New form of the time-energy uncertainty relation

Gislason, Eric A.; Sabelli, Nora H.; Wood, John W.

doi:10.1103/physreva.31.2078

Cited by 58 publications

(44 citation statements)

References 30 publications

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“…The t Ϸ 0 regime was excluded from the initial applications [6a, 6b] because this domain seems to be undefinable, in a rigorous sense, as regards preparation and measurement of an isolated unstable state. However, given the fact that a number of calculations and arguments using models have dealt with this region (e.g., [18,22,[35][36][37][38][39][40]), we thought that information coming from ab initio cal-culations on real states would provide additional insight as well as quantitative knowledge into these issues from a different angle.…”

Section: "Stationarity Coefficients" and Short-time Deviations From Ementioning

confidence: 99%

Time asymmetry, nonexponential decay, and complex eigenvalues in the theory and computation of resonance states

Nicolaides

2002

Int J of Quantum Chemistry

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Stationary-state quantum mechanics presents no difficulties in defining and computing discrete excited states because they obey the rules established in the properties of Hilbert space. However, when this idealization has to be abandoned to formulate a theory of excited states dissipating into a continuous spectrum, the problem acquires additional interest in many fields of physics. In this article, the theory of resonances in the continuous spectrum is formulated as a problem of decaying states, whose treatment can entail time-dependent as well as energy-dependent theories. The author focuses on certain formal and computational issues and discusses their application to polyelectronic atomic states. It is argued that crucial to the theory is the understanding and computation of a multiparticle localized wavepacket, ⌿ 0 , at t ϭ 0, having a real energy E 0 . Assuming this as the origin, without memory of the excitation process, the author discusses aspects of time-dependent dynamics, for t Ϸ 0 as well as for t 3 ϱ, and the possible significance of nonexponential decay in the understanding of time asymmetry. Also discussed is the origin of the complex eigenvalue Schrödinger equation (CESE) satisfied by resonance states and the state-specific methodology for its solution. The complex eigenvalue drives the decay exponentially, with a rate ⌫, to a good approximation. It is connected to E 0 via analytic continuation of the complex selfenergy function, A(z), (z is complex), into the second Riemann sheet, or, via the imposition of outgoing wave boundary conditions on the stationary state Schrödinger equation satisfied by the Fano standing wave superposition in the vicinity of E 0 . If the nondecay amplitude, G(t), is evaluated by inserting the unit operator I ϭ ͐dE|EϾϽE| G(t) ϭ Ͻ ⌿ 0 |e ϪiHt |⌿ 0 Ͼ, then the resulting spectral function is real, g(E) ϭ|Ͻ⌿ 0 |EϾ| 2 , and does not differentiate between positive and negative times. The introduction of time asymmetry, which is associated with irreversibility, is achieved by starting from Ͻ ⌿ 0 |(t)e ϪiHt |⌿ 0 Ͼ, where (t) is the step function at the discontinuity point t ϭ 0. In this case, the spectral function is complex. Within the range of validity of exponential decay, the complex spectral function is the same as the coefficient of ⌿ 0 in the theory of the CESE. A calculation of G(t) using the simple pole approximation and the constraints that t Ͼ 0 and E Ͼ 0 results in a nonexponential decay (NED) correction for t ӷ 1/⌫ that is different than when a real g(E) is used, representing the contribution of both "in" and "out" states. Earlier formal and computational work has shown that resonance states close to threshold are good candidates for NED to acquire nonnegligible magnitude. In this context, a pump-probe laser experiment in atomic physics is proposed, using as a paradigm the He Ϫ 1s2p

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Section: "Stationarity Coefficients" and Short-time Deviations From Ementioning

confidence: 99%

Time asymmetry, nonexponential decay, and complex eigenvalues in the theory and computation of resonance states

Nicolaides

2002

Int J of Quantum Chemistry

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“…It appears in several important contexts such as quantum mechanics [3], statistical estimation [4], and information theory [11].…”

Section: Theorem 1 Let F (X) Be a Continuously Differentiable Probabmentioning

confidence: 99%

Inequalities for characteristic functions involving Fisher information

Zhang

2007

Comptes Rendus. Mathématique

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“…The treatment is formulated in terms of explicit states and is suitable for calculations where only numerical data are available. The physics of the problem leads us to a different definition of the evolution time from that employed previously [1][2][3] and we relate this time to the energy uncertainty.…”

Section: Introductionmentioning

confidence: 99%

“…One such issue is the more conceptually difficult energy-time uncertainty relation, which has been the topic of much recent research [1][2][3]. The problem most often studied involves the evolution of an initial state under a Hamiltonian which is not an explicit function of time.…”

Section: Introductionmentioning

confidence: 99%

“…Hilgevoord [1] emphasizes that this relation is very different from the more familiar positionmomentum relation and Uffink [3] shows that it can be written in terms of the energy width of the distribution and the overlap of the initial and final wavefunctions. Gislason, Sabelli and Wood [2] discuss uncertainty relations and evolution times for several common distributions.…”

Section: Introductionmentioning

confidence: 99%

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Evolution time and energy uncertainty

Boykin¹,

Kharche²,

Klimeck³

2007

Eur. J. Phys.

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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 firstorder 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.

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New form of the time-energy uncertainty relation

Cited by 58 publications

References 30 publications

Time asymmetry, nonexponential decay, and complex eigenvalues in the theory and computation of resonance states

Time asymmetry, nonexponential decay, and complex eigenvalues in the theory and computation of resonance states

Inequalities for characteristic functions involving Fisher information

Evolution time and energy uncertainty

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