Statistical physics (2005)

Now consider the two particles. For a given microscopic configuration, the total energy is εi + εj . After having recalled the results for two distinguishable particles, we will have to separately consider the case of fermions and that of bosons.

since there is no limitation to the levels occupation by the second particle.

Fermions cannot be more than one to occupy the same state. Consequently, as soon as the level of energy εi is occupied by the first particle, the second one cannot be on it. Thus

In the case of bosons, in addition to the configurations possible for the fermions, the two particles can also lie on the same level :

It was easy to determine the levels available for the particles in this example because they were only two. For three particles one must then consider which restrictions appear for the third one because of the levels occupied by the first two particles. For N particles one should again proceed step by step, but this method is no longer realistic!

6.1.2Description in the Grand Canonical Ensemble

To avoid complicated conditions on the occupation numbers nk one will work in the grand canonical ensemble, assuming in a first step that the total number

134 Chapter 6. General Properties of the Quantum Statistics

of particles of the system is arbitrary. Thus one will calculate ZG and the grand potential A ; then the value of the Lagrange parameter α = βµ (or that of the chemical potential µ) will be fixed in such a way that the average number of particles N , deduced from the statistical calculation, coincides with the

real number of particles in the system. We know (§2.5.4) that the relative

√

fluctuation on N introduced by this procedure is of the order of 1/ N , that is, of the order of a few 10−12 for N of the order of the Avogadro number.

6.2Factorization of the Grand Partition Function

6.2.1Fermions and Bosons

By definition

where N is one of the numbers of particles and where α and the chemical potential are related by α = βµ. The canonical partition function ZN (β) corresponding to N particles is given by

Let us introduce the occupation numbers nk of the quantum states εk :

k

with

k

and the energy for this configuration of the N particles being given by

k

This geometrical series can be summed only if exp(α − βεk ) is smaller than unity, a condition to be fulfilled by all the states εk. One will thus have to verify that µ < ε1, where ε1 is the one-particle quantum state with the lowest energy, i.e., the fundamental state. This condition is also expressed through

α = βµ as α < βε1.

Then the grand partition function takes the form :

6.2.4Chemical Potential and Number of Particles

The grand canonical partition function ZG is associated with the grand potential A such that

As specified in §3.5.3, the partial derivatives of the grand partition function provide the entropy, the pressure and the average number of particles N :

In particular the constraint on the total number of particles of the system is expressed from :

6.3Average Occupation Number; Grand Potential

From the grand partition function ZG one deduces average values at temperature T of physical parameters of the whole system, but also of the state εk . In particular the average value of the occupation number nk of the state of energy εk is obtained through

nk

For fermions, the average occupation number, the so-called “Fermi-Dirac distribution,” is given by

For bosons, this average number is “the Bose-Einstein” distribution which takes the form

Note : Remember the sign di erence in the denominator between fermions and bosons! Its essential physical consequences are developed in the next chapters.

The average occupation numbers are related to factors in the grand partition function ZG : indeed one will verify that

These occupation numbers yield the average values, at given temperature, thus at fixed β, of physical parameters.

Thus the average number of particles is related to α and β, which appear in f (ε), through

The grand potential A is expressed, like the ZG factors, versus the occupation numbers of the states of energy εk :

k

a state of given phase propagates toward increasing x’s for kx > 0, toward decreasing x’s for kx < 0 ; the space wave function is a constant for kx = 0.

ii) or

corresponds to the same energy εx or the same pulsation ωx as in (6.39).

The solution of progressive-wave type (6.39) is found again if one chooses

√

A = 1/ Lx, B = iA. For real coe cients A and B, when changing the space origin in x there is indeed separation of the space and time variables in (6.41). Then a state of given space phase does not propagate in time, such a wave is stationary.

At three dimensions the kinetic energy terms along the three coordinates add up in the hamiltonian; the energy is equal to

and the wave function is the product of three wave functions, each concerning a single coordinate.

Now we are going to express, through boundary conditions on the wave function, that the particle is confined within the volume Ω. This will yield quantization conditions on the wave vector and the energy.

b) Boundary Conditions

The volume Ω containing the particles has a priori an arbitrary shape. However, it can be understood that each particle in motion is most of the time at a large distance from the container walls, so that the volume properties are not sensitive to the surface properties as soon as the volume is large enough. It is shown, and we will admit it, that the Statistical Physics properties of a macroscopic system are indeed independent of the shape of the volume Ω. For convenience we will now assume that Ω is a box (Lx, Ly , Lz), with macroscopic dimensions of the order of a mm or a cm.

We consider a one-dimension problem, on a segment of length Lx. The presence probability of the particle is zero outside the interval [O, Lx], in the region where the particle cannot be found. There are two ways to express this condition :

i) Stationary boundary conditions :

To indicate that the particle cannot leave the interval, one assumes that in x = 0 and x = Lx potential barriers are present, which are infinitely high and thus impossible to overcome. Thus the wave function vanishes outside the interval [O, Lx] and also at the extremities of the segment, to ensure its continuity at these points; since it vanishes at x = 0, it must be of the form ψ(x) = A sin kxx (Fig. 6.1). The cancellation at x = Lx yields

One understands that the physically distinct allowed values are restricted to nx > 0 : indeed taking nx < 0 is just changing the sign of the wave function, which does not change the physics of the problem. The value nx = 0 is to be rejected since a wave function has to be normalized to unity and thus cannot cancel everywhere. In the same way ky and kz are quantized by the conditions of cancellation of the wave function on the surface : finally the allowed vectors

k for the particle confined within the volume Ω are of the form