The Density Matrix

A statistical mixture of states

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

Example:

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 |yk>. .

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.

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 pk of being in the state |yk>, 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

.

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 |yk(t)>, and the density matrix with matrix elements

 .

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 |yk(t)> can also be made knowing rk(t).

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

We have now shown that knowing rk(t) suffices to characterize the quantum state of the system.

Properties of a pure state density operator:

rk(t) is a projector.  It is a Hermitian operator, .

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 |yk(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:


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

Example:

For such a system we introduce the density operator

and the density matrix with matrix elements

.

In terms of the density operator we express:

Properties of r(t):

r(t) is not a projector in general; r(t)2 r(t) in general.

Example:

r(t) is a Hermitian operator (rT=r from the definition of r ) and

.

r(t) is a positive operator.


The physical meaning of the matrix elements of the density operator

.

 is the probability that for a system in the state |yk> a measurement of the observable whose eigenbasis is {|un>} will leave the system in the state |un>rnn(t) therefore represents the average probability of finding the system in the state |un>. rnn(t) is called the population of the state |un>.

expresses interference effects between the states |un> and |up> which appear when |yk> is a linear superposition of states.  rnp(t)is the average of these interference effects.  It can be zero even if is not zero.  In a statistical mixture of states the averaging can cancel out interference effects.  The off-diagonal matrix elements of r(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 hwa and hwb 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 |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 |y1>=2-1/2(|a> + |b>) with 25% probability,

the system is in the state |y2>=2-1/2(|a> - |b>) with 25% probability,

the system is in the state |y3>=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.