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Physical Review A

Physical constraints on nonstationary states and nonexponential decay

Cleanthes A. Nicolaides

Published Aug 27, 2002• 24 citations
Publisher

Abstract

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, $P(t),$ 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 $E>~0,$ the constraint of $t>~0$ must be accounted for at the same time. It results from the discontinuity at $t=0$ 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 ${P}_{\mathrm{NED}}(t)$ obtained from semiclassical path-integral calculations agree with the quantum-mechanical ones when both physical constraints $E>0$ and $t>0$ are taken into account. As an example of the difference in the magnitude of the ${P}_{\mathrm{NED}}(t)$ 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.

DOI
10.1103/physreva.66.022118

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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.
The survival amplitude G(t)G(t)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 gL(E){g}^{L}(E)gL(E) and a modified Lorentzian G(E)=gL(E)E1∕2G(E)={g}^{L}(E){E}^{1∕2}G(E)=gL(E)E1∕2, and one is the complex version of gL(E),gcL(E){g}^{L}(E),{g}_{c}^{L}(E)gL(E),gcL​(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)G(t)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 gL(E){g}^{L}(E)gL(E) and gcL(E){g}_{c}^{L}(E)gcL​(E) are employed, are interpreted using concepts such as friction and fluctuation. On the other hand, when gL(E){g}^{L}(E)gL(E) is multiplied by an energy-dependent factor, as in G(E)G(E)G(E), the results are, as expected, more complicated and the interpretability of the differential equation satisfied by the NED propagator loses clarity.
The Hermitian properties of the Hamiltonian are related not only to the operator itself but also to the functions on which it acts. Hermitian Hamiltonians operate on functions in the L 2 Hilbert space which correspond to boundary conditions which vanish at infinity. In this chapter, in order to move into the non-Hermitian domain, we will impose on the solutions to the time-independent Schrödinger equation (TISE) different boundary conditions which lead to solutions which can be associated with different types of the complex poles of the scattering matrix. These solutions will contain information which was not available within the scope of functions in L 2 . By imposing outgoing boundary conditions on the eigenfunctions of the timeindependent Hamiltonian complex eigenvalues, E res = ε − (i/2)Г, are obtained. These complex energies are associated with decaying resonance states which were discussed in the previous two chapters. The bound states (if they exist) appear as real eigenvalues since they result from exactly such outgoing boundary conditions which appear under the threshold energy. When incoming boundary conditions are imposed two kind of solution are obtained. One type of solution is the complex conjugates of the decay resonance solutions mentioned above. In scattering theory text books (see Taylor for example) the physical resonance solutions are associated with the poles of the scattering matrix which are embedded in the lower half of the complex energy plane. However, in nuclear physics the complex poles embedded in the upper half of the complex energy plane, so-called virtual states , are denoted as capture resonances .
By solving rigorously and accurately the time-dependent Schr\"odinger equation, we have obtained numerical results for the decay probability, P(t),P(t),P(t), of real, multiparticle systems, in the time domain of $t\ensuremath{\approx}0.$ Three different types of atomic nonstationary states were examined, the ${\mathrm{He}}^{\ensuremath{-}}{1s2p}^{2}{}^{4}P,$ the Ca KLM 3d5p3Fo,3d5p{}^{3}{F}^{o},3d5p3Fo, and the ${\mathrm{He}}^{\mathrm{\ensuremath{-}}}$ 1s2s2p4P5/2,1s2s2p{}^{4}{P}_{5/2},1s2s2p4P5/2​, the last one being metastable and decaying via spin-spin interactions. The main results are that there is a t2{t}^{2}t2 dependence of $P(t\ensuremath{\approx}0)$ and that a time-dependent short-time decay rate can be calculated. The computed coefficients of the t2{t}^{2}t2 term reflect the degree of stability of the state, (i.e., the degree of proximity to the notion of the standard stationary state of quantum mechanics), and are named the stationarity coefficients. These, together with the conventional quantity of the lifetime, corresponding to the exponential decay regime, constitute intrinsic properties of each real unstable state. For the herein studied metastable state the onset of exponential decay occurs after about $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\mathrm{s},$ i.e., after a duration which is achievable experimentally with laser pulses.
As discussed in the previous chapter, the poles of the S-matrix are identified with discrete eigenvalues of the time-independent Schrödinger equation, where the asymptotes of the corresponding eigenfunctions are either purely outgoing waves or purely incoming waves. More specifically, the bound and decay resonance poles are obtained by imposing the outgoing boundary conditions on the solutions of the time-independent Schrödinger equation, while the anti-bound and virtual states (sometimes referred to as capture resonances) are associated with the solutions obtained under the requirement of the incoming boundary conditions. Except for the bound states, all other poles of the S-matrix are associated with exponentially divergent wavefunctions which by definition do not belong to the Hilbert space of conventional Hermitian quantum mechanics. This fact represents a major difficulty for the development of a non-Hermitian quantum mechanical formalism. Consequently, one may wonder, for example, how to properly define an inner product in non-Hermitian quantum mechanics (NHQM) if the wavefunctions diverge asymptotically. We recall in this context that the concept of an inner product constitutes a fundamental building block of standard (Hermitian) quantum mechanics (QM), by means of which one defines the quantum mechanical expectation values of physically observable quantities over the quantum states under consideration. An inner product for NHQM is necessary in order to accommodate the tools of conventional QM. Furthermore, one might anticipate that an appropriate NHQM inner product would also facilitate practical numerical calculation of the S-matrix poles for those cases where the eigenvalues of the Hamiltonian do not possess an analytical closed form expression (unlike the cases studied in the previous chapter).
An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. We find that the instantaneous energy of the unstable state for a large class of models of unstable states tends to the minimal energy of the system Emin{\cal E}_{min}Emin​ as t→∞t\rightarrow\inftyt→∞ which is much smaller than the energy of this state for ttt of the order of the lifetime of the considered state. Analyzing the transition time region between exponential and non-exponential form of the survival amplitude we find that the instantaneous energy of the considered unstable state can take large values, much larger than the energy of this state for ttt from the exponential time region. Taking into account results obtained for a model considered, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as σσσ meson as well as having astrophysical and cosmological consequences.