FOUNDATIONS OF QUANTUM MECHANICS

46 CHAPTER III. THE POSTULATES
A state which cannot be decomposed according to (III. 16) is called a pure state, otherwise it
is called a mixed state. The pure states are the states ρ concentrated on a single point of Γ, the δ -
‘functions’. Generally, the elements of a convex set which cannot be written in the form (III. 16),
with w 1 , w 2 ≠ 0, are called extreme elements of that set, therefore in our case the extreme elements
are the pure states. Every element of a convex set can always be written as a convex sum of extreme
elements. This corresponds to the expansion of ρ to δ - functions,
∫
ρ(q, p) = ρ(q ′ , p ′ ) δ(q − q ′ ) δ(p − p ′ ) dq ′ dp ′ . (III. 17)
Γ
The dynamics of an arbitrary state follows from the Hamiltonian equations of motion of the pure
states, found by calculating the path of least energy. This holds for conservative systems, which
the quantum mechanical states in these lecture notes are assumed to be. We will come back to the
derivation of the equations in section VI. 5.
The Hamiltonian equations of motion are
˙q = ∂H
∂p
and
ṗ = − ∂H . (III. 18)
∂q
To find the equation of motion in terms of ρ we use Liouville’s theorem which states that for points
moving in phase space obeying the Hamiltonian equations of motion the time evolution of the probability
distribution ρ(q, p, t) is constant. Using (III. 18) and the Poisson brackets
{H, ρ} :=
( ∂H
∂q
∂ρ
∂p − ∂ρ
∂q
∂H
∂p
the Liouville equation, the equation of motion for the state ρ
)
, (III. 19)
equals
dρ
dt = ∂ρ
∂t + ∂ρ
∂q
∂ρ ˙q + ṗ = 0, (III. 20)
∂p
∂ρ
∂t
= {H, ρ}. (III. 21)
Now we will consider, analogous to the classical case, a probability distribution of the state vectors
in H. With help of the state ρ we introduce a mapping µ of subsets ∆ of Γ to R,
∫
µ(∆) := ρ(q, p) dq dp with ∆ ⊆ Γ. (III. 22)
∆
This mapping µ is additive
µ (∪ i ∆ i ) = ∑ i=1
µ(∆ i ) (III. 23)
for every countable sequence of disjoint ∆ i ⊂ Γ. Furthermore,
0 µ(∆) 1, µ(∅) = 0 and µ(Γ) = 1. (III. 24)