Download Part II. Statistical mechanics Chapter 9. Classical and quantum

Part II. Statistical mechanics
Chapter 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
1 1
pi =
=
W 6
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
ρ(xi , pi ,t)
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. ρ is the
classical density function.
c. Quantum systems: here the microstate is specified by the density operator or density
matrix ρ̂(t) . The probability that the system is in quantum state “i” with state |i> is given
by
pi (t) = ∫ dxΨ *i (x)ρ̂(x,t)Ψ i (x) =< i | ρ̂(t) | i > .
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 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 = 1…3n 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 in time as the
trajectory moves along.
In statistical mechanics, the system must satisfy certain constraints: (e.g. all xi(t) must lie
within a box of volume V). The density function ρ(xi , pi ,ti ; constraints) that we usually
care about in statistical mechanics is the probability density of ALL systems satisfying the
constraints:
1 W
ρ = ∑ ρi ,
W i =1
where the sum is over all possible systems in the W microstates “i” satisfying the
constraint. Unlike ρ i, which is a moving spike in phase space, ρ 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. The
different ρi for each microstate are spikes at different positions in the box moving with
different momenta. Averaging over all microstates yields a ρ that is uniformly spread
over the positions in the box (independent of x) with a Maxwell-Boltzmann distribution
2
of velocities: ρ ~ e− p /2 mkB T . How does ρ evolve in time?
Each coordinate qi = xi evolves according to Newton’s law, which can be recast as
Fi = m
xi
∂V d
d ⎛ ∂K ⎞
= (mxi ) = ⎜
∂xi dt
dt ⎝ ∂xi ⎟⎠
if the force is derived from a potential V and where
1
1
pi2
2
K = ∑ mi xi = ∑
2 i
2 i 2mi
is the kinetics energy. We can define the Lagrangian L = K – V and rewrite the above
equation as
d ⎛ ∂L(xi , xi ,t) ⎞ ∂L
⎟⎠ − ∂x = 0 .
dt ⎜⎝
∂xi
i
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
−
tf
⎧⎪
⎫⎪
min ⎨S = ∫ L(xi , xi )dt ⎬
t0
⎪⎩
⎪⎭
where S is the action (not to be confused with the entropy!). Let’s say the particle moves
from from 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= xi and L(xi , xi ) . The actual trajectory followed by the
classical particles is the one that minimizes the above integral, called “the action.”
We have
∂L
= pi ( pi = mxi )
∂xi
Thinking of L as L( xi ) and of pi as the derivatives, we can Legendre transform to a new
representation H
N
−H ≡ L − ∑ xi pi .
i =1
It will become obvious shortly why we define H with a minus sign. According to the
rules for Legendre transforms,
∂H
∂H
= xi and
= pi .
∂pi
∂xi
These are Hamilton’s equation of motion for a trajectory in phase space. They are
equivalent to solving Newton’s equation. Evaluating H,
p
H ( xi , pi ) = −K + V + ∑ i pi
i mi
= K +V
The Hamiltonian is sum of kinetic and potential energy, i.e. the total energy, and is
conserved if H is not explicitly time-dependent:
dH (x, p) ∂H dx ∂H dp ∂H dH ∂H dH
=
+
=
−
= 0.
dt
∂x dt ∂p dt
∂x dp ∂p dx
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 H (qi , pi ) instead of using xi
(cartesian coordinates).
Let Â(qi , pi ,t) be any dynamical variable (many  s of interest do not depend
explicitly on t, but we include it here for generality). Then
⎛ ∂ dH ∂ dH ⎞ ∂Â
dÂ
∂ dqi ∂ dpi ∂Â
∂Â
=∑
+
+
= ∑⎜
−
+
= [ Â, H ] p +
⎟
dt
∂pi dt
∂t
∂pi dqi ⎠ ∂t
∂t
i ∂qi dt
i ⎝ ∂qi dpi
gives the time dependence of  . []P is the Poisson bracket.
Now consider the density ρ(qi , pi ,t) . Because trajectories cannot be destroyed, we
can normalize
∫∫ dqi dpi ρ(qi , pi ,t) = 1 .
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
d
dρ
∂ρ ⎫
⎧
dqi dpi ρ = 0 = ∫∫ dqi dpi
= ∫∫ dqi dpi ⎨[ ρ, H ] + ⎬ = 0
∫∫
dt
dt
∂t ⎭
⎩
∂ρ
⇒
= −[ ρ, H ] p .
∂t
This is the Liouville equation, it describes how the density propagates in time. To
calculate the average value of an observable Â(qi , pi ) in the ensemble of systems
described by ρ, we calculate
A(t) =
∫∫ dq dp ρ(q , p ,t)Â(q , p ) =
i
i
i
i
i
i
A
ens
Thus if we know ρ, we can calculate any average observable. For certain systems which
are left unperturbed by outside influences (closed systems)
∂ρ
lim
= 0.
t →∞ ∂t
ρ reaches an equilibrium distribution ρeq (qi , pi ) and
[ ρeq , H ] p = 0 (definition of equilibrium)
In such a case, A(t)  A, the equilibrium value of the observable. The goal of
equilibrium statistical mechanics is to find the values A for a ρ subject to certain
imposed constraints; e.g. ρeq (qi , pi ,U,V ) is the set of all possible system trajectories such
that U and V are constant. The more general goal of non-equilibrium statistical
mechanics is to find A(t) given an initial condition ρ0 (qi , pi ; constraints) . So much for
the classical picture.
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
ρ̂i (t) = ψ i (t) ψ i (t) = t t .
This density operator projects onto the microstate “i” of the system, ψ i (t) . If we have
an ensemble of W systems, we can define the ensemble density operator ρ̂ subject to
some constraints as
1 W
∑ ρ̂I (t; constraints)
W I =1
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
ρ̂(t; constraints) =
classical probability density over microstates subject to constraints. To obtain the
equation of motion for ρ̂ , we first look at the wavefunction.
Its equation of motion is
∂
Hψ = i ψ ,
∂T
which by splitting it into ψ r and iψ i , can be written
1
1
Ĥψ r = ψ i and Ĥψ i = −ψ r .


