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Physical Review A

Nonexponential decay propagator and its differential equation for real and complex energy distributions of unstable states

Theodosios Geo Douvropoulos, Cleanthes A. Nicolaides

Published Mar 16, 2004• 12 citations
Publisher

Abstract

The survival amplitude $G(t)$ of a nonstationary state decaying into a purely continuous spectrum is treated in terms of an integral transform of an energy distribution with $\ensuremath{\infty}>E\ensuremath{\geqslant}0$. We examine three such distributions. Two are real functions, the Lorentzian ${g}^{L}(E)$ and a modified Lorentzian $G(E)={g}^{L}(E){E}^{1∕2}$, and one is the complex version of ${g}^{L}(E),{g}_{c}^{L}(E)$. Real distributions are associated with Hermitian treatments while complex ones result from non-Hermitian treatments. The difference between the two has repercussions on the $G(t)$ for nonexponential decay (NED) and on the understanding of irreversible decay at the quantum level. For all three distributions, we derive analytically amplitudes (propagators) for NED and then show that these satisfy differential equations, from which additional insight into the decay process for very long and very short times can be obtained. By making analogy with the classical Langevin equation, the terms of the differential equation that are derived when the simpler ${g}^{L}(E)$ and ${g}_{c}^{L}(E)$ are employed, are interpreted using concepts such as friction and fluctuation. On the other hand, when ${g}^{L}(E)$ is multiplied by an energy-dependent factor, as in $G(E)$, the results are, as expected, more complicated and the interpretability of the differential equation satisfied by the NED propagator loses clarity.

