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

- 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 p_{k} of being
in a state |ψ_{k}>. (∑_{k}p_{k} = 1.)

We say that we are dealing with a **statistical mixture of
states**. We now want to know the probability of measuring the eigenvalue a_{n}
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 ∑_{k}c_{k}ψ_{k}, then its initial state is exactly known.
If the system has probability p_{k}
of being in the state |ψ_{k}>, then its
initial state is not exactly known. Even if p_{k} = c_{k} the
probability of obtaining the eigenvalue a_{n} 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.

To introduce the density operator, let us first consider a

|ψ

The system is in a well defined state, but if more than one of the c

Let <ψ

The mean value of an observable A at time t is

<A>(t) = <ψ_{k}(t)|A||ψ_{k}(t)> = ∑_{np}c_{n}^{k}*(t)c_{p}^{k}(t)A_{np}, where A_{np} = <u_{n}|A|u_{p}>.

We now introduce the **pure state density operator** ρ^{k}(t)
= |ψ_{k}(t)><ψ_{k}(t)|,

i.e. the projector onto the state |ψ_{k}(t)>,
and the **density matrix** with **matrix
elements
**ρ

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 |ψ_{k}(t)>
can also be made knowing ρ^{k}(t).

In terms of the density operator ρ^{k}(t)
we express:

- (a)
**conservation of probability**,

<ψ_{k}|ψ_{k}> = ∑_{n}<ψ_{k}|u_{n}><u_{n}|ψ_{k}> = ∑_{n}<u_{n}|ψ_{k}><ψ_{k}|u_{n}> = ∑_{n}ρ_{np}^{k}(t) = Tr{ρ^{k}(t)} = 1.

(d/dt)Tr{ρ^{k}(t)} = 0, - (b)
**the mean value of the observable A**,

<A> = ∑_{np}c_{n}^{k}*(t)c_{p}^{k}(t)A_{np}= ∑_{np}ρ_{pn}^{k}(t)A_{np}= ∑_{np}<u_{p}|ρ^{k}(t)|u_{p}><u_{n}|A|u_{p}> = ∑_{p}<u_{p}|ρ^{k}(t)A|u_{p}> = Tr{ρ^{k}(t)A}, - (c)
**the time evolution of the system**,

(d/dt)ρ^{k}(t) = (d|ψ_{k}>/dt) <ψ_{k}| + |ψ_{k}>(d<ψ_{k}|/dt)

(1/(iħ)H|ψ_{k}><ψ_{k}| - (1/(iħ)|ψ_{k}><ψ_{k}|H

= (1/(iħ))[H,ρ^{k}(t)], - (d)
**the probability that a measurement of A will yield the eigenvalue a**._{n}

P_{k}(a_{n}) = <ψ_{k}(t)|P_{n}|ψ_{k}(t)> = Tr{ρ^{k}(t)P_{n}},

where P_{n}is the projector into the eigensubspace of A associated with the eigenvalue a_{n}. The probability is the mean value of P_{n}.

We have therefore shown that knowing ρ^{k}(t) suffices to characterize the quantum state of the system.

It is a

**Why is the density matrix formulation useful?
**The important equations,

Tr{ρ

<A> = Tr{ρ

P

(d/dt)ρ

are all

d|ψ(x)|

<A> = ∫ψ*(x) A ψ(x)dx

P

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 {|u
_{n}>} is an eigenbasis of H and |ψ_{k}(t)> is an eigenstate of H, then the matrix ρ_{np}^{k}(t) of the pure state density operator is diagonal. All off-diagonal elements are zero.

|ψ_{k}> = |u_{k}>, ρ_{k }= |u_{k}><u_{k}|. - If |ψ
_{k}(t)> is a coherent state then ρ_{np}^{k}(t) has off-diagonal elements.

| ψ_{k}> = ∑_{n}c_{n}|u_{n}>, ρ^{k}= ∑_{nn'}c_{n}*c_{n'}|u_{n'}><u_{n}|. - We can always choose {|u
_{n}>} to be an eigenbasis of the density operator itself.

Then ρ_{np}^{k}(t) is diagonal.

