396 Quantum ensembles
solution grows exponentially with the number of degrees of freedom. If eqn. (10.1.4)
were to be solved on a spatial grid withMpoints along each spatial direction, then
the total number of points needed would beM^3 N. Thus, even on a very coarse grid
with justM= 10 points, forN∼ 1023 particles, the total number of grid points would
be on the order of 10^10
23
points! But even for a small molecule of justN= 10 atoms
in the gas phase, after we subtract out translations and rotations, Ψ is still a function
of 24 coordinates and time. The size of the grid needed to solve eqn.(10.1.4) is large
enough that the calculation is beyond the capability of current computing resources.
The same is true for theN-particle eigenvalue equation
[
−
̄h^2
2 m
∑N
i=1∇^2 i+U(r 1 ,...,rN)]
ψ{k,ms}(x 1 ,...,xN) =E{k,ms}ψ{k,ms}(x 1 ,....,xN)(10.1.5)
(see Problem 9.9). Here{k,ms}≡k 1 ,...,kN,ms 1 ,...,msN are the 4Nquantum num-
bers, includingSˆzvalues, needed to characterize the eigenfunctions and eigenvalues. In
fact, explicit solution of the eigenvalue equation for just 4 or 5 particles is considered
atour de forcecalculation. While such calculations yield a wealth of highly accurate
dynamical information about small systems, if one wishes to move beyond the limits
of the Schr ̈odinger equation and the explicit calculation of the eigenvalues and eigen-
functions ofHˆ, statistical methods are needed. Now that we have a handle on the
magnitude of the many-body quantum mechanical problem, we proceed to introduce
the basic principles of quantum equilibrium ensemble theory.
10.2 The ensemble density matrix
Quantum ensembles are conceptually very much like their classical counterparts. Our
treatment here will follow somewhat the development presented byFeynman (1998).
We begin by considering a collection ofZquantum systems, each with a unique state
vector|Ψ(λ)〉,λ= 1,...,Z, corresponding to a unique microscopic state. At this stage,
we imagine that our quantum ensemble is frozen in time, so that the state vectors are
fixed. (In Section 10.3 below, we will see how the ensemble develops in time.) As in the
classical case, it is assumed that the microscopic states of the ensemble are consistent
with a set of macroscopic thermodynamic observables, such as temperature, pressure,
chemical potential, etc. The principal goal is to predict observables in the form of
expectation values. Therefore, we define the expectation value of an operatorAˆas the
ensemble average of expectation values with respect to each microscopic state in the
ensemble. That is,
〈Aˆ〉=1
Z
∑Z
λ=1〈Ψ(λ)|Aˆ|Ψ(λ)〉. (10.2.1)Since each state vector is an abstract object, it proves useful to work in a particular
basis. Thus, we introduce a complete set of orthonormal vectors|φk〉on the Hilbert
space and expand each state of the ensemble in this basis accordingto
|Ψ(λ)〉=∑
kC(kλ)|φk〉, (10.2.2)