Time in quantum mechanics: A fresh look at the continuity equation

This paper considers in detail the empirical two-band model formulated by Nelson et al. [Phys. Rev. B 35, 7770 (1987)] for the electronic states of semiconductor quantum wells and superlattices. The model is also extended to the case where the structure of interest is in an external potential. It is shown that one can define probability and probability current densities such that a continuity equation is satisfied and that solutions corresponding to different energies are orthogonal. Expressions are derived for the oscillator strengths of interband transitions and of intersubband transitions within the conduction band. The pair of coupled first-order differential equations resulting from the model can be recast into a single, second-order Schr\"odinger equation with an energy- and position-dependent effective mass. For a uniform electric field, it is shown that analytic solutions to this equation can be obtained with an error of order (\ensuremath{\gamma}F${)}^{2}$, where \ensuremath{\gamma} is the nonparabolicity parameter and F is the electric field. For a 200-\AA{} rectangular GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As quantum well, results are presented for electric-field-dependent conduction-subband energies, envelope functions, interband oscillator strengths, and tunneling resonance widths. These results are compared with the corresponding results obtained by a direct numerical integration of the two-band-model Schr\"odinger equation and with results obtained using the single-band envelope-function approximation.

Many theories of quantum gravity suggest that there should exist a minimal scale with which it is possible to measure distances and time. We present a formalism of quantum mechanics exchibiting a minimal length and time scale. The work is based on the Page-Wootters formalism which gives a possible solution to the problem of time in quantum mechanics. This formalism is an example of a relational approach to quantum dynamics. It is based on the idea that reference frames with respect to which motion is described are themselves physical systems interacting with the degrees of freedom they wish to describe. Quantum dynamics is described in terms of quantum correlations between a clock and a system. In our work we modify the commutation relations between the time and frequency operators leading to a minimal uncertainty of time measurement. This results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We further investigate the e ects of such modi cation on a couple simple quantum systems.

We analyze the effect of pairing on particle transport in time-dependent theories based on the Hartree-Fock-Bogoliubov (HFB) or BCS approximations. The equations of motion for the HFB density matrices are unique and the theory respects the usual conservation laws defined by commutators of the conserved quantity with the Hamiltonian. In contrast, the theories based on the BCS approximation are more problematic. In the usual formulation of time-dependent Hartree-Fock $(\mathrm{TDHF})+\mathrm{BCS}$, the equation of continuity is violated and one sees unphysical oscillations in particle densities. This can be ameliorated by freezing the occupation numbers during the evolution in $\mathrm{TDHF}+\mathrm{BCS}$, but there are other problems with the BCS approximation that make it doubtful for reaction dynamics. We also compare different numerical implementations of the time-dependent HFB equations. The equations of motion for the $U$ and $V$ Bogoliubov transformations are not unique, but it appears that the usual formulation is also the most efficient. Finally, we compare the time-dependent HFB solutions with numerically exact solutions of the two-particle Schr\"odinger equation. Depending on the treatment of the initial state, the HFB dynamics produces a particle emission rate at short times similar to that of the Schr\"odinger equation. At long times, the total particle emission can be quite different, due to inherent mean-field approximation of the HFB theory.

Finding ‘‘real’’ time in quantum mechanics

Local conservation of probability, expressed as the continuity equation, is a central feature of non-equilibrium Statistical Mechanics. In the existing literature, the continuity equation is always motivated by heuristic arguments with no derivation from first principles. In this work we show that the continuity equation is a logical consequence of the laws of probability and the application of the formalism of inference over paths for dynamical systems. That is, the simple postulate that a system moves continuously through time following paths implies the continuity equation. The translation between the language of dynamical paths to the usual representation in terms of probability densities of states is performed by means of an identity derived from Bayes’ theorem. The formalism presented here is valid independently of the nature of the system studied: it is applicable to physical systems and also to more abstract dynamics such as financial indicators, population dynamics in ecology among others.

Time has always been a fundamental yet enigmatic concept in physics. From the arrow of time in thermodynamics to the timeless nature of quantum mechanics, understanding time's essence has challenged the foundations of modern physics. In this paper, we propose a novel interpretation of time in quantum mechanics, distinguishing between real time as observable events (wave function collapse or address changes) and imaginary time as the continuous evolution of the quantum state. By connecting this framework with the holographic principle, we aim to provide insights into the nature of time, its arrow, and its relationship with quantum mechanics. Keywords Thermodynamics; Wave function; Quantum mechanics; Holographic principle

Notes from the lectures by the author at the 7th Jerusalem Winter School 1990 on Quantum Cosmology and Baby Universes. The lectures covered quantum mechanics for closed systems like the universe, generalized quantum mechanics, time in quantum mechanics, the quantum mechanics spacetime, and practical quantum cosmology. References have not been updated.

Relativistic equations were derived of the motion of a continuous, electrically and mesically charged substance obeying the simple diffusion law and moving in a constant external electrical scalar potential. The relevant equation of continuity is presupposed with the right side different from zero, corresponding to the internal sources of matter. It is shown that if the matter distribution satisfies this equation of continuity, the gradient of the external potential is just compensated by the inner potential gradient of the electrically charged matter. In this case, the acceleration of the substance is zero and all dynamical variables are time independent. Thus, the validity of the equation of continuity has to be considered as a necessary and sufficient condition for the existence of steady, time independent distribution of the continuous substance density. Our considerations are limited to continua with spins equal zero because the osmotic velocity satisfies the simple diffusion law. The simplest way to find a solution of the nonlinear equation of continuity is its linearization, leading directly to the Schrödinger equation for steady bound states, from which the eigenvalues can be obtained. Substituting the latter into the nonlinear equation of continuity, one can reveal its solution.

