Page 43 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

Quantum mechanical density matrix 29

The interchange of the order of summation and integration should be proved to be
legitimate in a rigorous treatment. If we define
1 τ
∗
ρ ν ν = lim a (t)a ν (t)dt (2.7)

ν
τ→∞ τ 0
then (2.6) becomes
¯ ¯
φ = ρ ν ν φ νν (2.8)

ν,ν
or in matrix notation
¯ ¯
φ = Trρφ (2.9)
where ρ and φ are the (generally infinite dimensional) square matrices ρ ν ν and

φ νν . (The inversion of the order of indices in (2.7) is made in order to permit

(2.8) to be written as a matrix multiplication.) Tr means trace. Equation (2.7) is the
quantum version of (1.15) while (2.9) is the quantum form of (1.16). The matrix
ρ is called the density matrix. Just as the classical distribution function can be
described in terms of various sets of canonical coordinates and momenta related
by contact transformations, the density matrix can be expressed in terms of various
complete sets of orthonormal functions which span the space of wave functions and
thesearerelatedbyunitarytransformations.Justastheclassicalρ(q, p)dependedin
principle on all of the initial classical conditions p 0 , q 0 , the density matrix depends,
again in principle, on the initial wave function ψ(q, t 0 ) or, equivalently, on all of the
coefficients a ν (t = 0). Here we come, however, to a significant practical difference:
whereas in the classical case, the amount of data associated with specifying the
initial conditions, while large, is in principle countable and finite (6N numbers),
the initial condition specifying the wave function requires at the numerical level
an uncountably large amount of data, even for a system with a modest number of
particles. This apparently academic distinction has the very practical effect that
simulations of classical systems of thousands of particles are feasible while for
quantum systems they are extremely difficult and not very reliable, even for simple
systems.
As in the classical case, the density matrix may be interpreted in various ways.
For example, one obtains the probability interpretation if one supposes only that
( j) ( j)
the probability that the system has wave function (q) = a ν ψ ν (q)is P j ,
ν
thus avoiding any assumptions about the dynamics (but consistent with the known
¯ ¯
dynamics of quantum mechanics). Then the value of the double average φ would
be

¯ ¯ ( j)∗ ( j) 3N
φ = P j (q)φ op (q)d q = ρ ν ν φ νν (2.10)

j ν,ν