Time-dependent tunnelling via path integrals. Connection to results of the quantum mechanics of decaying states

Douvropoulos, TG; Nicolaides, CA

dc.contributor.author	Douvropoulos, TG	en
dc.contributor.author	Nicolaides, CA	en
dc.date.accessioned	2014-03-01T01:18:27Z
dc.date.available	2014-03-01T01:18:27Z
dc.date.issued	2002	en
dc.identifier.issn	0953-4075	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/15014
dc.subject.classification	Optics	en
dc.subject.classification	Physics, Atomic, Molecular & Chemical	en
dc.subject.other	Eigenvalues and eigenfunctions	en
dc.subject.other	Electron tunneling	en
dc.subject.other	Fourier transforms	en
dc.subject.other	Green's function	en
dc.subject.other	Probability	en
dc.subject.other	Spherical harmonic oscillators (SHO)	en
dc.subject.other	Quantum theory	en
dc.title	Time-dependent tunnelling via path integrals. Connection to results of the quantum mechanics of decaying states	en
heal.type	journalArticle	en
heal.identifier.primary	10.1088/0953-4075/35/21/310	en
heal.identifier.secondary	http://dx.doi.org/10.1088/0953-4075/35/21/310	en
heal.language	English	en
heal.publicationDate	2002	en
heal.abstract	The probability, P (t), of the irreversible dissipation into a continuous spectrum of an initially (t = 0) localized (Psi(0)) 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) equivalent to <Psi(0)\delta (H- E) \Psi(0)>, or a complex energy distribution, G (E) equivalent to <Psi(0)\(H - E + i0)\Psi(0)>. 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, that the potential can support. The Delta(n) are given by E-n + Delta(n) - (i/2)Gamma(n), where E-n are the real eigenvalues of the corresponding bound potential, Gamma(n) are the energy widths and Delta(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+) 1sigma(g)(2 1)Sigma(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.	en
heal.publisher	IOP PUBLISHING LTD	en
heal.journalName	Journal of Physics B: Atomic, Molecular and Optical Physics	en
dc.identifier.doi	10.1088/0953-4075/35/21/310	en
dc.identifier.isi	ISI:000179437100012	en
dc.identifier.volume	35	en
dc.identifier.issue	21	en
dc.identifier.spage	4453	en
dc.identifier.epage	4473	en