arXiv subjects

Klaus Renziehausen

Publications and source records attributed to Klaus Renziehausen.

4 recordsLinked to original sources

arXiv2019

How to generalize the Ehrenfest theorem and the uncertainty principle

The Ehrenfest theorem and the Robertson uncertainty relation are well-known basic equations in quantum mechanics. However, there exist problematic cases, where the Ehrenfest theorem and the Robertson uncertainty relation are not correct. These cases occur when the azimuthal angle in polar or spherical coordinates is used within these equations. The purpose of this paper is to present and discuss a generalized Ehrenfest theorem and a generalized uncertainty relation which are still valid for these cases. Hereby, we define and use a mathematical operation called expectation commutator.

quant-ph↗

arXiv2019

The connection between Bohmian mechanics and many-particle quantum hydrodynamics

Bohm developed the Bohmian mechanics (BM), in which the Schrödinger equation is transformed into two differential equations: A continuity equation and an equation of motion similar to the Newtonian equation of motion. This transformation can be executed both for single-particle systems and for many-particle systems. Later, Kuzmenkov and Maksimov used basic quantum mechanics for the derivation of many-particle quantum hydrodynamics (MPQHD) including one differential equation for the mass balance and two differential equations for the momentum balance, and we extended their analysis in a prework [K. Renziehausen, I. Barth, Prog. Theor. Exp. Phys. 2018, 013A05 (2018)] for the case that the particle ensemble consists of different particle sorts. The purpose of this paper is to show how the differential equations of MPQHD can be derived for such a particle ensemble with the differential equations of BM as a starting point. Moreover, our discussion clarifies that the differential equations of MPQHD are more suitable for an analysis of many-particle systems than the differential equations of BM because the differential equations of MPQHD depend on a single position vector only while the differential equations of BM depend on the complete set of all particle coordinates.

quant-ph↗

arXiv2017

Many-particle quantum hydrodynamics: exact equations and pressure tensors

In the first part of this paper, the many-particle quantum hydrodynamics (MPQHD) equations for a system containing many particles of different sorts are derived exactly from the many-particle Schrödinger equation. It includes the derivation of the many-particle continuity equations (MPCE), many-particle Ehrenfest equations of motion (MPEEM), and many-particle quantum Cauchy equations (MPQCE) for any of the different particle sorts and for the total particle ensemble. The new point in our analysis is that we consider a set of arbitrary particles of different sorts in the system. In MPQCEs, there appears a quantity called pressure tensor. In the second part of this paper, we analyze two versions of this tensor in depth -- the Wyatt pressure tensor and the Kuzmenkov pressure tensor. There are different versions because there is a gauge freedom for the pressure tensor similar to that for potentials. We find that the interpretation of all quantities contributing to the Wyatt pressure tensor is understandable but for the Kuzmenkov tensor, it is difficult. Furthermore, the transformation from Cartesian coordinates to cylindrical coordinates for the Wyatt tensor can be done in a clear way, but for the Kuzmenkov tensor, it is rather cumbersome.

quant-ph↗

arXiv2012

On the divergence of time-dependent perturbation theory applied to laser-induced molecular transitions: Analytical calculations for the simple algorithm

Shaped laser pulses are a powerful tool to induce population transfer between electronic molecular states, and time-dependent perturbation theory is suitable for a description of such a transfer in weak external fields. The application of perturbation theory in numerical simulations of field matter interactions can lead to divergences. In a recent paper [K. Renziehausen et. al., J. Phys. B: At. Mol. Opt. Phys., 42:195402, 2009] we explained that the arising error in the norm of the wave function can be split into two parts. The first part is related to numerical errors caused by the discretisation of time that is required in the simulation and can be suppressed for a sufficiently small time step or abolished for an adequate numerical implementation of perturbation theory. The second part may cause divergences and is associated with the perturbative expansion order. We presented numerical evidence without any analytical proof. Here we are focussing on the derivation of analytical expressions to interpret the behavior of what we have called in the above mentioned paper 'simple algorithm'. The derivation of analytical expressions for the interpretation of what we have called in the above mentioned paper 'improved algorithm' are given in another paper [K. Renziehausen. arXiv, in preparation, 2012]. Moreover, we introduce here a gedankenexperiment to illustrate the influence of the different orders on the field-molecule interaction.

math-ph↗