Time in quantum mechanics

We suggest a mathematical potential well with spherical symmetry and apply to the 1d Schr\"odinger equation. We use some well-known techniques as Stationary Perturbation Theory and WKB to gain insight into the solutions and compare them to each other. Finally, we solve the 1d Schr\"odinger equation using a numerical approach with the so-called Numerov technique for comparison. It can be a good exercise for undergrad students to grasp the above-cited techniques in a quantum mechanics course.

Several modifications of the Schr\"odinger equation that dynamically reduce the off-diagonal elements of the density operator in the position representation (localization) have received recent attention in the literature. Models of quantum open systems provide localization for the system of interest by considering the effects of an environment on the effective master equation that describes the dynamics of the reduced density operator for the system of interest. In contrast, localization is introduced as a fundamental modification to the Schr\"odinger equation in quantum mechanics with spontaneous localization (QMSL) [G. C. Ghirardi, A. Rhimini, and T. Weber, Phys. Rev. D 34, 470 (1986)]. We compare several environment models and discuss some of the anomalous features of the effective master equations noted by Ballentine [L. E. Ballentine, Phys. Rev. A 43, 9 (1991)] by examining the Ehrenfest relations for position, energy, and momentum. Insight is provided into the relative importance of various terms in the effective master equations for several environment models. We also briefly discuss the implications of similar anomalous features present in the QMSL master equation in light of our analysis of the open-system models.

An analysis of the Hamiltonian of a molecule interacting with time-dependent external fields leads to the identification of various time scales governing the motion of the electrons and the nuclei. The most important role is played by the Born-Oppenheimer approximation, which considers the nuclear kinetic energy to be bounded and small compared to the gradient of the electronic-potential surfaces. The relative electronic-excitation operator is defined and is shown to be slowly dependent on time. This is a consequence of the Born-Oppenheimer approximation in the context of the time-dependent Schr\"odinger equation and the interaction with an external electric field. We show how to capitalize on this identified slow motion in the presence of rapid Rabi oscillations driven by the external field. The stroboscopic-integration technique is used in an illustrative calculation for the case of two electronic states coupled by the external field.

We systematically study the influence of laser intensities on the dissociation of $\mathrm{H}{}_{2}^{+}$ in strong laser fields by numerically solving the time-dependent Schr\"odinger equation. In Ti:sapphire infrared (IR) laser fields, $\mathrm{H}{}_{2}^{+}$ mainly dissociates along $1s{\ensuremath{\sigma}}_{g}$ and $2p{\ensuremath{\sigma}}_{u}$ states. The mixture of these dissociative states results in the laser-intensity-dependent electron asymmetric localization. When the dissociation process is dressed by a laser field with a very long wavelength, the potential surfaces $1s{\ensuremath{\sigma}}_{g}$ and $2p{\ensuremath{\sigma}}_{u}$ degenerate due to the Stark effect, hence the kinetic energy release is changed. When $\mathrm{H}{}_{2}^{+}$ is exposed to a strong ultraviolet (UV) pulse, $\mathrm{H}{}_{2}^{+}$ is pumped to several excited states by absorbing a few energetic photons, resulting in the molecular dissociation ending with several kinetic energy peaks. The discrepancy of simulation results from the numerical models with and without Born-Oppenheimer approximation are analyzed.

We investigate spin effects in four-quark systems consisting of two heavy anti-bottom quarks and two light up/down quarks. To this end we use the Born-Oppenheimer approximation. We utilize potentials of two static antiquarks in the presence of two quarks of finite mass computed via lattice QCD and solve a coupled-channel Schr\"odinger equation for the anti-bottom-anti-bottom separation. Without taking heavy quark spins into account this approach predicted a $u d \bar b \bar b$ tetraquark bound state with quantum numbers $I(J^P) = 0(1^+)$. We now extend this Born-Oppenheimer approach with coupled channel Schr\"odinger equations allowing us to incorporate effects due to the heavy $\bar b$ spins. We confirm the existence of the $u d \bar b \bar b$ tetraquark.

The multiphoton dissociation branching ratios for ${{\text{H}}_{2}}^{+}$ and ${{\text{D}}_{2}}^{+}$ as a function of laser peak intensity and pulse length are investigated by solving the time-dependent Schr\"odinger equation in the Born-Oppenheimer approximation, neglecting nuclear rotation. An 800 nm laser pulse with peak intensities from $8\ifmmode\times\else\texttimes\fi{}{10}^{9}\text{ }\text{W}/{\text{cm}}^{2}$ to ${10}^{14}\text{ }\text{W}/{\text{cm}}^{2}$ and pulse lengths from 5 to 7.5 fs is used. We also investigate the viability of identifying zero-, one-, two-, and three-photon processes based only on the nuclear kinetic energy release spectrum, and check these identifications with a rigorous Floquet-like method.