DOI
10.1103/physreva.69.032105

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We study the properties of the survival probability of an unstable quantum state described by a Lee Hamiltonian. This theoretical approach resembles closely Quantum Field Theory (QFT): one can introduce in a rather simple framework the concept of propagator and Feynman rules, Within this context, we re-derive (in a detailed and didactical way) the well-known result according to which the amplitude of the survival probability is the Fourier transform of the energy distribution (or spectral function) of the unstable state (in turn, the energy distribution is proportional to the imaginary part of the propagator of the unstable state). Typically, the survival probability amplitude is the starting point of many studies of non-exponential decays. This work represents a further step toward the evaluation of the survival probability amplitude in genuine relativistic QFT. However, although many similarities exist, QFT presents some differences w.r.t. the Lee Hamiltonian which should be studied in the future.
For the understanding of irreversibility at the quantum level, the formation and decay of transient (unstable) states play a fundamental role. If the system is treated within Hermitian quantum mechanics, the resulting energy distribution of the resonance state, whose Fourier transform yields the time-dependent probability of decay, P(t),P(t),P(t), is real. The physical constraint of the lower bound in the energy spectrum introduces ``memory,'' and causes nonexponential decay (NED) to set in after $t\ensuremath{\gg}\ensuremath{\tau},$ where \ensuremath{\tau} is the lifetime defined by exponential decay. The closer to threshold the decaying state is, the earlier NED appears. Apart from the constraint of E> 0,E>~0,E> 0, the constraint of t> 0t>~0t> 0 must be accounted for at the same time. It results from the discontinuity at t=0t=0t=0 of the solution of the time-dependent Schr\"odinger equation, which breaks the unitarity associated with the S matrix and gives rise to a complex energy distribution, as a manifestation of the non-Hermitian property of the decaying states. For a narrow isolated resonance state, for which the self-energy is essentially energy-independent, analytic results for PNED(t){P}_{\mathrm{NED}}(t)PNED​(t) obtained from semiclassical path-integral calculations agree with the quantum-mechanical ones when both physical constraints E>0E>0E>0 and t>0t>0t>0 are taken into account. As an example of the difference in the magnitude of the PNED(t){P}_{\mathrm{NED}}(t)PNED​(t) when using a real and a complex energy distribution, application is made to the decay of the unstable ${\mathrm{He}}_{2}^{2+}{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ ground molecular state.
\nWe study the properties of the survival probability of an unstable quantum state described by a Lee Hamiltonian. This theoretical approach resembles closely Quantum Field Theory (QFT): one can introduce in a rather simple framework the concept of propagator and Feynman rules, Within this context, we re-derive (in a detailed and didactical way) the well-known result according to which the amplitude of the survival probability is the Fourier transform of the energy distribution (or spectral function) of the unstable state (in turn, the energy distribution is proportional to the imaginary part of the propagator of the unstable state). Typically, the survival probability amplitude is the starting point of many studies of non-exponential decays. This work represents a further step toward the evaluation of the survival probability amplitude in genuine relativistic QFT. However, although many similarities exist, QFT presents some differences w.r.t. the Lee Hamiltonian which should be studied in the future.\n
The probability, P ( t ), of the irreversible dissipation into a continuous spectrum of an initially ( t = 0) localized (Ψ o ) nonstationary state acquires, as time increases, 'memory' due to the lower energy bound of the spectrum, and eventually follows a nonexponential decay (NED). Regardless of the degree of dependence on energy, the magnitude of this deviation from exponential decay depends on the degree of proximity to threshold, and on whether the theory employs a real energy distribution, one form of which is g ( E ) ≡ ⟨Ψ o |δ( H − E )|Ψ o ⟩, or a complex energy distribution, G ( E ) ≡ ⟨Ψ o |( H − E + i0)|Ψ o ⟩. It is the latter that is physically consistent, since it originates from the singularity at t = 0, which breaks the S -matrix unitarity, in accordance with the non-Hermitian character of decaying states. In order to test the quantum mechanical theory, we carried out semiclassical path integral calculations of the P ( t ) for an isolated narrow tunnelling state, whereby the truncated Fourier transform of a semiclassical Green function, G sc ( E ), is obtained. The results are in agreement with the analytic results of quantum mechanics when energy and time asymmetry are taken into account. It is shown that the analytic structure of G sc ( E ) is [ D regular + D pole ], where D pole is a finite sum over complex poles, which are the complex eigenvalues, W n , that the potential can support. The W n are given by E n + Δ n − (i/2)Γ n , where E n are the real eigenvalues of the corresponding bound potential, Γ n are the energy widths and Δ n are the energy shifts, both expressed in terms of computable semiclassical quantities. The spherical harmonic oscillator (SHO) with and without angular momentum, and unstable ground states of diatomic molecules, are treated as particular cases. The exact spectrum of the SHO is recovered only when the Kramers–Langer semiclassical expression for the centrifugal potential is used, thereby bypassing the difficulty of the singularity at r = 0. The spectrum from the use of the quantal form l ( l + 1) reduces to that of l ( l + 1/2) 2 in the limit of large l , i.e., for orbits far from r = 0. Using previously computed energies and widths for the vibrational levels of He 2 2+ 1σ g 2 1 Σ g + , the application of two formulae for P ( t ), one derived from a Lorentzian real energy distribution and the other from the corresponding complex energy distribution, shows that, for the lowest level, in the former case NED starts after about 193 lifetimes, and in the latter after about 102 lifetimes. The fact that this difference is large should have consequences for the deeper understanding of irreversibility at the quantum level.
An eect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. In 1957 Khalfin proved that this amplitude tends to zero as t ! 1 more slowly than any exponential function oft. This can be described in terms of the time-dependent decay rate (t) which, when considered with the Khalfin result, means that this (t) is not a constant for large t but that it tends to zero as t!1. We find that a similar conclusion can be drawn for a large class of models of unstable states for a quantity, which can be interpreted as the ”instantaneous energy” of the unstable state. This energy should be much smaller for suitably larger values of t than when t is of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long (t ! 1) and relatively short (lifetime of the state) time regions, are dierent. This is a purely quantum mechanical eect. It is hypothesized that there is a possibility to detect this eect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters.
The Fock-Krylov formalism for the calculation of survival probabilities of unstable states is revisited paying particular attention to the mathematical constraints on the density of states, the Fourier transform of which gives the survival amplitude. We show that it is not possible to construct a density of states corresponding to a purely exponential survival amplitude. The survival probability P(t)P(t)P(t) and the autocorrelation function of the density of states are shown to form a pair of cosine Fourier transforms. This result is a particular case of the Wiener-Khinchin theorem and forces P(t)P(t)P(t) to be an even function of time which in turn forces the density of states to contain a form factor which vanishes at large energies. Subtle features of the transition regions from the nonexponential to the exponential at small times and the exponential to the power-law decay at large times are discussed by expressing P(t)P(t)P(t) as a function of the number of oscillations nnn performed by it. The transition at short times is shown to occur when the survival probability has completed one oscillation. The number of oscillations depends on the properties of the resonant state and a complete description of the evolution of the unstable state is provided by determining the limits on the number of oscillations in each region.