Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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7.1. RELATION BETWEEN QUANTUM AND CLASSICAL MECHANICS.141<br />
purpose to illustr<strong>at</strong>e th<strong>at</strong> a connection between quantum mechanical density<br />
oper<strong>at</strong>ors and classical distribution functions can be made. In general, the<br />
expressions one needs for the distribution function W are more complic<strong>at</strong>ed<br />
than the form given by Wigner.<br />
Classical density m<strong>at</strong>rix.<br />
In order to give the formulas for classical st<strong>at</strong>istical mechanics we consider<br />
the canonical ensemble first. Since the Hamiltonian in the classical limit is<br />
diagonal in X, we find th<strong>at</strong> the density m<strong>at</strong>rix is diagonal too, and th<strong>at</strong><br />
< X|ρ| X >= ρ( X) = Ce −βH( X)<br />
(7.7)<br />
In many ways this form of ρ can be directly compared with the function W<br />
introduced in the previous paragraph.<br />
Volume per st<strong>at</strong>e.<br />
The normaliz<strong>at</strong>ion constant C follows again from the requirement th<strong>at</strong> Trρ =<br />
1. This trace implies a sum over all possible st<strong>at</strong>es. In phase space, this is<br />
replaced by an integral. We do have to multiply this integral by a volume factor,<br />
though. The magnitude of this volume factor follows from the old quantum<br />
mechanical quantiz<strong>at</strong>ion condition<br />
<br />
pdq = nh (7.8)<br />
where p is the generalized momentum corresponding to the generalized coordin<strong>at</strong>e<br />
q. This contour integral is an integral multiple of Planck’s constant h. In<br />
two-dimensional phase space [p, q] the area inside a quantum orbit is nh, hence<br />
the area between two quantum orbits is h itself. In other words, the area per<br />
orbit is h. This value is consistent with the Heisenberg uncertainty rel<strong>at</strong>ion.<br />
If we construct a wave-package from plane-waves, we find th<strong>at</strong> the position is<br />
specified within ∆x and the corresponding momentum coordin<strong>at</strong>e within ∆p.<br />
Also, ∆x∆p > α for a value of α which is about one. If we represent the st<strong>at</strong>es<br />
of a system by a grid of points in the p-x plane the distance between the points<br />
in the x-direction has to be about ∆x. If grid points are closer together than<br />
∆x they could not be distinguished and if they are further apart they do not<br />
cover all space. Similarly, the distance of the grid points in p-space should be<br />
∆p and hence the volume per point in one dimension should be ∆x∆p, or about<br />
α. The formula above shows th<strong>at</strong> we have to take α = 2π.<br />
Classical integrals.<br />
If the N particles in our system are not identical, we therefore have the<br />
rule