The rate and extent of the exploration of the available phase space of a bound quantum mechanical system are shown to depend on the repulsion of energy eigenstates. Central to the argument is the Fourier transform relating the survival probability (in time) of an initially prepared nonstationary state and the (frequency) autocorrelation function of the excitation spectrum. Strong repulsion of states, as in the Wigner surmise, leads to a rapid dephasing of the initially coherently prepared state. The rate and extent of sampling of phase space depend not only on the spacing distribution but also on the intensity fluctuations. The rate of dephasing is equal to that inferred from the width of the spectral autocorrelation function.
The rate and extent of the exploration of the available phase space of a bound quantum mechanical system are shown to depend on the repulsion of energy eigenstates. Central to the argument is the Fourier transform relating the survival probability (in time) of an initially prepared nonstationary state and the (frequency) autocorrelation function of the excitation spectrum. Strong repulsion of states, as in the Wigner surmise, leads to a rapid dephasing of the initially coherently prepared state. The rate and extent of sampling of phase space depend not only on the spacing distribution but also on the intensity fluctuations. The rate of dephasing is equal to that inferred from the width of the spectral autocorrelation function.
Nonstationary phase-space distributions \ensuremath{\rho}(t) are shown to sweep out phase space at a rate scrR which can be determined from the autocorrelation function, P(t)=hNTr[\ensuremath{\rho}\ensuremath{\rho}(t)]. It is demonstrated that, after the initial decay of P(t), subsequent recurrences (increases) of P(t) cause permanent slowdowns in the rate scrR. Such slowdowns may cause localization of quantum eigenstates, whereas no such implications apply to the classical stationary distributions. The localization is discussed in terms of the fraction of the available phase space which the dynamics accesses, and the phase-space distribution of individual eigenfunctions.
We present an analysis of quantum energy flow and localization in molecules from the point of view of a scaling approach. This scaling approach is based on earlier scaling theories of Anderson localization in disordered metals. The picture provides a simple general framework for describing both energy flow and the localization of eigenstates. This framework can be applied to molecules whose Hamiltonians are described by local random matrices, in which transitions between states nearby in quantum space occur more easily than those between distant states. In this theory the central quantity of interest is the survival probability. It is the squared overlap of a wave function at a later time with its initial condition. Its long time behavior varies depending on whether the eigenstates are localized or not. In the scaling description the survival probability approaches its long time limit varying inversely with time raised to some power. Different power laws are valid in different time regimes, depending on the degree of localization and the size of the molecule’s phase space. Near the localization transition we find that the survival probability decays as t−1. This decay is much slower than when energy flows easily, for a large number of dimensions participating in the energy flow.
Quantum mechanical wave function predicts probabilities of finding a ‘particle’ at different points in space, but at the time of detection a particle is detected only at one place. The question is: how this place gets decided, and can be predicted. To seek answer to this, we assume here that a ‘particle’ has some “diameter”, in stead of being a ‘point-particle’ of mathematical zero dimension; and depending upon the relative velocity between this particle and observer, its “diameter” experiences ‘Relativistic length-contraction’. Then we Fourier-transform this ‘length-contraction’ in ‘space-domain’ into ‘spectral-expansion’ ∆ω in ‘frequency-domain’, and find that momentum of a particle can be expressed as: m v = h ∆ω / 2 π c, and de Broglie’s wavelength, λB = 2 π c /∆ω ; as was derived in [ref.1 and 2. In the ref-2 it was shown that: in fact it is the ‘expansion of spectrum’ in the frequency-domain, which is the physical-cause for the Relativistic length-contraction.] Then we notice that the frequency-domain translation of the particle’s length in space-domain has a continuous spectrum; i.e. it contains a set of frequencies ranging from ωmax to ωmin . Therefore, as we found in ref. [3], this wide set of waves coherently add only at discrete points in space, and mutually nullify their amplitudes at rest of the places. And the place at which all the spectral-components of the wide band of waves will add constructively, will depend on the relative phase of all the spectral components. It is proposed here, that we need to know the relative phase angles of every spectral-component contained in the wide set of waves contained in the expanded wide band, for predicting the exact place of detection of the ‘particle’.
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.
Quantum speed limits set an upper bound to the rate at which a quantum system can evolve, and as such can be used to analyze the scrambling of information. To this end, we consider the survival probability of a thermofield double state under unitary time evolution which is related to the analytic continuation of the partition function. We provide an exponential lower bound to the survival probability with a rate governed by the inverse of the energy fluctuations of the initial state. Further, we elucidate universal features of the nonexponential behavior at short and long times of evolution that follow from the analytic properties of the survival probability and its Fourier transform, both for systems with a continuous and for systems with a discrete energy spectrum. We find the spectral form factor in a number of illustrative models; notably, we obtain the exact answer in the Gaussian unitary ensemble for any N with excellent agreement with recent numerical studies. We also discuss the relationship of our findings to models of black hole information loss, such as the Sachdev-Ye-Kitaev model dual to AdS2, as well as higher-dimensional versions of AdS/CFT.
Inspired by quantum cosmology, in which the wave function of the universe is annihilated by the total Hamiltonian, we consider the internal dynamics of a simple particle system in an energy eigenstate. Such a system does not possess a uniquely defined time parameter, and all physical questions about it must be posed without reference to time. We consider in particular the following question: what is the probability that the system's trajectory passes through a set of regions of configuration space without reference to time? We first consider the classical case, where the answer has a variety of forms in terms of a phase-space probability distribution function. We then consider the quantum case, and we analyze this question using the decoherent histories approach to quantum theory, adapted to questions which do not involve time. When the histories are decoherent, the probabilities approximately coincide with the classical case, with the phase-space probability distribution replaced by the Wigner function of the quantum state. For some initial states, decoherence requires an environment, and we compute the required influence functional and examine some of its properties. Special attention is given to the inner product used in the construction (the induced or Rieffel inner product), the construction of class operators describing the histories, and the extent to which reparametrization invariance is respected. Our results indicate that simple systems without an explicit time parameter may be quantized using the decoherent histories approach, with the expected classical limit extracted. The results support, for simple models, the usual heuristic proposals for the probability distribution function associated with a semiclassical wave function satisfying the Wheeler-DeWitt equation.