Often we have incomplete information about a system, we do not know the initial state of the system perfectly well.

#### Example:

• Two beams of linearly polarized Laser light with the same frequency are polarized perpendicular to each other and are combined on a target.  We do not know the polarization of an individual photon striking the target, since our detector can only count photons.  We only know that the probability that the photon has one or the other polarization, since we know the intensities of the two beams.  We have incomplete information about the individual photons.

In general we handle incomplete information about the initial state using the concept of probability.  We can say that the system has a probability pk of being in a state |ψk>.  (∑kpk = 1.)

We say that we are dealing with a statistical mixture of states.  We now want to know the probability of measuring the eigenvalue an if a measurement of the observable A is made.

Note: Probabilities enter at two levels.

• The initial information about the system is given in terms of probabilities.
• The predictions of Quantum Mechanics are probabilistic.

A statistical mixture of states is not equivalent to a linear superposition of states.  If a system is in a state  ∑kckψk, then its initial state is exactly known.  If the system has probability pk of being in the state |ψk>, then its initial state is not exactly known.  Even if pk = ck the probability of obtaining the eigenvalue an when measuring A is in general not the same for the two systems.  Interference effects are absent for a statistical mixture.  We cannot describe a statistical mixture using an "average state vector".  However an "average operator", called the density operator permits a simple description of a statistical mixture.

#### 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> = ∑nρnpk(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)|up><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
= (1/(iħ))[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) = (1/(iħ))[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) = (1/(iħ))[H,ρk(t)] implies that (d/dt)ρ(t) = (1/(iħ))[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
|<ψa|ρ||ψ> = ∑kpk<ψ|ψk><ψk|ψ> = ∑kpk|<ψ|ψk>|2 ≥ 0.
ρ(t) is a positive operator.

#### The physical meaning of the matrix elements of the density operator ρnn(t) =  ∑kpkρnnk(t) =  ∑kpk|cnk(t)|2. cnk(t)|2 is the probability that for a system in the state |ψk> a measurement of the observable whose eigenbasis is {|un>} will leave the system in the state |un>.  ρnn(t) therefore represents the average probability of finding the system in the state |un>. ρnn(t) is called the population of the state |un>.

ρnp(t) = ∑kpkcnk(t)cnk*(t) (n ≠ p) expresses interference effects between the states |un> and |up> which appear when |ψk> is a linear superposition of states.  ρnp(t) is the average of these interference effects.  It can be zero even if cnk(t)cnk*(t) is not zero.  In a statistical mixture of states the averaging can cancel out interference effects.  The off-diagonal matrix elements of ρ(t) are called coherences.  The populations and coherences obviously depend on the chosen basis.

#### Problem:

Consider a simple two-state quantum system with energy eigenvalues ħωa and ħωb and corresponding eigenkets |a> and |b> which are taken as the basis elements.
(a)  Write down the equation giving the time evolution of the eigenket |a> in the Schroedinger picture and in the Heisenberg picture.
(b)  Consider a pure state, which at time t = 0 is |Φ> = 2-1/2(|a> + i|b>).  Write down the expression for |Φ(t)> in the Schroedinger and in the Heisenberg picture.
(c)  Consider a mixed state, which at t = 0 is defined by:
the system is in the state |ψ1> = 2-1/2(|a> + |b>) with 25% probability,
the system is in the state |ψ2> = 2-1/2(|a> - |b>) with 25% probability,
the system is in the state |ψ3> = 2-1/2(|a> + i|b>) with 50% probability.
Find the density matrix ρ in the {|a>,|b>} basis at t = 0.
(d)  Explain how the density operator evolves in time in the Schroedinger picture and in the Heisenberg picture.  Find the density matrix at time t for the mixed state in part (c) in each picture.
(e)  Consider the operator X which has the property X|a> = |b> and X|b> = |a>.  Find the expectation value of X at time t when the system is in the mixed state of part (c).  You may use either the Schroedinger or the Heisenberg picture for your calculations, but specify which picture you are using.

Solution:

Let |aS> denote the state vector in the Schroedinger picture and let |aH> denote the state vector in the Heisenberg picture.

• (a) iħ∂|as>/∂t = H|as>,  |as(t)> = U(t,0)|as(0)> = exp(-iHt/ħ)as(0)>.
|aH> does not depend on time.
• (b)

.

.

• (c) The pure state density operator .  The density operator for a statistical mixture of states is is the probability of being in state |ψk>.

.

The density matrix is .

,

.

• (d)

=

=

=.

.

Therefore in the Schroedinger picture and in the Heisenberg picture

In the Heisenberg picture the density operator is constant.  Therefore the density matrix is constant.

• (e) The expectation value of any observable A is <A>(t) = Tr{ρA}.
In the Heisenberg picture we have

at t = 0.

at t = 0, or

.  Therefore

.