0. This expression is a function of the time t 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 t 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/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.