The Density Matrix

The density operator
To introduce the density operator, let us first consider a pure state, i.e. not a statistical mixture of states.  The initial conditions are completely known.  Let {|un>} be an orthonormal eigenbasis of some operator.  Most often this operator is chosen to be H.  Let the initial state of the system at time t be
k(t)> =  ∑ncnk(t)|un>,  cnk(t) = <unk(t)>.
The system is in a well defined state, but if more than one of the cnk(t) are non-zero, it is not in an eigenstate of H, but in a coherent state
Let <ψkk> = 1, then ∑n|cnk(t)|2 = 1.

The mean value of an observable A at time t is
<A>(t) = <ψk(t)|A|ψk(t)> =  ∑npcnk*(t)cpk(t)Anp, where Anp = <un|A|up>.

We now introduce the pure state density operator ρk(t) = |ψk(t)><ψk(t)|,
i.e. the projector onto the state |ψk(t)>, and the density matrix with matrix elements
ρnpk(t) = <unk(t)|up> = <unk(t)><ψk(t)|up> = cnk(t)cpk*(t).

We will now show that knowing the density operator (or the elements of the density matrix) completely suffices to characterize the state of the system.  All of the predictions of Quantum Mechanics that can be made knowing |ψk(t)> can also be made knowing ρk(t).

In terms of the density operator ρk(t) we express:

Properties of a pure state density operator
ρk(t) is a projector
It is a Hermitian operator, ρk(t) = ρk(t)  with ρk(t)2 = ρk(t) and  Tr{ρk(t)} = 1.

Why is the density matrix formulation useful?
The important equations,
Tr{ρk(t)} = 1,
<A> = Tr{ρk(t)A},
Pk(an) = <ψk(t)|Pnk(t) = Tr{ρk(t)Pn},
(d/dt)ρk(t) = (iħ)-1[H,ρk(t)],
are all linear, while the corresponding equations for the first three of these equations in terms of |ψk(t)>,
d|ψ(x)|2/dt = 0,
<A> = ∫ψ*(x) A ψ(x)dx
Pk(an) = |<ψa|A||ψ>|2 = |∫ψa*(x) A ψ(x) dx|2
are quadratic.  The equation corresponding to the fourth equation is the Schroedinger equation.

The linear equations can easily be generalized to describe a statistical mixture of states and express all physical predictions about such a system in terms of a density operator.

Note:


In experiments, we are often dealing with systems about which we have incomplete information.

Example:

For such a system we introduce the density operator ρ(t) =  ∑kpkρk(t), and the density matrix with matrix elements
ρpn(t) = ∑kpk<upk(t)|un> = = ∑kpkcpk(t)cnk*(t).
In terms of the density operator we express:

Properties of ρ(t)
ρ(t) is not a projector in general; ρ(t)2 ≠ ρ(t) in general.

Example:

ρ(t) is a Hermitian operator (ρ = ρ from the definition of ρ) and
a|ρ||ψ> = ∑kpk<ψ|ψk><ψk|ψ> = ∑kpk|<ψ|ψk>|2 ≥ 0.
ρ(t) is a positive operator.