Statistical Mechanics - Physics at Oregon State University

Chapter 6
Density matrix formalism.
6.1 Density operators.
Different ways to look at statistical mechanics.
The theory of statistical mechanics is easily formulated in a quantum mechanical
frame-work. This is not the only possibility, however. All original
developments were, of course, in terms of classical mechanics. Different questions
arise and we will address some of those questions in the next chapter. But
even in quantum mechanics there are different ways to obtain information about
a system. Using the time-independent Schrödinger equation, like we did, is one
way. We can also use matrices, like Heisenberg, or more general, operators.
In statistical mechanics we start with a definition of the entropy. A different
formulation of statistical mechanics, as given in this chapter, essentially is based
on a different definition of the entropy. Our task is to show that the resulting
theories are equivalent. In order to make efficient use of the definition of entropy
introduced in this chapter we have to introduce density matrices.
Density matrices are an efficient tool facilitating the formulation of statistical
mechanics. A quantum mechanical description of a system requires you to solve
for the wave-functions of that system. Pure states are represented by a single
wave function, but an arbitrary state could also contain a mixture of wave
functions. This is true in statistical mechanics, where each quantum state is
possible, and the probability is given by the Boltzmann factor. Density matrices
are designed to describe these cases.
Suppose we have solved the eigenvalue system
H |n〉 = En |n〉 (6.1)
with normalization 〈n ′ | n〉 = δn ′ n and closure
n |n〉 〈n| = 1. If the probability
for the system of being in state |n〉 is pn, the expectation value of an arbitrary
operator A is given by
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