It is shown how the semiclassical theory of path integrals can be implemented in a practical manner for the analysis of a potential that combines the two-state system of a double well potential ͑DWP͒ with decay into a continuous spectrum. This potential may correspond to a variety of physical situations in physics and chemistry. The structure of the formalism and of the results is such that it allows computation not only for analytic but also for numerically given potentials. The central theme is the determination of the energy-dependent Green's function, which is shown to consist of a regular part and a part containing simple and double complex poles. These poles represent the position of the energy levels, as well as the energy widths and shifts due to the interaction with the continuous spectrum. When applied to the bound DWP without tunneling, the theory is shown to reduce in certain limits to known results from the Jeffreys-Wentzel-Kiamers-Bhrillouin approximation. If the system is taken to be prepared in the first well, the interactions with the remaining of the potential lead to two types of transition rates. One represents the transient motion toward a virtual equilibrium state of the DWP. It emerges as a positive imaginary part of the self-energy. The other represents the decay into the continuum and emerges as a negative imaginary part of the pole. Comparison of the two mechanisms of nonstationarity is made for different magnitudes of the second barrier relative to the first one. Since the system decays to the continuum while oscillating, the theory obtains a correction to the frequency of oscillation in the DWP due to the interaction with the continuum. This phenomenon is observable in real two-state systems, if an external perturbation which affects mainly one state converts it into a resonance state.

An approximate formula for the period of pendulum motion beyond the small amplitude regime is obtained based on physical arguments. Two different schemes of different accuracy are developed: in the first less accurate scheme, emphasis is given on the non-quadratic form of the potential in connection to isochronism, and a specific form of a generic formula that is met in many previous works is produced, while the second and main result contains the Struve function which is further approximated by a simple sinusoidal expression based on its maximum value. The accuracy of the final formula gives a relative error of less than 0.2% for angles up to 140°. In addition, a simple relation between the Struve function and the complete elliptic integral of the first kind is produced, since they both constitute solutions of the pendulum period. This relation makes it possible for someone to connect different areas in physics and solve a difficult task by comparison with another much more simple one. As an example, a connection between the pendulum period and the acoustical radiation impedance is proposed through impedance matching and some interesting relations are produced. This paper is intended for undergraduate students to be useful for analysing pendulum experiments in introductory physics labs and it is also believed to offer valuable insight into some properties of the simple pendulum in undergraduate courses on general physics.

The survival amplitude G͑t͒ of a nonstationary state decaying into a purely continuous spectrum is treated in terms of an integral transform of an energy distribution with ϱϾE ജ 0. We examine three such distributions. Two are real functions, the Lorentzian g L ͑E͒ and a modified Lorentzian G͑E͒ = g L ͑E͒E 1/2 , and one is the complex version of g L ͑E͒ , g c L ͑E͒. Real distributions are associated with Hermitian treatments while complex ones result from non-Hermitian treatments. The difference between the two has repercussions on the G͑t͒ 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 ͑E͒ and g c L ͑E͒ are employed, are interpreted using concepts such as friction and fluctuation. On the other hand, when g L ͑E͒ is multiplied by an energy-dependent factor, as in G͑E͒, the results are, as expected, more complicated and the interpretability of the differential equation satisfied by the NED propagator loses clarity.

ABSTRACT:A recently published methodology based on semiclassical path integral (SCPI) theory was implemented in the case of a model of a double-well potential perturbed by a static electric field, with application to the inversion frequency of NH 3 . This model was chosen as an idealized case for testing of the present approach, as well as for quantum mechanical models that might be applied in the future. The calculations were concerned with the variation of the frequency of inversion as a function of field strength, F, and of distance, x f (from the symmetric point x o ϭ 0), where the field is "felt." It is found that this variation occurs sharply in very small regions of values of these parameters, and the system switches from internal oscillation to diffusion to the continuum. The fact that the theory is in analytic form allows the extraction of results and conclusions not only at the full SCPI level, but also at the Jeffreys-Wentzel-KramersBrillouin (JWKB) level. Comparison shows that the discrepancy sets in as the field strength increases, in accordance with the well-known limitations of the JWKB method regarding its dependence on the degree of variation of the potential as a function of position.

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 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) ≡ ⟨Ψo |δ(H − E)|Ψo ⟩, or a complex energy distribution, G(E) ≡ ⟨Ψo |(H − E + i0)|Ψo ⟩. 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, Gsc(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 Gsc(E) is [Dregular + Dpole], where Dpole is a finite sum over complex poles, which are the complex eigenvalues, Wn, that the potential can support. The Wn are given by En + Δn − (i/2)Γn, where En are the real eigenvalues of the corresponding bound potential, Γn are the energy widths and Δn 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 He22+1σ g2 1Σ 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.