1. 5. Formulation of Quantum Statistics
Quantum Mechanical Ensemble Theory: The Density Matrix
Statistics of the Various Ensembles
Examples
Systems Composed of Indistinguishable Particles
The Density Matrix & the Partition Function of a System of Free Particles

2. Statistics Particle type Math Object
Classical
Distinguishable
Phase space density
Quantum
Indistinguishable
Density matrix
Advantage of using density matrix :
Quantum & ensemble averaging are combined into one averaging.

3. Classical Statistical Mechanics
(Probability) density function  ( p,q,t ) :
Caution:
Some authors, e.g., Landau-Lifshitz, use a normalized version of  .
Liouville’s theorem :
Microcanonical ensemble :
Canonical ensemble :
Grand canonical ensemble :

4. Quantum Statistical Mechanics (To be Proved)
Ensemble = phase space
Classical mechanics :
Ensemble = Hilbert space
Quantum mechanics :
PE = projection operator onto the N-D subspace of states with energy E.
Microcanonical :
Canonical :
Grand canonical :

5. Pure State Density Operator
Orthonormal basis { | n  } is complete :
Expectation value of f :
Density operator for |   :

6. r-Representation 
f is a 1-particle operator 

7. Mixed State Density Operator
Averaged value of f :
Orthonormal basis { | n  } is complete :
Density operator :
Skip to
ensembles
Ex: Derive the quantum Liouville eq.

8. 5.1. Quantum Mechanical Ensemble Theory: The Density Matrix
Consider ensemble of N identical systems labelled by k = 1, 2,..., N.
Each system is described by
i = 1,2,..., N
 k runs through all independent solutions of this Schrodinger eq.
Let
be the wave function of the kth system in the ensemble.
Let
be a set of complete orthonormal basis that spans the Hilbert space of H & satisfies the relevant B.C.s. 
with

9.  
where
H can be t-dep  
 k

10. Density Operator Density operator :
pk = weighting (or probability) factor with
Matrix elements : 
n or  d ~ quantum averaging
 ens ~ k ~ ensemble averaging

11.  
where
H can be t-dep

12. Equilibrium Ensemble System in equilibrium  ensemble stationary :
i.e. 
and
Energy representation :  
System in equilibrium  
In a general basis ,  is hermitian
 detailed balance

13. Expectation Values Expectation value of a physical quantity G :
( Quantum + ensemble av. )
Assuming k normalized, i.e.,   
i.e.
k normalized :

14. 5.2. Statistics of the Various Ensembles
Microcanonical ensemble : Fixed N, V, E or
( quantum statistics: no Gibbs’ paradox )
( N, V, E;  ) = # of accessible microstates
Equal a priori probabilities postulate 
Energy representation:    
i.e.

15. Pure State Only 1 state  p is accessible   3rd law  
Energy representation : 
Thus
i.e.
idempotent
(  is a projector )
In another representation with basis { m } so that
,  normalized  

16. Mixed State Multiple states are accessible, i.e.  > 1.
Any representation :
= set of accessible
state indices
Let K be the subspace spanned by the accessible  k ’s.
Consider any orthonormal basis {n } such that
Since { k } is a basis of K, its completeness means 
(  is diagonal w.r.t. {n } )

17. Let 
k = ensemble member index 
So that
Postulate of a priori random phases

18. Canonical Ensemble E-representation : i.e.
Canonical ensemble : Fixed N, V, T.   
By definition

19. Grand Canonical Ensemble
Grand canonical ensemble : Fixed , V, T
Er, s = Er (Ns )
= E of r th state of Ns p’cle sys 

20. 5.3. Examples An Electron in a Magnetic Field signed
Single e with spin
& magnetic moment
Pauli matrices : 
A diagonal      
agrees with §

21. A Free Particle in a Box
Free particle of mass m in a cubical box of sides L. 
with
Periodic B.C : 
with

22. ( r - representation ) 
with 
( see next page )

23.  

24. 
 is symmetric 
Location uncertainty :
Particle density at r :

25. Alternatively
Uising
& integrate by parts twice : 

26. A Simple Harmonic Oscillator
n = 0,1,2,...
Hermite polynomials :
Rodrigues’ formula

27. is real
Kubo, “Stat Mech.”, p.175
Mathematica 

28. Probability density :
 q is a Gaussian with dispersion ( r.m.s. deviation ) :

29. Classical limit :
(purely thermal)  
Quantum limit :
(non-thermal)  
= Probability density of ground state

30.  

31. 5.4. Systems Composed of Indistinguishable Particles
N non-interacting particles subject to the same 1-particle hamiltonian h. 
i = label of the eigenstate assumed by the i th particle.
Let n = # of particles occupying the  th eigenstate. 
L( , j ) = label of the j th particle that occupies the  th eigenstate.

32. Note: [ ... ] = 1 if n = 0.
Let P denote a permutation of the particle labels :

33. Distinguishable particles :
permutations within the same  counted as the same.
permutations across different ’s counted as distinct.
 # of distinct microstates is
Indistinguishable particles :
Boltzmannian
( distinguishable p’cles)

34. Indistinguishable Particles
Particles indistinguishable  Physical properties unchanged
under particle exchange
i.e.  

35. Anti-symmetric : 
Pauli’s exclusiion principle
i.e. 
Fermi-Dirac statistics
Symmetric : 
Bose-Einstein statistics

36. 5.5. The Density Matrix & the Partition Function of a System of Free Particles
N non-interacting, indistinguishable particles :
Let i stands for ri , & i  for ri .
e.g.,
Goal: To write or

37. Non-interacting particles 
Periodic B.C. 
Bosons
Fermions
Mathematica

38. Consider the N ! permutations among { ki } associated with a given K.
 E is unchanged 
nk > 1 cases neglected
(measure 0)

39. arbitrary P 
P  = I
2-p'cle

40. from § 5.3
= thermal ( de Broglie ) wavelength

41. mean inter-particle distance = n = particle density
Let
with  
Mathematica
mean inter-particle distance =
n = particle density  

42. Resolution of problems in classical statistics:
Gibbs correction factor ( 1 / N! ).
Phase space volume per state
Classical limit :
Non-classical systems are said to be degenerate.
 n 3 = degeneracy discriminant
( no spatial correlation )
Classical limit

43. Exchange Correlation
Let N = 2 : 

44. Classical limit

45. Statistical Potential
Mathematica