We consider a system having decaying states with complex energies ${E}_{j}\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$, and we define the ``width-weighted spectrum'' as $D\ensuremath{\Gamma}(E)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\Gamma}}_{j}\ensuremath{\delta}({E\ensuremath{-}E}_{j})$. We derive a semiclassical formula for this width-weighted spectrum, a formula which is analogous to the periodic-orbit representation of the density of states. The formula applies if classical motion is regular and if decay occurs by tunneling through a barrier. The semiclassical formula involves not periodic orbits or closed orbits, but action integrals associated with irreducible loops on the ``extremal torus,'' on which the particle climbs up the barrier and hangs at the top. Calculations confirm the validity of the formula.
We consider a system having decaying states with complex energies Ej\ensuremath−i\ensuremathΓj/2, and we define the width-weighted spectrum'' as $D\ensuremath{\Gamma}(E)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\Gamma}}_{j}\ensuremath{\delta}({E\ensuremath{-}E}_{j})$. We derive a semiclassical formula for this width-weighted spectrum, a formula which is analogous to the periodic-orbit representation of the density of states. The formula applies if classical motion is regular and if decay occurs by tunneling through a barrier. The semiclassical formula involves not periodic orbits or closed orbits, but action integrals associated with irreducible loops on the extremal torus,'' on which the particle climbs up the barrier and hangs at the top. Calculations confirm the validity of the formula.
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 semiclassical limit of the coherent state propagator $⟨{\mathbf{z}}^{\ensuremath{''}}\ensuremath{\mid}{e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}T∕\ensuremath{\hbar}}\ensuremath{\mid}{\mathbf{z}}^{\ensuremath{'}}⟩$ involves complex classical trajectories of the Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}(\mathbf{u},\mathbf{v})=⟨\mathbf{v}\ensuremath{\mid}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\mid}\mathbf{u}⟩$ satisfying $\mathbf{u}(0)={\mathbf{z}}^{\ensuremath{'}}$ and $\mathbf{v}(T)={\mathbf{z}}^{\ensuremath{''}}*$. In this work we study mostly the case ${\mathbf{z}}^{\ensuremath{'}}={\mathbf{z}}^{\ensuremath{''}}$. The propagator is then the return probability amplitude of a wave packet. We show that a plot of the exact return probability brings out the quantal images of the classical periodic orbits. Then we compare the exact return probability with its semiclassical approximation for a soft chaotic system with two degrees of freedom. We find two situations where classical trajectories satisfying the correct boundary conditions must be excluded from the semiclassical formula. The first occurs when the contribution of the trajectory to the propagator becomes exponentially large as $\ensuremath{\hbar}$ goes to zero. The second occurs when the contributing trajectories undergo bifurcations. Close to the bifurcation the semiclassical formula diverges. More interestingly, in the example studied, after the bifurcation, where more than one trajectory satisfying the boundary conditions exist, only one of them in fact contributes to the semiclassical formula, a phenomenon closely related to Stokes lines. When the contributions of these trajectories are filtered out, the semiclassical results show excellent agreement with the exact calculations.
Gutzwiller's trace formula has a central place in quantum chaos because it provides semiclassical approximations for quantum energy levels in classically chaotic systems by linking them to classical periodic orbits. In this didactic article, we discuss a derivation of the trace formula starting from the Feynman path integral. We then describe how the trace formula is used to explain universal features in the distribution of the quantum energy levels that are described by random matrix theory, and we give an overview of related work.
A semiclassical method of complex trajectories for the calculation of the tunneling exponent in systems with many degrees of freedom is further developed. It is supplemented with an easily implementable technique, which enables one to single out the physically relevant trajectory from the whole set of complex classical trajectories. The method is applied to semiclassical transitions of a bound system through a potential barrier. We find that the properties of physically relevant complex trajectories are qualitatively different in the cases of potential tunneling at low energy and dynamical tunneling at energies exceeding the barrier height. Namely, in the case of high energies, the physically relevant complex trajectories describe tunneling via creation of a state close to the top of the barrier. The method is checked against exact solutions of the Schrodinger equation in a quantum mechanical system of two degrees of freedom.
A semiclassical method of complex trajectories for the calculation of the tunneling exponent n systems with many degrees of freedom is further developed. It is supplemented with an easily implemented technique that enables one to single out the physically relevant trajectory from the whole set of complex classical trajectories. The method is applied to semiclassical transitions of a bound system through a potential barrier. We find that the properties of physically relevant complex trajectories are qualitatively different in the cases of potential tunneling at low energy and dynamical tunneling at energies exceeding the barrier height. Namely, in the case of high energies, the physically relevant complex trajectories describe tunneling via creation of a state close to the top of the barrier. The method is checked against exact solutions of the Schrödinger equation in a quantum-mechanical system of two degrees of freedom.
Semiclassical initial value representation (IVR) formulas for the propagator have difficulty describing tunneling through barriers. A key reason is that these formulas do not automatically reduce, in the classical limit, to the version of the Van Vleck-Gutzwiller (VVG) propagator required to treat barrier tunneling, which involves trajectories that have complex initial conditions and that follow paths in complex time. In this work, a simple IVR expression, that has the correct tunneling form in the classical limit, is derived for the propagator in the case of one-dimensional barrier transmission. Similarly, an IVR formula, that reduces to the Generalized Gaussian Wave Packet Dynamics (GGWPD) expression [D. Huber, E. J. Heller, and R. Littlejohn, J. Chem. Phys. 89, 2003 (1988)] in the classical limit, is derived for the transmitted wave packet. Uniform semiclassical versions of the IVR formulas are presented and simplified expressions in terms of real trajectories and WKB penetration factors are described. Numerical tests show that the uniform IVR treatment gives good results for wave packet transmission through the Eckart and Gaussian barriers in all cases examined. In contrast, even when applied with the proper complex trajectories, the VVG and GGWPD treatments are inaccurate when the mean energy of the wave packet is near the classical transmission threshold. The IVR expressions for the propagator and wave packet are cast as contour integrals in the complex space of initial conditions and these are generalized to potentially allow treatment of a larger variety of systems. A steepest descent analysis of the contour integral formula for the wave packet in the present cases confirms its relationship to the GGWPD method, verifies its semiclassical validity, and explains results of numerical calculations.