1. Statistical Description of Macroscopic Systems of Particles

2. Laws of Classical or Quantum Mechanics ≡ Statistical Mechanics
Now, we are ready to talk about
PHYSICS
In the rest of the course, we’ll combine statistical ideas with the
Laws of Classical or Quantum Mechanics ≡ Statistical Mechanics
We can use either a classical or a quantum description of a system. Of course, which is valid obviously depends on the problem!!

3. Statistical Description of a System with many particles:
Four Necessary Ingredients for a
Statistical Description of a System with many particles:
1. Specify the System “Macrostate”.
2. Choose a Statistical Ensemble
3. Formulate a Basic
Postulate about à-priori Probabilities.
4. Do Probability Calculations

4. A specification of the system’s macroscopic parameters
1. Specify the System “Macrostate”.
From your undergrad course, you should recall what is meant by “Macrostate” & that this is very different than the system “Microstate”! We’ll quickly review these concepts.
Macrostate  Macroscopic System State
A specification of the system’s macroscopic parameters

5. Specification of the System State
Microstate  Microscopic System State
A quantum description of the System: This means specifying a (large!)
set of quantum numbers.
Classical Description of the System: This means specifying a point in a
large dimensional phase space.

6. subset of the quantum states
Quantum Description
of the System:
For an isolated system, this means specifying a
subset of the quantum states
of the system.
The system is described by macroscopic parameters (that can be measured).

7. 2. Statistical Ensemble:
We need to decide exactly which ensemble to use. This is also discussed in this chapter.
In either Classical Mechanics
or Quantum Mechanics:
If we had a detailed knowledge of all positions & momenta of all system particles & if we knew all inter-particle forces, we could (in principle) set up & solve the coupled, non-linear differential equations of motion, we could find EXACTLY the behavior of all particles for all time!

8. Statistical/Probabilistic Methods.
If we could set up & solve the coupled, non-linear differential equations of motion, we could (in principle) find EXACTLY the behavior of all particles for all time!
In practice we don’t have enough information to do this. Even if we did, such a problem is
Impractical, if not Impossible to solve!
Instead, we’ll use
Statistical/Probabilistic Methods.

9. Require choosing an Ensemble
Statistical/Probabilistic Methods:
Require choosing an Ensemble
Now, lets think of doing MANY (≡ N) similar experiments on the system of particles we are considering. In general, the outcome of each experiment will be different.
So, we ask for the PROBABILITY of a particular outcome. This PROBABILITY ≡ the fraction of cases out of N experiments which have that outcome.
This is how probability is determined by experiment.
A goal of STATISTICAL MECHANICS
is to predict this probability theoretically.

10. 3. A Basic Postulate about
Next, we need to start somewhere, so we need to assume
3. A Basic Postulate about
à-priori Probabilities.
“à-priori” ≡ prior (based on our prior knowledge of the system).
Our knowledge of a given physical system leads is to expect that there is NOTHING in the laws of mechanics (classical or quantum) which would result in the system preferring to be in any particular one of it’s
Accessible (micro) States.

11. Webster’s on-line Dictionary:
Definition of “à-priori”
1a : Deductive
1b: Relating to or derived by reasoning from self-evident propositions.
a synonym to “à-posteriori”
1c: Presupposed by experience.
2a : Being without examination or analysis. analysis : Presumptive
2b : Formed or conceived beforehand

12. Accessible Microstates.
3. Basic Postulate about
à-priori Probabilities.
There is NOTHING in the laws of mechanics (classical or quantum) which would result in the system preferring to be in any particular one of it’s
Accessible Microstates.

13. that the system is in ANY ONE of it’s accessible microstates.
3. Basic Postulate about
à-priori Probabilities.
So, (if we have no contrary experimental evidence) we make the hypothesis that:
it is equally probable
(or equally likely)
that the system is in ANY ONE
of it’s accessible microstates.

14. Physics is an experimental science!!
The hypothesis is that it is equally probable (equally likely) that the system is in ANY ONE of it’s accessible microstates.
This postulate is reasonable & doesn’t contradict any laws of mechanics (classical or quantum). Is it correct?
That can only be confirmed by checking theoretical predictions & comparing those to experimental observation!
Physics is an experimental science!!
Sometimes, this postulate is called
The Fundamental Postulate of (Equilibrium) Statistical Mechanics!

15. Fundamental Postulate, Probability Theory
4. Probability Calculations
Finally, we can do some calculations!
Once we have the
Fundamental Postulate,
we can use
Probability Theory
to predict the outcome of experiments.
Now, we will go through steps 1., 2., 3., 4. again in detail!

