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

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

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 , then its initial state is exactly known.
If the system has probability *p _{k}*
of being in the state

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 {*|u _{n}>*} be an orthonormal eigenbasis of some
operator. Most often this operator is chosen to be

.

The system is in a well defined state, but not in an eigenstate of *H*, if more
than one of the are non zero.
If it
is not in an eigenstate of *H*, it is in a **coherent state**.
Let

.

The mean value of an observable *A* at time* t* is

where

.

We now introduce the **pure state density operator**

,

i.e. the projector onto the state |*y _{k}(t)*>,
and the

.

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 |*y _{k}(t)*>
can also be made knowing

In terms of the density operator *r** _{k}(t)
*we express:

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

,

- (b)
**the mean value of the observable A**,,

- (c)
**the time evolution of the system**,,

- (d)
**the probability that a measurement of**.*A*will yield the eigenvalue*a*_{n},

where*P*is the projector into the eigensubspace of_{n}*A*associated with the eigenvalue*a*. The probability is the mean value of_{n}*P*._{n}

We have now shown that knowing *r** _{k}(t)*
suffices to characterize the quantum state of the system.

*r _{k}(t) *is a

**Why is the density matrix formulation useful?**

The important equations,

- ,
- ,
- ,
- ,

are all **linear**, while the corresponding equations for the first three
of these equations in terms of *|y _{k}(t)>*,

- ,
- ,

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 {
*|u*} is an eigenbasis of_{n}>*H*and*|y*is an eigenstate of_{k}(t)>*H*, then the matrix of the pure state density operator is diagonal. All off-diagonal elements are zero. (|*y*_{k}>=|*u*>,_{k}*r*_{k}=|*u*><_{k}*u*|.)_{k} - If
*|y*is a coherent state then has off-diagonal elements._{k}(t)> - We can always choose {
*|u*} to be an eigenbasis of the density operator itself. Then is diagonal._{n}>

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

- A
*C*beam emerges from an accelerator. The^{4+}*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*|y*with probability_{k}>*p*, . We are dealing with a_{k}**statistical mixture of states**.

For such a system we introduce the density operator

and the density matrix with matrix elements

.

In terms of the density operator we express:

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

- (b)
**the probability that a measurement of the observable**,*A*will yield the eigenvalue*a*_{n},

where*P*is the projector into the eigensubspace of_{n}*A*associated with the eigenvalue*a*,_{n} - (c)
**the mean value of the observable A**,,

since, ,

where {

*|a*} is an eigenbasis of_{n}>*A*. - (d)
**the time evolution of the system**.The linearity of implies that .

Properties of *r**(t):*

*r(t) *is not a projector in general; *r**(t) ^{2}¹^{ }r(t)* in general.

- Let
. .

*r(t) *is a Hermitian operator (*r ^{T}=r* from the definition of

.

*r(t) *is a **positive operator**.

.

is the probability that for a
system in the state |*y*_{k}> a
measurement of the observable whose eigenbasis is {*|u _{n}>*} will leave
the system in the state

expresses
interference effects between the states* |u _{n}>* and

Consider a simple two-state quantum system with
energy eigenvalues hw* _{a}*
and hw

(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 |f>=2^{-1/2}(|*a*> + i|*b*>).
Write down the expression for* |*f*(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 |y_{1}>=2^{-1/2}(|*a*>
+ |*b*>) with 25% probability,

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

the system is in the state |y_{3}>=2^{-1/2}(|*a*>
+ i|*b*>) with 50% probability.

Find the density matrix *r*
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*denote the state vector in the Schroedinger picture and let |_{S}>*a*> denote the state vector in the Heisenberg picture._{H}- (a) .

|*a*> does not depend on time._{H} - (b)
.

.

- (c) The pure state density operator .
The density operator for a statistical mixture of states is is the probability of being in state |
*y*_{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*{*rA*}. In the Heisenberg picture we have.

since

.

- (a) .