#### 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 |ψk(t)> =  ∑ncnk(t)|un>,  cnk(t) = <un|ψk(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 <ψk|ψk> = 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:

• (a) conservation of probability,
kk> = ∑nk|un><unk> = ∑n<unk><ψk|un> = ∑nρnnk(t) = Tr{ρk(t)} = 1.
(d/dt)Tr{ρk(t)} = 0.
• (b) the mean value of the observable A,
<A> =  ∑npcnk*(t)cpk(t)Anp = ∑npρpnk(t)Anp =  ∑np<upk(t)|un><un|A|up>
= ∑p<upk(t)A|up> = Tr{ρk(t)A}.
• (c) the time evolution of the system,
(d/dt)ρk(t) = (d|ψk>/dt) <ψk| +  |ψk>(d<ψk|/dt)
= (1/(iħ)H|ψk><ψk| - (1/(iħ)|ψk><ψk|H
= (iħ)-1[H,ρk(t)].
• (d) the probability that a measurement of A will yield the eigenvalue an.
Pk(an) = <ψk(t)|Pnk(t)> = Tr{ρk(t)Pn},
where Pn is the projector into the eigensubspace of A associated with the eigenvalue an.  The probability is the mean value of Pn.
We have therefore shown that knowing ρk(t) suffices to characterize the quantum state of the system.
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:

• If {|un>} is an eigenbasis of H and |ψk(t)> is an eigenstate of H, then the matrix ρnpk(t) of the pure state density operator is diagonal.  All off-diagonal elements are zero.
k> = |uk>, ρk = |uk><uk|.
• If |ψk(t)> is a coherent state then ρnpk(t) has off-diagonal elements.
| ψk> = ∑ncn|un>, ρk = ∑nn'cn*cn'|un'><un|.
• We can always choose {|un>} to be an eigenbasis of the density operator itself.
Then ρnpk(t) is diagonal.

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

#### Example:

• A C4+ beam emerges from an accelerator.  The C ions have been stripped of most of their electrons by having been past through a thin foil.  Most ions are in their ground state, but some are in an excited, metastable state.  For each ion we only know that it may be in a state |ψk> with probability pk,  ∑kpk = 1.  We are dealing with a statistical mixture of states.

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:

• (a) conservation of probability,
Tr{ρ(t)} = Tr{∑kpkρk(t))} = ∑kpkTr{ρk(t)) = ∑kpk = 1.
• (b) the probability that a measurement of the observable A will yield the eigenvalue an,
P(an) = ∑kpkPk(an) = ∑kpkTr{ρk(t)Pn} = Tr{∑kpkρk(t)Pn} = Tr{ρ(t)Pn},
where Pn is the projector into the eigensubspace of A associated with the eigenvalue an.
• (c) the mean value of the observable A,
<A>(t)  = ∑nanP(an) = ∑nanTr{ρ(t)Pn} = Tr{ρ(t)∑nanPn} = Tr{ρ(t)A},
since ∑nanPn = A,  A|ψ> =  ∑nA|an><an|ψ> = ∑nan|an><an|ψ> = ∑nanPn|ψ>,
where {|an>} is an eigenbasis of A.
• (d) the time evolution of the system.
The linearity of (d/dt)ρk(t) = (iħ)-1[H,ρk(t)] implies that (d/dt)ρ(t) = (iħ)-1[H,ρ(t)].

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

#### Example:

• Let |ψk> = |uk> with probability pk, ρ =  ∑k'pk'|uk'><uk'|,
ρkk =  <uk|∑k'pk'|uk'><uk'|uk> = pk,  ρkn(k ≠ n) = 0, Tr{ρ} = ∑kpk = 1, Tr{ρ2} = ∑kpk2  ≤ 1.

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.