16. Statistical Formulation of the Mechanical Problem
Specification of the System State
≡ Microstate
Consider any system of particles. We know that the particles will obey the laws of Quantum Mechanics (we’ll discuss the Classical description shortly).
We’ll emphasize the Quantum treatment.

17. Ψ(q1,q2,….qf,t), Consider any system of particles.
Using the Quantum treatment, consider a system with f degrees of freedom can be described by a (many particle!) wavefunction
Ψ(q1,q2,….qf,t),
where q1,q2,….qf ≡ Set of f generalized
coordinates required to characterize the system (needn’t be position coordinates!)

18. Ψ(q1,q2,….qf,t), Schrödinger Equation
For a system with f degrees of freedom, the many particle wavefunction is formally:
Ψ(q1,q2,….qf,t),
q1,q2,….qf ≡ a set of f generalized coordinates which are required to characterize the system.
A particular quantum state (macrostate) of the system is specified by giving values of some set of f quantum numbers. If we specify Ψ at a given time t, we can (in principle) calculate it at any later time by solving the appropriate
Schrödinger Equation

19. Now, briefly look at some simple examples, which might also review some elementary
Quantum Mechanics.

20. 2 possible states of the system.
Example 1
Single particle, fixed in position,
intrinsic spin s = ½
Intrinsic angular momentum = ½ћ.
In the Quantum Description of this system, the state of the particle is specified by specifying the projection m of this spin along a fixed axis
(which we usually call the z-axis).
The quantum number m can thus have 2 values:
½ (“spin up”) or -½ (“spin down”)
So, there are
2 possible states of the system.

21.  There are (2)N unique states of the system! With N ~ 1024,
Example 2
N particles (non-interacting), fixed in position. Each has intrinsic spin ½ so EACH particle’s quantum number mi (i = 1,2,…N) can have one of the 2 values ½.
Suppose that N is HUGE: N ~ 1024.
The state of this system is then specified by specifying the values of EACH of the quantum numbers: m1,m2, .. mN.
 There are (2)N unique states of the system! With N ~ 1024,
this number is HUGE!!!

22. or μz = -μ, for spin “down”
Example 3
A system with 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”

23. Possible States of a 3 Spin System in Any One of These 8 States.
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.

24. ε+ ≡ - μH for spin “up” ε- ≡ μH 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!!

25. “States Accessible to the System” ≡
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!

26. (+,+,-) (+,-,+) (-,+,+) Example 4 E ≡ - μH
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:
(+,+,-) (+,-,+) (-,+,+)

27. So the Quantum Energy of this system is:
Example 5
The system is a quantum mechanical, one-dimensional, simple harmonic oscillator, with position coordinate x & classical frequency ω.
So the Quantum Energy of this system is:
En = ћω(n + ½), (n = 0,1,2,3,….).
The quantum states of this oscillator are then specified by specifying the quantum number n. So, there are essentially an
 NUMBER of such states!

28. Example 6 Ei = ћωi(ni + ½), (ni = 0,1,2,3,….).
The system is N quantum mechanical, one-dimensional, simple harmonic oscillators, at positions xi, with classical frequencies ωi (i = 1,2,.. N).
The Quantum Energies of each particle in this system are:
Ei = ћωi(ni + ½), (ni = 0,1,2,3,….).
The system’s quantum states are specified by specifying the values of each quantum number ni.
Here also, there are essentially an
 NUMBER of such states.
But, there are also a larger number of these than in Example 3!

29. How do we specify the state of the Classical system?
What about the Classical Description
of the state of a many particle System? The
Quantum Description is always correct!
But, it is often useful & convenient to
make the
Classical Approximation.
How do we specify the state of the Classical system?

30. A Single Particle in 1 Dimension:
Start with a very simple case:
A Single Particle in 1 Dimension:
In classical mechanics, it can be completely described in terms of it’s generalized coordinate q & it’s momentum p. The usual case: consider the
Hamiltonian Formulation
of classical mechanics, where we talk of generalized coordinates q & generalized momenta p, rather than the
Lagrangian Formulation,
where we talk of coordinates q & velocities (dq/dt).

31. Hamilton’s Equations of Motion.
The particle obeys
Newton’s 2nd Law
under the action of the forces on it.
Equivalently, it obeys
Hamilton’s Equations of Motion.
q & p completely describe the particle classically. Given q, p at any initial time (say, t = 0), they can be determined at any other time t by integrating the equations of motion.

32.  q & p completely describe the particle
q & p completely describe the particle classically. Given q, p at any initial time (say, t = 0), they can be determined at any other time t by integrating the
Newton’s 2nd Law
Equations of Motion
forward in time.  Knowing q & p at t = 0 in principle allows us to know them for all time t.
 q & p completely describe the particle
for all time. This situation can be abstractly represented in a geometric way discussed on the next page.

