Abstract

CHAPTER 1. OVERVIEW 11
in many different states at once. While one could write a wavefunction that would
describe the state of the ensemble of electrons, the wavefunction would be written
as ψ(¯x1, ¯x2,...,¯xn), where ¯xi would be the spatial coordinate of the ith electron.
This becomes quite computationally burdensome when dealing with more than a few
electrons. Therefore, other formalisms of quantum mechanics would be better suited
to handle the description of an ensemble of electrons. One such formalism is the
density matrix formalism [5], [12]. The Wigner function is derived from the density
matrix, so we shall first explain the density matrix.
Let {ψi(q)} be a collection of the possible states which the electrons can exists and
{¯ji} be the corresponding probabilites that an electron is found in these states. The
density matrix stores all this information into one compact form. The density matrix
is a function of two position variables and is given by ρ(q, r) =
i ¯jiψi(q)ψ ∗ i (r). The
time evolution of the density matrix can be derived from Schrödinger’s equation as
follows:
∂ρ(q, r)
∂t
∂
=
∂t
i
= ∂
¯ji
i
¯jiψi(q)ψ ∗ i (r)
∂t [ψi(q)ψ ∗ i (r)]
=
¯ji[ ∂ψi(q)
Schrödinger’s equation tells us that ∂ψ(q)
∂t
i
∂t ψ∗ i (r)+ψi(q) ∂ψ∗ i (r)
∂t
1
=
i [−2
2m∗ ∂2ψ(q) ∂q2 +U(q)ψ(q)] and ∂ψ∗ (r)
∂t =
− 1
i [−2
2m∗ ∂2ψ ∗ (r)
∂r2 + U(r)ψ∗ (r)]. Substituting these into the above equation and rear-
]