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

Abstract

The probability, P(t),of the irreversible dissipation into a continuous spectrum of an initially (t = 0) localized(Ψo)nonstationary state acquires, as time increases, ‘memory’ due to the lower energybound of the spectrum, and eventually follows a nonexponential decay (NED).Regardless of the degree of dependence on energy, the magnitude of this deviationfrom exponential decay depends on the degree of proximity to threshold, and onwhether the theory employs a real energy distribution, one form of which is g(E) ≡ ⟨Ψo |δ(H − E)|Ψo ⟩,or a complex energy distribution, G(E) ≡ ⟨Ψo |(H − E + i0)|Ψo ⟩. It is thelatter that is physically consistent, since it originates from the singularity at t = 0, which breaksthe S-matrixunitarity, in accordance with the non-Hermitian character of decaying states. In order totest the quantum mechanical theory, we carried out semiclassical path integral calculationsof the P(t)for an isolated narrow tunnelling state, whereby thetruncated Fourier transform of a semiclassical Green function,Gsc(E),is obtained. The results are in agreement with the analytic resultsof quantum mechanics when energy and time asymmetry aretaken into account. It is shown that the analytic structure ofGsc(E) is [Dregular + Dpole],where Dpoleis a finite sum over complex poles, which are the complex eigenvalues,Wn, that the potentialcan support. The Wnare given by En + Δn − (i/2)Γn, whereEnare the real eigenvalues of the corresponding bound potential, Γn are the energywidths and Δnare the energy shifts, both expressed in terms of computable semiclassicalquantities. The spherical harmonic oscillator (SHO) with and without angularmomentum, and unstable ground states of diatomic molecules, are treatedas particular cases. The exact spectrum of the SHO is recovered onlywhen the Kramers–Langer semiclassical expression for the centrifugalpotential 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 ofl(l + 1/2)2 in thelimit of large l, i.e., fororbits far from r = 0.Using previously computed energies and widths for the vibrationallevels of He22+1σg2 1Σg+, the applicationof two formulae for P(t),one derived from a Lorentzian real energy distribution and the other from thecorresponding complex energy distribution, shows that, for the lowest level, in theformer case NED starts after about 193 lifetimes, and in the latter afterabout 102 lifetimes. The fact that this difference is large should haveconsequences for the deeper understanding of irreversibility at the quantum level.