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Teubner-Texte zur Physik

Density Matrices and Density Operators

H. Eschrig

Published Jan 1, 1996• 0 citations
Publisher

Abstract

As was repeatedly seen in the last chapter, for computing most of the usual matrix elements only a few of the many coordinates of an N -particle wavefunction are used, where due to the symmetry properties it is irrelevant which ones. Density matrices are a tool to extract the relevant information out of such monstrous constructs as are N -particle wavefunctions. (For reviews, see [Erdahl and Smith, 1987, Coleman, 1963, McWeeny, 1960]; in the text below we try to keep the notation as canonical as possible.) These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-322-97620-8_3

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As was repeatedly seen in the last chapter, for computing most of the usual matrix elements only a few of the many coordinates of an N -particle wavefunction are used, where due to the symmetry properties it is irrelevant which ones. Density matrices are a tool to extract the relevant information out of such monstrous constructs as are N -particle wavefunctions. (For reviews, see [Erdahl and Smith, 1987, Coleman, 1963, McWeeny, 1960]; in the text below we try to keep the notation as canonical as possible.) These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
After having started in Chapter 1 with the N -particle wavefunction, in the preceding chapter it was shown that the single-particle and two-particle density matrices suffice to calculate most groundstate properties of a many-body system described by the Hamiltonian (1.4). If one were able to classify the set of all admissible two-particle density matrices—the corresponding single-particle density matrices follow from them via (2.13)—then the groundstate energy, E , of the many-body system could be obtained as the absolute minimum of (2.46) over this set in a variational way, without reference to many-body wavefunctions as needed in (1.3). These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In the formalism of reduced density matrices and functions, using the irreducible tensor-operator technique and the space-spin separation scheme, the matrix elements of one of the main spin-relativistic corrections of the Breit-Pauli Hamiltonian, the spin-orbit interactions, are expressed in a form suitable for numeric implementation. A comparison with other methods is made and the advantages of such an approach are discussed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.
The FT has many applications, particularly in the fields of quantum computation and quantum information. In these fields, the FT is often called the quantum Fourier transform . Traditionally, notation used in quantum physics (i.e., the Dirac notation) to denote vectors is different from that used in mathematics. In this chapter, we introduce the Dirac notation and describe the FT in terms of the new notation. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.