Discrete approaches have long been used in numerical modelling of physical systems in both research and teaching. Discrete versions of the Schrödinger equation employing either one or several basis functions per mesh point are often used by senior undergraduates and beginning graduate students in computational physics projects. In studying discrete models, students can encounter conceptual difficulties with the representation of the current and its divergence because different finite-difference expressions, all of which reduce to the current density in the continuous limit, measure different physical quantities. Understanding these different discrete currents is essential and requires a careful analysis of the current operator, the divergence of the current and the continuity equation. Here we develop point forms of the current and its divergence valid for an arbitrary mesh and basis. We show that in discrete models currents exist only along lines joining atomic sites (or mesh points). Using these results, we derive a discrete analogue of the divergence theorem and demonstrate probability conservation in a purely localized-basis approach.
Discrete approaches have long been used in numerical modelling of physical systems in both research and teaching. Discrete versions of the Schrödinger equation employing either one or several basis functions per mesh point are often used by senior undergraduates and beginning graduate students in computational physics projects. In studying discrete models, students can encounter conceptual difficulties with the representation of the current and its divergence because different finite-difference expressions, all of which reduce to the current density in the continuous limit, measure different physical quantities. Understanding these different discrete currents is essential and requires a careful analysis of the current operator, the divergence of the current and the continuity equation. Here we develop point forms of the current and its divergence valid for an arbitrary mesh and basis. We show that in discrete models currents exist only along lines joining atomic sites (or mesh points). Using these results, we derive a discrete analogue of the divergence theorem and demonstrate probability conservation in a purely localized-basis approach.
Discrete versions of continuous models are central to numerical calculations in physics and engineering. A very common problem in setting up a discrete model is how to handle derivatives. There are, for example, three common approximations for the first derivative, and each embeds different properties in the discrete model. Discretizing continuous expressions simplified using rules of calculus is especially problematic, since many different discretizations can stand for the same continuous expression depending on the stage of simplification at which the discretization is carried out. The problems are resolved by requiring that the discrete model satisfies discrete versions of the properties satisfied by the continuous original. We illustrate by using some examples from undergraduate-level one-dimensional quantum mechanics.
A standard way to solve a Schrödinger equation is to discreteize the radial coordinates and apply a numerical method for a differential equation, such as the Runge-Kutta method or the Numerov method. Here I employ a discrete basis formalism based on a finite mesh method as a simpler alternative, with which the numerical computation can be easily implemented by ordinary linear algebra operations. I compare the numerical convergence of the Numerov integration method to the finite mesh method for calculating penetrabilities of a one-dimensional potential barrier, and show that the latter approach has better convergence properties.
A technique for simulating quantum dynamics in phase space i discussed. It makes use of ensembles of classical trajectories to approximate the distribution functions and their derivatives by implementing Adaptive Kernel Density Estimation. It is found to improve the accuracy and stability of the simulations compared to more conventional particle methods. Formulation of the method in higher dimensions is straightforward. Complete analytical description of physical systems exist only for the most simple ideal cases. The vast majority of physical problems require the solution of approximate equations most fre- quently by using numerical calculations. To this end, much effort has been devoted to the devel- opment of suitable techniques to be used in computer simulations. The increasing computational power makes possible the description of more complicated, that is, less ideal systems allowing for more precise predictions of their properties as well as for a better understanding of the underlying phenomena. Among the wealth of numerical approaches to solve dynamical equations by means of discretization, particle methods have gained attention in the last few decades as a conceptu- ally appealing alternative. Conventional numerical solvers rely on a regular discretization of the independent variables to facilitate the evaluation of derivatives to a given order of accuracy. As a consequence, a mesh needs to be defined as the base of the simulations, parts of which remain empty at any given simulation time even with the most sophisticated mesh schemes. The cen- tral idea of particle methods consists of representing functions by sets of particles conveniently located and weighted such that the function is appropriately sampled by the particles. Evolution of the functions is then accomplished by an equivalent evolution of the particles. This meshless or Lagrangian type of discretization has some intrinsic computational efficiency advantages because there is no wasted space. The advantage, however, comes at the expense of having to evaluate derivatives of functions on unstructured arrays of sampling points. The issues about convergence and stability of the resulting algorithms have been the subject of active and fruitfull research. Currently there are some well established techniques to solve dynamical equations with particle methods. Our interest is to provide an alternative ingredient to further improve the performance of numerical solutions of dynamical problems of physical relevance. In the present work we discuss, in the context of particle methods, the implementation of a technique for representing functions and derivatives with adaptive parameters. Adaptiveness require the parameters to be automatically ad- justed to optimal values according to the location of the particles. As shown below, the proposed scheme is conceptually simple and has been tested in a number of different situations. In some simple test models the present work focuses on quantum dynamics, previously treated with more
In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.
The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.