SECTION VII. The Density Matrix, Reactions and Related Processes, and Introduction to Irreversible Thermodynamics
The Density Matrix
Introduction
In dealing with systems whose eigenstates are known, such as ideal spin systems in a magnetic field, it is sometimes useful to introduce a formalism involving the density matrix. Knowledge of the density matrix operator permits the expectation values of other operators to be determined in a way that is straightforward in principle. Weshow below that the ensemble average expectation value for an operator A of a system is given by (A) = Tr(pA), where p is the density matrix and Tr is the trace or diagonal sum of the product matrix. The density matrix contains statistical information about the system.
The density matrix is the quantum mechanical analogue of the classical phase space density of representative points description of an ensemble of systems introduced in Chapter 4. For systems consisting of large numbers of particles, the density matrix will, in general, be very large. To simplify matters, systems for which the density matrix can be written in compact form are used to introduce the subject. Generalization to other systems follows directly. The basic ideas are developed with application to an ideal spin system. The form of the density matrix for the ideal gas case is briefly considered. The density matrix approach is extremely powerful and can, for example, provide insight into how systems that are perturbed in some controlled way tend toward equilibrium. Simple examples involving particle beams are used to illustrate the method.
The Density Matrix Formalism
The density matrix formalism emerges from the basic quantum mechanical ideas that are briefly reviewed here. Consider a large system with energy in the range from E to E+dE. Let a particular eigenstate of the system be ф). This state may be written in terms of the complete set of eigenstates I/) of the Hamiltonian of the system as I
It is convenient to consider the products Cj C, as forming a matrix representation of an operator P, with matrix elements (iPj)C]Ci. Substitution forCJC, in Equation 18.1 gives
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.
In statistical physics, we are generally interested in ensemble averages, which for the operators we consider are denoted as ^A^. From Equation 18.1, we obtain
The ensemble average of the products C)C, gives the matrix representation of the density matrix
operator p, with elements (ipj) = C,C* . p is the ensemble average of the operator P that is introduced in Equation 18.2. Examples of the form of the density matrix in the various statistical ensembles are given in Section 18.3. Equation 18.3 becomes
Note that p is an Hermitian operator with (i Ipj) = (j Ipl/)*. If
i
Equation 18.4 provides the basis for density matrix calculations.
For systems that are not in equilibrium, the density matrix is time dependent, and it is therefore useful to allow for this possibility in our discussion. The time-dependent Schrodinger equation is ifid/ dt
Hamiltonian to be time dependent. Substitution for |(p'j results in /Л^Э/ df(c, |y')j = ^C, Jf(r)|y).
i j
Multiplying by (i I and making use of the orthonormality property give
Now,
and with Equation 18.5 substituted into Equation 18.6, we obtain
In operator form, the evolution of the density matrix with time is described by the equation
where [p, Jf] is the commutator of the two operators.
When the Hamiltonian H is time independent, the solution to Equation 18.7 is
where p(0) is the density matrix at time / = 0.
Exercise 18.1
Show by expanding the exponential operators that Equation 18.8 is a solution to Equation 18.7.
The exponential operators are expanded as follows:
e±(i/K)M _ i+(j7s)jfH-(l/2!)(/7S)2 J{2t2 ±.....Differentiation of p(t) with respect to time gives
Returning to exponential form, we obtain d/d p (t) = i/Й [p, J{], as required.
From Equation 18.8, the matrix elements of p are (f|p(0|./) = or in terms
of the wave functions, we have ju“p(t)ujdx = j р(0)е("г')м'и,б1. With the use of the
series expansion of the exponential operator, it follows that
Equation 18.9 shows that the diagonal elements of the density matrix are time independent, provided Л is time independent, whereas the off-diagonal elements oscillate with frequencies
(0,j=(Ej -E,)/h.
For a system in equilibrium, no time dependence of observable properties is expected. This implies that for a system in equilibrium, all off-diagonal elements of p(0) are zero, that is, (ip{0)j) = 0 for all /V/ and consequently (ip(t)j) = 0 for all times t. By separating the real and imaginary parts, the coefficients C, in Equation 18.3 may be written as C,elal, where a, is а
phase factor. The ensemble average of the product of coefficients is C,Cj = |С,||Су|И“-'~“' If
the phases are assumed to be statistically independent, C,C) = |p| y) = 0. Vanishing of the off- diagonal elements of the density matrix is a consequence of what is called the random phase hypothesis, according to which there is no phase correlation between different members of the large ensemble.
Following the introduction to the density matrix given in this section, which is based on fundamental quantum mechanical concepts and relationships, we are in a position to apply the formalism to various systems such as the ideal gas and the ideal spin system. Before we do this, we consider the form that the density matrix takes for the three statistical ensembles introduced in Chapter 10.
