1. 10.6. Cluster Expansion for a Quantum Mechanical System
Define the probability density operator as  

2. WN 
Mathematica  

3. Since HN is invariant under particle exchange,
bosons
fermions 
bosons & fermions
A unitary transformation leaves the trace of an operator invariant.
For WN , there are 2 sets of bases, { |   } & { | 1, ..., N  }.
Thus, a unitary transformation on { |   } that leaves the set { | 1, ..., N  } untouched will leave the diagonal elements WN(1,...,N) invariant.

4. If can be divided into two groups A & B, i.e.,
such that 1. 2.
Then
r0 = effective range of u(r)
Difficult to prove mathematically, but reasonable physically.

5. Wl ~ Ul
For N = 2 :
Let 
Cluster functions Ul are defined by :

6. Ul ~ Wl 

7. bl
Cluster integrals :
Properties of bl :
dimensionless. 

8. 
Analogous to § 

9. Classically, bl is obtained by calculating a few fij integrals ( see §10.1) .
The quantum counterpart requires calculating all j-body interaction Uj with j l.
Lee-Yang scheme : bl calculated by successive approximations.
Ul calculated using Boltzmann statistics & unsymmetrized .
Ul expanded in powers of a binary (2-body) kernel B.
Better approach: Quantum field theory (see Chap 11)
Reminder :
Classical :
Quantum ( § 7.1 & 8.1 ) :

10. 10.7. Correlations & Scattering
Uncorrelated
Correlated
Short-Range
Long-Range
vapor
liquid,
paramagnets
solid,
ferro-magnets
critical pt.
power law
liquid crystal

11. n1 Number densities nj : 1-body (local) number density
Translationally invariant system :

12. n2
2-body number density
= probability of finding one particle within ( r, r+d r )
& another within ( r, r +d r )
Translationally invariant system :
Ideal gas ( u = 0 ) : 
Pair correlation function g :

13. = probability of finding a particle
in spherical shell of radius r & thickness dr.
for classical ideal gas ( Prob.10.7 )
Exact solution of the Percus-Yevick approximation for hard sphere gas with diameter D & 

14. No particle can be found inside core 
This causes oscillation in g
 = correlation length

15. P Classical fluid with potential energy Canonical ensemble :
( configurational partition function )
where 

16. From § 3.7 :
d = dimension 
Virial equation of state 

17. For hard spheres :
Prob.10.14
Also

18. 10.7.A. Static Structure Factor
Incoming plane wave is scattered by particle at ri .
Scattered wave is detected at R with
Born approximation :
1(k) = amplitude of scattering wave
f (k) = single particle scattering form factor
Scattering of N particles :

19. Scattering intensity :
static structure factor
... = average over ri & rj
Uniform fluid ( translationally invariant & isotropic ) : 
( S ~ Fourier transform of g )

20. 
d = 1
d = 2
d = 3
fluctuation-compressibility relation
inverse FT of S
l-Ar n-scatt.

21. 10.7.B. Scattering from Crystalline Solids
Simple crystal : identical atom at each (Bravais) lattice site R.
Simple cubic lattice :
Reciprocal lattice vectors G :
Simple cubic lattice :
Perfect Bravais lattice (no vibrations)

22. (no vibrations)
Thermal excitation :
Gaussian excitation :
Motion uncorrelated for far-apart atoms :
Debye-Wallace factor 
No long range order in 2-D lattice with short range interaction.
power-law singularity instead of  in S.
Solid / liquid transition is continuous (Kosterlitz-Thouless.)