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

- A C
^{4+}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 |ψ_{k}> with probability p_{k}, ∑_{k}p_{k}= 1. We are dealing with a**statistical mixture of states**.

For such a system we introduce the density operator ρ(t) = ∑_{k}p_{k}ρ^{k}(t), and the density matrix with matrix elements

ρ_{pn}(t) = ∑_{k}p_{k}<u_{p}|ρ^{k}(t)|u_{n}>
= = ∑_{k}p_{k}c_{p}^{k}(t)c_{n}^{k}*(t).

In terms of the density operator we express:

- (a)
**conservation of probability**,

Tr{ρ(t)} = Tr{∑_{k}p_{k}ρ^{k}(t))} = ∑_{k}p_{k}Tr{ρ^{k}(t)) = ∑_{k}p_{k}= 1. - (b)
**the probability that a measurement of the observable A will yield the eigenvalue a**,_{n}

P(a_{n}) = ∑_{k}p_{k}P_{k}(a_{n}) = ∑_{k}p_{k}Tr{ρ^{k}(t)P_{n}} = Tr{∑_{k}p_{k}ρ^{k}(t)P_{n}} = Tr{ρ(t)P_{n}}.

where P_{n}is the projector into the eigensubspace of A associated with the eigenvalue a_{n}, - (c)
**the mean value of the observable A**,

<A>(t) = ∑_{n}a_{n}P(a_{n}) = ∑_{n}a_{n}Tr{ρ(t)P_{n}} = Tr{ρ(t)∑_{n}a_{n}P_{n}} = Tr{ρ(t)A},

since ∑_{n}a_{n}P_{n}= A, A|ψ> = ∑_{n}A|a_{n}><a_{n}|ψ> = ∑_{n}a_{n}|a_{n}><a_{n}|ψ> = ∑_{n}a_{n}P_{n}|ψ>,

where {|a_{n}>} is an eigenbasis of A. - (d)
**the time evolution of the system**.

The linearity of (d/dt)ρ^{k}(t) = (1/(iħ))[H,ρ^{k}(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)

- Let |ψ
_{k}> = |u_{k}> with probability p_{k}, ρ = ∑_{k'}p_{k'}|u_{k'}><u_{k'}|,

ρ_{kk}= <u_{k}|∑_{k'}p_{k'}|u_{k'}><u_{k'}|u_{k}> = p_{k}, ρ_{k}_{n}(k ≠ n) = 0, Tr{ρ} = ∑_{k}p_{k}= 1, Tr{ρ^{2}} = ∑_{k}p_{k}^{2}≤ 1.

ρ(t) is a Hermitian operator (ρ^{† }= ρ from the definition of ρ)
and

|<ψ_{a}|ρ||ψ> = ∑_{k}p_{k}<ψ|ψ_{k}><ψ_{k}|ψ>
= ∑_{k}p_{k}|<ψ|ψ_{k}>|^{2} ≥ 0.

ρ(t) is a **positive operator**.

c

ρ

ρ

ρ_{np}(t) = ∑_{k}p_{k}c_{n}^{k}(t)c_{n}^{k}*(t)
(n ≠ p) expresses
interference effects between the states |u_{n}> and |u_{p}> which appear when |ψ_{k}> is a linear
superposition of states. ρ_{np}(t) is
the average of these interference effects. It can be zero even if c_{n}^{k}(t)c_{n}^{k}*(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.

Consider a simple two-state quantum system with
energy eigenvalues ħω_{a}
and ħω_{b}
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 |ψ_{1}> = 2^{-1/2}(|a>
+ |b>) with 25% probability,

the system is in the state |ψ_{2}> = 2^{-1/2}(|a>
- |b>) with 25% probability,

the system is in the state |ψ_{3}> = 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 |a_{S}> denote the state vector in the Schroedinger
picture and let |a_{H}> denote the state vector in the Heisenberg
picture.

- (a) iħ∂|a
_{s}>/∂t = H|a_{s}>, |a_{s}(t)> = U(t,0)|a_{s}(0)> = exp(-iHt/ħ)a_{s}(0)>.

|a_{H}> 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 |ψ
_{k}>..

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

.