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9. Classical and quantum dynamics of density matrices

Statistical mechanics makes the connection between macroscopic dynamics and equilibriums states based on microscopic dynamics.  For example, while thermodynamics can manipulate equations of state and fundamental relations, it cannot be used to derive them.  Statistical mechanics can derive such equations and relations from first principles.

Before we study statistical mechanics, we need to introduce the concept of the density, referred to in classical mechanics as density function, and in quantum mechanics as density operator or density matrix.  The key idea in statistical mechanics is that the system can have “microstates,” and these microstates have a probability.  For example, there may be a certain probability that all gas atoms are in a corner of the room, and this is probably much lower than the probability that they are evenly distributed throughout the room.  Statistical mechanics deals with these probabilities, rather than with individual particles.  Three general contexts of probability are in common use:

 

a. Discrete systems: and example would be rolling dice.  A die has 6 faces, each of which is a “microstate.”  Each outcome is equally likely if the die is not loaded, so

                                                                   

is the probability of being in microstate “i” where W=6 is the total number of microstates.

 

b. Classical systems: here the microstate has to be specified by the positions xi and the momenta pi of all particles in the system, so

                                                                     

is the time dependent probability of finding the particles at xi and pi.  Do not confuse the momentum here with the probability in a.!  It should be clear from the context.  r is the classical density function.

 

c. Quantum systems: here the microstate is specified by the density operator or density matrix .  The probability that the system is in quantum state “i” with state |i> is given by

                                           .

Of course the probability does not have to depend on time if we are in an equilibrium state.

 

In all three cases, statistical mechanics attempts to evaluate the probability from first principles, using the Hamiltonian of the closed system.  In an equilibrium state, the probability does not depend on time, but still depends on x and p (classical) or just x (quantum).

 

We need to briefly review basic concepts in classical and quantum dynamics to see how the probability evolves in time, and when it does not evolve (reach equilibrium).

 

Mechanics:  classical

 

Definition of phase space: Phase space is the 6n dimensional space of the 3n coordinates and 3n momenta of a set of n particles, which, taken together constitute the system.

 

Definition of a trajectory: The dynamics of a system of N degrees of freedom (N = 6n for n particles in 3-D) are specified by a trajectory {xj(t), pj(t)}, j = 13n in phase space.

 

Note: a system of n particles is phase space is defined by a single point {x1(t), x2(t), ... x3n(t), p1(t), p2(t), ... p3n(t)} that evolves in time.  For that specific system, the density function is a delta function in time centered at the single point, and moving along in time as the phase spacve trajectory trajectory moves along. The phase space trajectory is not to be confused with the 3-D trajectories of individual particles.

 

In statistical mechanics, the system must satisfy certain constraints:  (e.g. all xi(t) must lie within a box of volume V; all speeds must be less than the speed of light; etc).  The density function  that we usually care about in statistical mechanics is the average probability density of ALL systems satisfying the constraints. We call the group of systems satisfying the same constraints an “ensemble”, and the average density matrix the “ensemble denisty matrix”:

                                                                   ,

where the sum is over all W possible systems in the microstates “m” satisfying the constraint.  Unlike an individual rm, which is a moving spike in phase space, r sums over all microstates and looks continuous (at least after a small amount of smoothing).  For example, consider a single atom with position and momentum {x, p} in a small box.  Each different rm for each microstate “m” is a spike moving about in the phase space {x, p}.  Averaging over all such microstates yields a r that is uniformly spread over the positions in the box (independent of position x), with a Maxwell-Boltzmann distribution of velocities:  (assuming the walls of the box can equilibrate the particle to a temperature T).  How does r evolve in time?

 

Each coordinate  evolves according to Newton’s law, which can be recast as

                                                                      

                                                      

if the force is derived from a potential V and where

                                                      

is the kinetics energy.  We can define the Lagrangian L = KV and rewrite the above equation as

                                                       .

One can prove using variational calculus (see appendix A) that this differential equation (Lagrange’s equation) is valid in any coordinate system, and is equivalent to the statement

                                                           

where S is the action (not to be confused with the entropy!).  Let’s say the particle moves from from positions xi(t0) at t=t0 to xi(tf) at tf.  Guess a trajectory xi(t).  The trajectory xi(t) can be used to compute velocity=∂xi/∂t= and .  The actual trajectory followed by the classical particles is the one that minimizes the above integral, called “the action.”

We have, from the definition of the Lagrangian given nin Cartesian coordinates above,

                                                            

Thinking of  and of  as the derivatives (slope), of  we can Legendre transform to a new representation H

                                                              .

You may want to review Legendre transforms from the thermodynamics lectures: they are used to transform a function y(x) into a function j(m) where j is the intercept and m is the slope or y. y(x) and j(m) contain equivalent information. It will become obvious shortly why we define H with a minus sign.  According to the rules for Legendre transforms,

                                                          .

 

These are Hamilton’s equation of motion for a trajectory in phase space.  They are equivalent to solving Newton’s equation.  Evaluating H,

The Hamiltonian is sum of kinetic and potential energy, i.e. the total energy, and is conserved if H is not explicitly time-dependent:

.

Thus Newton’s equations conserve energy.  This is because all particles are accounted for in the Hamiltonian H (closed system).  Note that Lagrange’s and Hamilton’s equations hold in any coordinate system, so from now on we will write  instead of using xi (cartesian coordinates).  Also note that the above derivation is not a rigorous proof because we started with Cartesian coordinates. In classical mechanics texts you will find the formal proof valid in more general coordinate systems.

Let  be any dynamical variable (many s of interest do not depend explicitly on t, but we include it here for generality).  Then

gives the time dependence of .

