Finite element descriptions of quantum mechanical systems

Nobuhiro YOSHIKAWA, Yasuyuki Mitsui, Osamu Kuwazuru +1 more

Finite Element Method， O ( N ) Methods

R. Goloskie, J. W. Kramer, L. R. Ram‐Mohan

Toby Young, George Maroulis, Theodore E. Simos

This paper presents a description of the abstractions required for the expression and solution of the linear single‐particle Schrödinger equation in a finite element basis. This paper consists of two disparate themes: First, to layout and establish the foundations of finite element analysis as an approximate numerical solution to extendable quantum mechanical systems; and second, to promote a high‐performance open‐source computational model for the approximate numerical solution to quantum mechanical systems. The structural foundation of the one‐and two‐dimensional time‐independent Schrödinger equation describing an infinite potential well is explored and a brief overview of the hierarchal design of the computational library written in C++ is given.

Carl M. Bender

Robert William Johnson

I describe, in the simplified context of finite groups and their representations, a mathematical model for a physical system that contains both its quantum and classical aspects. The physically observable system is associated with the space containing elements fxf for f an element in the regular representation of a given finite group G. The Hermitian portion of fxf is the Wigner distribution of f whose convolution with a test function leads to a mathematical description of the quantum measurement process. Starting with the Jacobi group that is formed from the semidirect product of the Heisenberg group with its automorphism group SL(2,F{N}) for N an odd prime number I show that the classical phase space is the first order term in a series of subspaces of the Hermitian portion of fxf that are stable under SL(2,F{N}). I define a derivative that is analogous to a pseudodifferential operator to enable a treatment that parallels the continuum case. I give a new derivation of the Schrodinger-Weil representation of the Jacobi group. Keywords: quantum mechanics, finite group, metaplectic. PACS: 03.65.Fd; 02.10.De; 03.65.Ta.

Domenico D'Alessandro

Quantum systems whose dynamics depends on one or more control functions will be named quantum control systems. In a typical situation the objects of control are charged particles while the control functions are appropriately shaped electromagnetic fields. Therefore, to derive the models, in this chapter we describe elements of the theory of field-particle interaction. We start with the classical theory of electrodynamics and show how to obtain the quantum mechanical description through a process called canonical quantization. We present physical examples of (finite dimensional) quantum control systems of interest in applications and we highlight the fact that such systems have a common mathematical structure, as bilinear systems whose evolution is determined by an element of the Lie group of unitary matrices. In a parallel to classical control theory, we identify a state, an input, and an output in our description of quantum control systems.