Using any complete basis H ϕ j = E j ϕ j , the trace of ρ̂ is conserved
Tr{ρi (t)} = ∑ < ϕ j ψ i (t) ψ i (t) ϕ j >= ∑ ψ i ϕ j ϕ j ψ i = ψ i (t) ψ i (t) = 1
j
j
if ψ j (t) is normalized, or
Tr{ρ̂i } = 1, and Tr{ρ̂} = 1 .
This basically means that probability density cannot be destroyed, in analogy to trajectory
conservation. Note that if
ρ̂i = ψ ψ
⇒ ρ̂i 2 = ρ̂i so Tr( ρ̂i 2 ) = 1
A state described by a wavefunction |Ψ> that satisfies the latter equation is a pure
state. Most states of interest in statistical mechanics are NOT pure states. If
1 W
ρ̂ = ∑ ρ̂i ,
W i =1
the complex off-diagonal elements tend to cancel because of random phases and
Tr( ρ̂ 2 ) < 1 , this an impure state.
Example:
Let ψ = c0i 0 + c1i 1 be an arbitrary wavefunction for a two-level system.
⇒ ρ = Ψ Ψ = c0i c0i * 0 0 + c0i c1i* 0 1 + c1i c0i * 1 0 + c1i c1i* 1 1
or, in matrix form,
⎛ ci 2 ci ci* ⎞
2
2
0
0 1
⎟ Tr ( ρ̂i ) = c0i + c1i = 1
ρ̂i = ⎜
2
⎜⎝ c0i c1i* c1i ⎟⎠
Generally, macroscopic constraints (volume, spin population, etc.) do not constrain the
phases of c0 = c0 eiϕ 0 and c1 = c1 eiϕ1 . Thus, ensemble averaging
⎛ 2
W
⎜ c0
1
ρ̂ = lim ∑ ρ̂ I = lim ⎜ −iϕ
W →∞ W
W →∞ e
⎜
i =1
⎜⎝
W
4
eiϕ ⎞
2
W ⎟ ⎛ c0
⎟ =⎜
⎜ 0
2 ⎟
c1 ⎟ ⎝
⎠
0 ⎞
2⎟
c ⎟⎠
1
4
⇒ Tr ( ρ̂ 2 ) = c0 + c1 < 1
Such a state is known as an impure state.
To obtain the equation of motion of ρ̂ , use the time-dependent Schrödinger equation
in propagator form,
ψ i (t) = e
i
− Ĥt

