Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTime in quantum mechanics: A fresh look at the continuity equation
176
0
0.0
(
0
)
Authors
Axel Schild
Ask ChatGPT about the research
No Arabic abstract
The local conservation of a physical quantity whose distribution changes with time is mathematically described by the continuity equation. The corresponding time parameter, however, is defined with respect to an idealized classical clock. We consider what happens when this classical time is replaced by a non-relativistic quantum-mechanical description of the clock. From the clock-dependent Schrodinger equation (as analogue of the time-dependent Schrodinger equation) we derive a continuity equation, where, instead of a time-derivative, an operator occurs that depends on the flux (probability current) density of the clock. This clock-dependent continuity equation can be used to analyze the dynamics of a quantum system and to study degrees of freedom that may be used as internal clocks for an approximate description of the dynamics of the remaining degrees of freedom. As an illustration, we study a simple model for coupled electron-nuclear dynamics and interpret the nuclei as quantum clock for the electronic motion. We find that whenever the Born-Oppenheimer approximation is valid, the continuity equation shows that the nuclei are the only relevant clock for the electrons.
rate research
Read More
Quantum hydrodynamics is a formulation of quantum mechanics based on the probability density and flux (current) density of a quantum system. It can be used to define trajectories which allow for a particle-based interpretation of quantum mechanics, commonly known as Bohmian mechanics. However, quantum hydrodynamics rests on the usual time-dependent formulation of quantum mechanics where time appears as a parameter. This parameter describes the correlation of the state of the quantum system with an external system -- a clock -- which behaves according to classical mechanics. With the Exact Factorization of a quantum system into a marginal and a conditional system, quantum mechanics and hence quantum hydrodynamics can be generalized for quantum clocks. In this article, the theory is developed and it is shown that trajectories for the quantum system can still be defined, and that these trajectories depend conditionally on the trajectory of the clock. Such trajectories are not only interesting from a fundamental point of view, but they can also find practical applications whenever a dynamics relative to an external time parameter is composed of fast and slow degrees of freedom and the interest is in the fast ones, while quantum effects of the slow ones (like a branching of the wavepacket) cannot be neglected. As an illustration, time- and clock-dependent trajectories are calculated for a model system of a non-adiabatic dynamics, where an electron is the quantum system, a nucleus is the quantum clock, and an external time parameter is provided, e.g. via an interaction with a laser field that is not treated explicitly.
We take a closer and new look at the effects of tidal forces on the free fall of a quantum particle inside a spherically symmetric gravitational field. We derive the corresponding Schrodinger equation for the particle by starting from the fully relativistic Klein-Gordon equation in order (i) to briefly discuss the issue of the equivalence principle and (ii) to be able to compare the relativistic terms in the equation to the tidal-force terms. To the second order of the nonrelativistic approximation, the resulting Schrodinger equation is that of a simple harmonic oscillator in the horizontal direction and that of an inverted harmonic oscillator in the vertical direction. Two methods are used for solving the equation in the vertical direction. The first method is based on a fixed boundary condition, and yields a discrete-energy spectrum with a wavefunction that is asymptotic to that of a particle in a linear gravitational field. The second method is based on time-varying boundary conditions and yields a quantized-energy spectrum that is decaying in time. Moving on to a freely-falling reference frame, we derive the corresponding time-dependent energy spectrum. The effects of tidal forces yield an expectation value for the Hamiltonian and a relative change in time of a wavepackets width that are mass-independent. The equivalence principle, which we understand here as the empirical equivalence between gravitation and inertia, is discussed based on these various results. For completeness, we briefly discuss the consequences expected to be obtained for a Bose-Einstein condensate or a superfluid in free fall using the nonlinear Gross-Pitaevskii equation.
We discuss the hot hand paradox within the framework of the backward Kolmogorov equation. We use this approach to understand the apparently paradoxical features of the statistics of fixed-length sequences of heads and tails upon repeated fair coin flips. In particular, we compute the average waiting time for the appearance of specific sequences. For sequences of length 2, the average time until the appearance of the sequence HH (heads-heads) equals 6, while the waiting time for the sequence HT (heads-tails) equals 4. These results require a few simple calculational steps by the Kolmogorov approach. We also give complete results for sequences of lengths 3, 4, and 5; the extension to longer sequences is straightforward (albeit more tedious). Finally, we compute the waiting times $T_{nrm H}$ for an arbitrary length sequences of all heads and $T_{nrm(HT)}$ for the sequence of alternating heads and tails. For large $n$, $T_{2nrm H}sim 3 T_{nrm(HT)}$.
The apparent unification of gauge couplings around 10^16 GeV is one of the strong arguments in favor of Supersymmetric extensions of the Standard Model (SM). In this contribution a new analysis, using the latest experimental data, is performed. The strong coupling alpha_s emerges as the key factor for evaluating the results of the fits, as the experimental and theoretical uncertainties in its measurements are substantially higher than for the electromagnetic and weak couplings. The present analysis pays special attention to numerical and statistical details. The results, combined with the current limits on the supersymmetric particle masses, favor a value for the SUSY scale <~ 150 GeV and for alpha_s = 0.118-0.119.
An alternative approach to the calculation of tunneling actions, that control the exponential suppression of the decay of metastable phases, is presented. The new method circumvents the use of bounces in Euclidean space by introducing an auxiliary function, a tunneling potential $V_t$ that connects smoothly the metastable and stable phases of the field potential $V$. The tunneling action is obtained as the integral in field space of an action density that is a simple function of $V_t$ and $V$. This compact expression can be considered as a generalization of the thin-wall action to arbitrary potentials and allows a fast numerical evaluation with a precision below the percent level for typical potentials. The method can also be used to generate potentials with analytic tunneling solutions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
