10 Quantum ensembles and the density matrix
10.1 The difficulty of many-body quantum mechanics
We begin our discussion of the quantum equilibrium ensembles by considering a system
ofNidentical particles in a container of volumeV. This is the same setup we studied in
Section 3.1 in developing the classical ensembles. In principle, the physical properties
of such a large quantum system can be obtained by solving the full time-dependent
Schr ̈odinger equation. Suppose the Hamiltonian of the system is
Hˆ=
∑N
i=1ˆp^2 i
2 m+U(ˆr 1 ,...,ˆrN). (10.1.1)Inddimensions, there will bedN position and momentum operators. All the posi-
tion operators commute with each other as do all of the momentum operators. The
commutation rule between position and momentum operators is
[ˆriα,pˆjβ] =i ̄hδijδαβ, (10.1.2)whereαandβindex thedspatial directions andiandjindex the particle number.
Given the commutation rules, the many-particle coordinate and momentum eigen-
vectors are direct products (also calledtensor products) of the eigenvectors of the
individual operators. For example, a many-particle coordinate eigenvector in three
dimensions is
|x 1 y 1 z 1 ···xNyNzN〉=|x 1 〉⊗|y 1 〉⊗|z 1 〉···|xN〉⊗|yN〉⊗|zN〉. (10.1.3)Thus, projecting the Schr ̈odinger equation onto the coordinatebasis, theN-particle
Schr ̈odinger equation in three dimensions becomes
[
−̄h^2
2 m∑N
i=1∇^2 i+U(r 1 ,...,rN)]
Ψ(x 1 ,...,xN,t) =i ̄h∂
∂tΨ(x 1 ,....,xN,t), (10.1.4)wherexi=ri,si, and the expectation value of a Hermitian operatorAˆcorresponding
to an observable is〈Aˆ〉t=〈Ψ(t)|Aˆ|Ψ(t)〉. The problem inherent in solving eqn. (10.1.4)
and evaluating the expectation value (which is adN-dimensional integral) is that,
unless an analytical solution is available, the computational overhead for a numerical