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CRC Press eBooks

The Density Matrix

Michael J.R. Hoch

Published Apr 19, 2016• 0 citations
Publisher

Abstract

It is convenient to consider the products Cj*Ci as forming a matrix representation of an operator P, with matrix elements 〈 〉i P j C Cj i= ∗ . Substitution for Cj*Ci in Equation 18.1 gives 〈 〉 〈 〉〈 〉 〈 〉A i P j j A i i PA i PA AP = = = =∑ ∑ , ( ) ( ).Tr Tr (18.2) Tr denotes the diagonal sum of the matrix elements. Equations 18.1 and 18.2 serve as our starting point for the introduction of the density matrix operator.

DOI
10.1201/9781439850541-22

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It is convenient to consider the products CjCi as forming a matrix representation of an operator P, with matrix elements 〈 〉i P j C Cj i= ∗ . Substitution for CjCi in Equation 18.1 gives 〈 〉 〈 〉〈 〉 〈 〉A i P j j A i i PA i PA AP = = = =∑ ∑ , ( ) ( ).Tr Tr (18.2) Tr denotes the diagonal sum of the matrix elements. Equations 18.1 and 18.2 serve as our starting point for the introduction of the density matrix operator.
Up until this point, we have concerned ourselves with a description of quantum mechanics centered upon how a state |ψ〉 evolves in time. Using this we can show that the expectation value of an operator evolves as 〈A(t)〉 = 〈ψ(t)| ˆA|ψ(t)〉 (6.1) However, for many instances, especially if we are interested in describing relaxation processes, it is useful to introduce the density operator ρˆ(t) = |ψ(t)〉〈ψ(t)| (6.2) taken as the outer product of the state vector with itself. From this definition we can write ρˆ = ∑ mn c∗ncm |m〉〈n| (6.3) = ∑ mn ρmn|m〉〈n| (6.4) where the ρmn are the density matrix elements. Expectation values of operator are then given by the trace 〈A(t)〉 = ∑ mn Amnρmn = Tr [ ˆAρ(t)] (6.5) where Tr [ ˆAρ(t)] denotes the trace operation: Tr [ ˆAρ(t)] = ∑ nn′ If ρ is diagonal, then Tr [ ˆAρ(t)] = ∑ Annρnn = ∑ 〈n|A|n〉Pn where Pn = ρnn is the statistical probability of finding the system in state n. These statistical weights must be such that Pn ≤ 1 and∑ Pn = 1 Hence, we conclude that knowing ρ we can compute the statistical average of an operator A.
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.
This paper examines attenuation processes from t h e view- point of c o l l i s i o n theory, using the o p t i c a l theorem t o connect c a t t e n u a t i o n cross-section CJ and r e f r a c t i v i t y n with diagonal elements of the T matrix.This approach provides resonance prof i l e s (for '*natural*' l i n e s shapes) of the form where t h e p r o f i l e parameters A, B, C , D, r, w 0 are given i n t e r m s of atomic matrix elements.P a r t I reviews the notion of resonances.Part I1 summarizes the r e l e v a n t r e s u l t s of c o l l i s i o n theory, s t r e s s i n g physical ' 1
A new type of ``intermediate'' state, \ensuremath{\Vert}\ensuremath{\Psi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{J}}$〉, is introduced in the representation of many-body propagators and related resolvent matrix elements. A simple algebraic procedure is presented for constructing the unitary transformation matrix Q that relates the intermediate states and the exact energy eigenstates, \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$〉, of the interacting system. Here the starting point is the matrix X of generalized spectroscopic amplitudes 〈${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$\ensuremath{\Vert}C${\mathrm{^}}_{\mathit{J}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉, where C${\mathrm{^}}_{\mathit{J}}$ denotes a physical excitation operator and \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉 is the N-electron ground state. A block QR decomposition [G. H. Golub and C. F. von Loan, MatrixB Computations (Johns Hopkins University, Baltimore, 1989)] of X according to the equation X=Q F allows one to determine explicit expressions for the subblocks of Q and the intermediate propagator representations constituted by a nondiagonal effective interaction matrix C and an effective spectroscopic matrix f. These effective quantities f and C are formulated entirely in terms of ground-state density-matrix elements and related energy expectation values. The relevance of the intermediate representations as a means for deriving computational schemes is based on the regularity and compactness of the perturbation expansions for the effective matrices f and C. These basic properties also establish that the intermediate representations are closed-form versions of the algebraic-diagrammatic construction approximation schemes derived previously as a reformulation of the original diagrammatic propagator perturbation series. The intermediate representations represent a link between algebraic and diagrammatic approaches in the field of propagator methods and are expected to be useful also in the development of nonperturbative approximations.
The methods developed by Prigogine and his colleagues are applied to the study of quantum mechanical systems. The density matrix is divided into a slowly varying and an oscillatory part, and asymptotic equations are established for the elements of this matrix. In the representation in which the unperturbed hamiltonian is diagonal, these equations take a particularly simple form for weakly coupled systems: the matrix elements lying on diagonals parallel to the principal diagonal undergo separate transformations. A particular case is that of the diagonal elements themselves; for these the well-known Pauli equations are obtained. As an illustrative example the problem of the frictional forces on an oscillator immersed in a thermostat is considered in detail. The frictional forces introduced by Langevin and Lorentz are special cases of this.