ψ i (0)
and the definition of ρ,
ρ̂i (t) = ψ i (t) ψ i (t)
to obtain the equation of motion
i
+ Ĥt ⎫
∂
∂ ⎧ − i Ĥt
ρ̂i (t) = ⎨e  ψ i (0) ψ i (0) e  ⎬
∂t
∂t ⎩
⎭
i
i
i
i
− Ĥt
+ Ĥt
− Ĥt
+ Ĥt ⎛
i
i ⎞
= − Ĥe  ψ i (0) ψ i (0) e  + e  ψ i (0) ψ i (0) e  ⎜ + Ĥ ⎟
⎝  ⎠

i
i
= − H ρi + ρi H


1
= [ ρi , H ]
i
This is known as the Liouville-von Neumann equation. 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
1 W
∂ρ̂
1
ρ̂i ⇒
= − [ ρ̂, Ĥ ]
∑
W i =1
∂t
i
This von-Neumann equation is the quantum equation of motion for ρ̂ . If ρ represent 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,
{
}
A(t) = ψ i (t)  ψ i (t) = ∑ ψ i (t) ϕ j ϕ j  ψ i (t) = ∑ ϕ j  ψ i (t) ψ i (t) ϕ j = Tr Âρ̂i (t)
j
j
Summing over ensembles,
⎧ 1
1 W
Tr Âρi = Tr ⎨ Â
∑
W i =1
⎩ W
In
particular,
⎫
{ }
∑ ρ̂ ⎬ ⇒ A(t) = T { Âρ̂(t)}
⎭
P = T { P̂ ρ̂(t)} = Tr { ϕ ϕ ρ̂(t)} = ϕ ρ̂(t) ϕ = ρ (t)
j
r
j
probability of being in state j at time t.
j
W
i =1
j
i
r
j
j
jj
is
the
∂ρ
=0⇒
∂t
⎡⎣ ρ̂eq , H ⎤⎦ = 0 (condition for equilibrium).
Finally, ρ̂ may evolve to long-time solutions ρ̂eq such that
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
⎛ E0 0 ⎞
⎛ ρ00 0 ⎞
H j = E j for j = 0,1 or H = ⎜
;
if
p̂
=
⎜⎝ 0 ρ ⎟⎠ ⇒ [ ρ̂, H ] = 0
⎝ 0 E1 ⎟⎠
11
Thus a diagonal density matrix of a closed system does not evolve. The equilibrium
density matrix must be diagonal so [ ρeq , H ] = 0 . 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
1
ρ̂ = ∑ ρi
W i
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 (e.g. a heat reservoir). ρ̂ depends on both i and j and can be written as a matrix
ρ̂ij,i1 j1 (t). 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
ρ̂ ( red ) = Trj {ρ̂}
ρ̂ ( red ) 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, ρ̂ ( red ) is still a 2x2 matrix.
We will show later that for a bath is at constant T.
⎛ e − E0 / k B T
⎜ Q
( red )
ρ̂ → ρeq = ⎜
⎜
⎜ 0
⎝
⎞
⎟
⎟
e− E1 / kB T ⎟
Q  ⎟⎠
0
Note that “reducing” was not necessary in the classical discussion because quantum
coherence and phases do not exist there. To summarize analogous entities:
Classical
ρ(qi , pi ,t;constr.)
Quantum
ρ̂(t;constr.)
Density function or operator
Â(qi , pi )
operator
∂
∂
H = q
H = − p
∂p
∂q
Â
Observable, hermition
1
1
Hψ r = ψ i Hψ i = −ψ r


Eq. of motion for traj./ ψ
T{ }
∫∫ dp dq
T { ρ̂} = 1
∫∫ dp dq ρ = 1
A(t) = ∫∫ dp dq Â(q , p )ρ(q , p ,t)
A(t) = T ( Â, ρ̂ )
i
i
r
Averaging
i
i
r
Conservation of probability
i
∂ρ
= − [ ρ, H ]P
∂t
⎡⎣ ρeq , H ⎤⎦ = 0
P
i
i
i
i
i
∂ρ̂ 1
= [ ρ, H ]
∂t i
⎡⎣ ρ̂eq , H ⎤⎦ = 0
r
Expectation value in ens.
Eq. of motion for p
Necessary eq. 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 ρ.