10.6. Cluster Expansion for a Quantum Mechanical System Define the probability density operator as
WN Mathematica
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.
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.
Wl ~ Ul For N = 2 : Let Cluster functions Ul are defined by :
Ul ~ Wl
bl Cluster integrals : Properties of bl : dimensionless.
Analogous to § 10.1
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 ) :
10.7. Correlations & Scattering Uncorrelated Correlated Short-Range Long-Range vapor liquid, paramagnets solid, ferro-magnets critical pt. power law liquid crystal
n1 Number densities nj : 1-body (local) number density Translationally invariant system :
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 :
= 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 &
No particle can be found inside core This causes oscillation in g = correlation length
P Classical fluid with potential energy Canonical ensemble : ( configurational partition function ) where
From § 3.7 : d = dimension Virial equation of state
For hard spheres : Prob.10.14 Also
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 :
Scattering intensity : static structure factor ... = average over ri & rj Uniform fluid ( translationally invariant & isotropic ) : ( S ~ Fourier transform of g )
d = 1 d = 2 d = 3 fluctuation-compressibility relation inverse FT of S l-Ar n-scatt.
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)
(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.)