 

Definition: []P is the Poisson bracket as defined by the above equation.  It is a convenient shorthand, and related to the commutator in quantum mechanics by the correspondence principle.

 

Now consider the ensemble density  as a specific example of a dynamical variable.  Because trajectories cannot be destroyed, we can normalize

.

Integrating the probability over all state space, we are guaranteed to find the system somewhere subject to the constraints.  Since the above integral is a constant, we have

                         

This is the Liouville equation, it describes how the density propagates in time.  To calculate the average value of an observable  in the ensemble of systems described by r, we calculate

 

Thus if we know r, we can calculate any average observable as a function of time.  For certain systems which are left unperturbed  by outside influences (closed systems)

                                                                    

r  reaches an equilibrium distribution  and

 (definition of equilibrium)

In such a case, A(t) à A, the average value of the observable approaches the equilibrium value of the observable. 

 

Now we can state the two goals of statistical mechanics succinctly. Given a system with Hamiltonian H,

  • The goal of equilibrium statistical mechanics is to find the values of any observable A for a  subject to to certain imposed constraints. E.g.  corresponds to the set of all possible system trajectories such that U and V are constant and  .
  • The more general goal of non-equilibrium statistical mechanics is to find A(t) given an initial condition  by solving the Liouville equation.

So much for the classical picture. It seems simple, but solving the equations is hard.

Mechanics: quantum

 

Now let us rehearse the whole situation again for quantum mechanics.  The quantum formulation is the one best suited to systems where the energy available to a degree of freedom becomes comparable or smaller than the characteristic energy gap of the degree of freedom.  Classical and quantum formulations are highly analogous.

 

A fundamental quantity in quantum mechanics is the density operator

                                                    

This density operator projects onto the microstate “m” of the system, .  If we have an ensemble of W systems, we can define the ensemble density operator  subject to some constraints as

 

For example, let the constraint be U = const.  Then we would sum over all microstates that are degenerate at the same energy U.  This average is analogous to averaging the classical probability density over microstates subject to constraints.  To obtain the equation of motion for , we first look at the wavefunction.

Its equation of motion is (using a dot for the time derivative)

                                                                    ,

which by splitting it into real and imaginary parts , can be written

.

In analogy to Hamilton’s equations of motion. Such pairs of differential equations conserve volume (classically in phase space, quantum mechanically the norm of the wavefunction) and energy, as we saw in the classical mechanics section. They are called “symplectic.” Using any complete basis , the trace of is conserved                

if  is normalized, or

.

This basically means that probability density cannot be destroyed, in analogy to trajectory conservation.  Note that if

                                                                    

                                                       

A state described by a wavefunction |Y> that satisfies the latter equation is a pure state.  Most states of interest in statistical mechanics are NOT pure states.  If

                                                                  ,

then the complex off-diagonal elements tend to cancel because of random phases and , this an impure state.

 

Example:  

Let  be an arbitrary wavefunction for a two-level system.                         

or, in matrix form,

Generally, macroscopic constraints (volume, spin population, etc.) do not constrain the phases of   Thus, ensemble averaging

 

Such a state is known as an impure state.

To obtain the equation of motion of , use the time-dependent Schrödinger equation in propagator form,

                                                    

and the definition of r,

                                                     

to obtain the equation of motion

              

This is known as the Liouville-von Neumann equation, again analogous to the Poisson bracket equation of classical mechanics that we saw earlier.  The commutator defined in the last line is equivalent to the Poisson bracket in classical dynamics.  Summing over all microstates to obtain the average density operator

                                                     

This von-Neumann equation is the quantum equation of motion for .  If r represents an impure state, this propagation cannot be represented by the time-dependent Schrödinger equation.

 

We are interested in average values of observables in an ensemble of systems.  Starting with a  pure state,

Summing over ensembles,

                                 

In particular,  is the probability of being in state j at time t.

 

Finally,  may evolve to long-time solutions  such that

(condition for equilibrium).

In that case, the density matrix has relaxed to the equilibrium density matrix, which no longer evolves in time.

 

Example:  Consider a two-level system again.  Let

           

Thus a diagonal density matrix of a closed system does not evolve.  The equilibrium density matrix must be diagonal so .  This corresponds to a completely impure state.  Note that unitary evolution cannot change the purity of any closed system.  Thus, the density matrix of a single closed system cannot evolve to diagonality unless

                                                                   

for the ensemble is already diagonal.  In reality, single systems still decohere because they are open to the environment:  let i denote the degrees of freedom of the system, and j of a “bath” or “environment” (e.g. a heat reservoir).   depends on both i and j and can be written as a matrix   For example, for a two-level system coupled to a large bath, indices i,j only go from 1 to 2, but indices i’ and j’ could go to 1020.  We can average over the bath by letting

 

 only has matrix elements for the system degrees of freedom, e.g. for a two level system in contact with a bath of 1020 states, is still a 2x2 matrix.

 

We will show later that for a bath is at constant T.

                                                

 

Note that “reducing” was not necessary in the classical discussion because quantum coherence and phases do not exist there.  To summarize analogous entities in the classical and quantum formulations:

 

 

 

 

 

 

Classical                                  Quantum

                                               Density function or operator

                                                                              Observable, hermitian operator

                            Eq. of motion for traj./

                                                                       Averaging

                                                        Conservation of probability

                Expectation value in ensemb.

                                               Eq. of motion for p

                                                    Necessary equil. condition                                                                                                      (closed sys.)

 

 

To illustrate basic ideas, we will often go back to simple discrete models with probability for each microstate pi, but to get accurate answers, one may have to work with the full classical or quantum probability r.