33. “State of the Particle”
Consider the (abstract) 2-dimensional space defined by q, p: ≡ “Classical Phase Space” of the particle.
At any time t, stating the (q, p) of the particle describes it’s
“State”
Specification of the
“State of the Particle”
is done by stating which point in this plane the particle “occupies”.
Of course, as q & p change in time, according to the
equation of motion, the point representing the particle
“State” moves in the plane.

34. 2-Dimensional Classical “Phase Space”
q, p are continuous variables, so an  number of points are in this
2-Dimensional Classical “Phase Space”
We’d like to describe the particle “State” classically in a way that the number of states is countable.
To do this, it is convenient to
subdivide the ranges of q & p
into very small rectangles of size:
q  p.
Think of this 2-d phase space as divided into small cells of equal area: qp ≡ ho
ho ≡ a small (arbitrary) constant with units of angular momentum .

35. “State” The phase space cell labeled by the (q,p) that
The 2-d phase space has a large number cells of area:
qp = ho.
The (classical) particle “State” is specified by stating which cell in phase space the q, p of the particle is in. Or, by stating that it’s coordinate lies between q & q + q & that it’s momentum lies between p & p + p.
“State”
The phase space cell
labeled by the (q,p) that
the particle “occupies”.

36. Heisenberg Uncertainty Principle:
This involves the “small” parameter ho, which is arbitrary. As a side note, however, we can use
Quantum Mechanics & the
Heisenberg Uncertainty Principle:
“It is impossible to SIMULTANEOUSLY
specify a particle’s position & momentum
to a greater accuracy than qp ≥ ½ћ”
So, the minimum value of ho is clearly ½ћ.
As ho ½ћ,
The classical description of the State approaches the quantum description & becomes more & more accurate.

37. MANY PARTICLE SYSTEM Multidimensional phase space.
Now!! Lets generalize all of this to a
MANY PARTICLE SYSTEM
1 particle in 1 dimension means we have to deal
with a 2-dimensional phase space.
The generalization to N particles is straightforward, but requires thinking in terms of a very abstract
Multidimensional phase space.
Consider a system with f degrees of freedom:
 The system is described classically by
f generalized coordinates: q1,q2,q3, …qf
f generalized momenta: p1,p2,p3, …pf.

38. f generalized coordinates: q1,q2,q3, …qf.
A complete description of the classical “State” of
the system requires the specification of:
f generalized coordinates: q1,q2,q3, …qf.
& f generalized momenta: p1,p2,p3, …pf
(N particles, 3-dimensions  f = 3N !!)

39. 2f-dimensional phase space
A complete description of the classical “State” of
the system requires the specification of:
f generalized coordinates: q1,q2,q3, …qf.
& f generalized momenta: p1,p2,p3, …pf
(N particles, 3-dimensions  f = 3N !!)
So, now lets think VERY abstractly in terms of a
2f-dimensional phase space
The system’s
& f generalized momenta: p1,p2,p3, …pf
are regarded as a point in the 2f-dimensional phase space of the system.

40. 2f-dimensional Phase Space: 2f-dimensional “differential volume”:
f q’s & f p’s:
Each q & each p label an axis (analogous to the 2-d phase space for 1 particle in 1 dimension).
Subdivide this phase space into small “cells” of
2f-dimensional “differential volume”:
q1q2q3…qfp1p2p3…p1f ≡ (ho)f
The classical “State” of the system is then ≡ the cell in this 2f-dimensional phase space that the system “occupies”.

41. ≡ “The Gibbs Viewpoint” ≡ “The Boltzmann Viewpoint”:
Reif, as all modern texts, takes the viewpoint that the system’s “State” is described by a 2f-dimensional phase space
≡ “The Gibbs Viewpoint”
The system “State” ≡ The cell in this phase space that the system “occupies”.
Older texts take a different viewpoint
≡ “The Boltzmann Viewpoint”:
In this viewpoint, each particle moves in
it’s own 6-dimensional phase space
In this view, specifying the system “State” requires specifying each cell in this phase space that each particle in the system “occupies”.

42. 2f-dimensional phase space
Summary
Specification of the System State:
In Quantum Mechanics:
Enumerate & label all possible system quantum states.
In Classical Mechanics:
Specify which cell in
2f-dimensional phase space
The system is in.
Need the coordinates & momenta of all particles & the “box” in the p-q plane the system occupies. As ho → ½ћ, the classical & quantum descriptions become the same.