Transcript Document

Section 2.2: Statistical Ensemble
A typical episode from the on-line comic series
“PhD Comics” by Jorge Cham
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“PhD Movie I” & “PhD Movie II”
• As we know, Statistical Mechanics deals with the
behavior of systems of a large number of particles.
• Because the number of particles is so huge, we give
up trying to keep track of individual particles.
• We can’t solve Schrödinger’s Eqtn in closed form for
helium (4 particles), so what hope do we have of solving
it for the gas molecules in this room (10f particles) ??
• Statistical Mechanics handles many particles by
Calculating the Most Probable
Behavior of the System as a Whole
rather than by being concerned with the behavior
of individual particles.
In Statistical Mechanics
• We assume that the more ways there are to
arrange the particles to give a particular distribution
of energies, the more probable that distribution is.
Example: 6 energy units, 3 particles to give it to
321
312
213
231
123
132
6 ways
411
141
114
3 ways
Most Probable
Distribution
Color Codes for
Energy Units
≡1
≡2
≡3
≡4
Another Example
• Assuming that
All Energy Distributions are
Equally Probable
• If E = 5 and N = 5 then
5
5
5
4
3
2
1
0
4
3
2
1
0
4
3
2
1
0
• All possible configurations have equal probability, but
the possible number of ways (weight) is different for each.
The Dominant Configuration
• For a large number of molecules & a large number of
energy levels, there is a Dominant Configuration.
• In the probability distribution, the weight of the
dominant configuration is much larger than the
weight of the other configurations.
Weight of the Dominant
Configuration
Weights
Wi
{ni}
 Configurations
The Dominant Configuration
If E = 5 and N = 5 then
W = 1 = (5!/5!)
5
5
5
4
3
2
1
0
4
3
2
1
0
4
3
2
1
0
W = 20 = (5!/3!)
W = 5 = (5!/4!)
• The difference in the W’s becomes larger as N increases!
• In molecular systems (N~1023) considering the most dominant
configuration is certainly enough to calculate averages.
The Principle of Equal à-priori
Probabilities
• Statistical thermodynamics is based on the
fundamental hypothesis of
Equal à-priori Probabilities.
• That is, all possible configurations of a given
system which satisfy the given boundary conditions
such as temperature, volume and number of
particles, are equally likely to occur. OR
The system is equally likely to be found
in any one of its accessible states.
Example
• Consider the orientations of three unconstrained
& distinguishable spin-1/2 particles.
• What is the probability that 2 are spin up &
1 is down at any instant?
Example
• Consider the orientations of three unconstrained
& distinguishable spin-1/2 particles.
• What is the probability that 2 are spin up &
1 is down at any instant?
Solution
• Of the eight possible spin configurations for the system:
↑↑↑, ↑↑↓, ↑↓↑, ↓↑↑, ↑↓↓, ↓↑↓, ↓↓↑, ↓↓↓
• The second, third, & fourth make up the subset
"two up and one down". Therefore,
The probability of occurrence of this
particular configuration is: P = 3/8
• In principle, the problem of a many particle system is
COMPLETELY deterministic:
• If we specify the many particle wavefunction Ψ (state) of
the system (or the classical phase space cell) at time t = 0,
we can determine Ψ for all other times t by solving
The Time-Dependent Schrödinger Equation
& from Ψ(t) we can calculate all observable quantities.
• Or, classically, if we specify the positions & momenta of
all particles at time t = 0, we can predict the future
behavior of the system by solving
The Coupled Many Particle
Newton’s 2nd Law Equations of Motion.
• Generally, we usually don’t have such a complete
specification of the system available.
• We need f quantum numbers, but f ≈ 1024!
• Actually, we usually aren’t interested in such a
complete microscopic description anyway.
Instead, we’re interested in predictions of
MACROSCOPIC properties.
 We use Probability & Statistics.
 To do this we need the concept of an
ENSEMBLE.
• A Statistical Ensemble is a LARGE number
(≡ N) of identically prepared systems.
• In general, the systems of this ensemble will
be in different states & thus will have
different macroscopic properties.
 We ask for the probability that a given
macroscopic parameter will have a certain value.
A Goal or Aim of Statistical
Mechanics is to
Predict this Probability.
Example
• Consider the spin problem again. But, now,
Let The System Have N = 3 Particles,
fixed in position, each with spin = ½
 Each spin is either “up” (↑, m = ½)
or “down” (↓, m = -½).
• Each particle has a vector magnetic moment μ.
• The projection of μ along a “z-axis” is either:
μz = μ, for spin “up”
or μz = -μ, for spin “down”
• Put this system into an External Magnetic Field H.
• Classical E&M tells us that a particle with
magnetic moment μ in an external field H has energy:
ε = - μ H
• Combine this with the Quantum Mechanical result:
 This tells us that each particle has 2 possible energies:
ε+ ≡ - μH for spin “up”
ε- ≡ μH for spin “down”
 So, for 3 particles, the State of the system
is specified by specifying each m = 
 There are (2)3 = 8 Possible States!!
Possible States of a 3 Spin System
• Given that we know no other information about this
system, all we can say about it is that
It has Equal Probability of Being Found
in Any One of These 8 States.
• However, if (as is often the case in real problems) we have a
partial knowledge of the system (say, from experiment),
then, we know that
The system can be only in any
one of the states which are
COMPATIBLE with our knowledge.
(That is, it can only be in one of it’s accessible states)
“States Accessible to the System” ≡
those states which are compatible with ALL of the
knowledge we have about the system.
Its important to use all of the information
that we have about the system!
Example
• For our 3 spin
system, suppose that
we measure the total
system energy & we
find E ≡ - μH
• This additional information limits the states which
are accessible to the system. Clearly, from the table,
Out of the 8 states, only 3 are
compatible with this knowledge.
 The system must be in one of the 3 states:
(+,+,-) (+,-,+) (-,+,+)