Cossey L. Introduction to Mathematical Physics

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where K

can take values , where a is lattice's period and n is an integer. Plot of E as

a function of k is represented in figure

figeneeleclib.

Quasi-free electron model

Let us show that if the potential is no more the potential of a periodic box, degeneracy at

is erased. Consider for instance a potential V(x) defined by the sum of the box

periodic potential plus a periodic perturbation:

In the free electron model functions

are degenerated. Diagonalization of Hamiltonian in this basis (perturbation method for

solving spectral problems, see section chapresospec) shows that degeneracy is erased by

the perturbation.

Thigh binding model

Tight binding approximation consists in approximating the state space by the space

spanned by atomic orbitals centred at each node of the lattice. That is, each eigenfunction

is assumed to be of the form:

ψ(r) =

∑

(r − R

)

Application of Bloch's theorem yields to look for ψ

such that it can be written:

ψ (r) = e u (r)

ikr

Identifying u (r)

and u (r + R )

k i

, it can be shown that . Once more, symmetry

considerations fully determine the eigenvectors. Energies are evaluated from the

expression of the Hamiltonian.

Exercises

Exercice:

Give the name of the symmetry group molecule NH

belongs to.

Exercice:

Parametrize the vibrations of molecule NH

and expand it in a sum of irreducible

representations.

Exercice:

Propose a basis for the study of molecule BeH

different from those presented at section

secnnne.

Exercice:

Find the eigenstates as well as their energies of a system constituted by an electron in a

square box of side a (potential zero for − a / 2 < x < a / 2 and − a / 2 < y < + a / 2,

potential infinite elsewhere). What happens if to this potential is added a perturbation of

value ε on a quarter of the box (0 < y < a / 2 and 0 < y < a / 2)? Calculate by perturbation

the new energies and eigenvectors. Would symmetry considerations have permitted to

know in advance the eigenvectors?

Introduction

Statistical physics goal is to describe matter's properties at macroscopic scale from a

microscopic description (atoms, molecules, etc\dots). The great number of particles

constituting a macroscopic system justifies a probabilistic description of the system.

Quantum mechanics (see chapter ---chapmq---) allows to describe part of systems made

by a large number of particles: Schr\"odinger equation provides the various accessible

states as well as their associated energy. Statistical physics allows to evaluate occupation

probabilities P

of a quantum state l. It introduces fundamental concepts as temperature,

heat\dots

Obtaining of probability P

is done using the statistical physics principle that states that

physical systems tend to go to a state of ``maximum disorder. A disorder measure is

given by the statistical entropy\footnote{ This formula is analog to the information

entropy chosen by Shannon in his information theory .}{entropy}

The statistical physics principle can be enounced as:

Postulate:

At macroscopic equilibrium, statistical distribution of microscopic states is, among all

distributions that verify external constraints imposed to the system, the distribution that

makes statistical entropy maximum.

{{IMP/rem| This problem corresponds to the classical minimization (or maximization)

problem presented at section chapmetvar) which can be treated by Lagrange multiplier

method.

Entropy maximalization

The mathematical problem associated to the calculation of occupation probability P

here presented:{maximalization} In general, a system is described by two types of

variables. External variables y

whose values are fixed at y

by the exterior and internal

variables X

taht are free to fluctuate, only their mean being fixed to . Problem to

solve is thus the following:

Problem:

Find distribution probability P

over the states (l) of the considered system that

maximizes the entropy

and that verifies following constraints:

Entropy functional maximization is done using Lagrange multipliers technique. Result is:

where function Z, called partition function, {partition function} is defined by:

Numbers λ

are the Lagrange multipliers of the maximization problem considered.

Example:

In the case where energy is free to fluctuate around a fixed average, Lagrange multiplier

is:

where T is temperature.{temperature} We thus have a mathematical definition of

temperature.

Example:

In the case where the number of particles is free to fluctuate around a fixed average,

associated Lagrange multiplier is noted βµ where µ is called the chemical potential.

Relations on means\footnote{They are used to determine Lagrange multipliers λ

from

associated means

} can be written as:

It is useful to define a function L by:

L = lnZ(y,λ ,λ ,...)

1 2

It can be shown\footnote{ By definition

thus

} that:

This relation that binds L to S is called a {\bf Legendre

transform}.{Legendre transformation}

L is function of the y

's and λ

's, S is a function of the y

's and 's.

Canonical distribution in classical

mechanics

Consider a system for which only the energy is fixed. Probability for this system to be in

a quantum state (l) of energy E

is given (see previous section) by:

Consider a classical description of this same system. For instance, consider a system

constituted by N particles whose position and momentum are noted q

and p

, described

by the classical hamiltonian H(q ,p )

i i

. A classical probability density w

is defined by:

Quantity w (q,p )dq dr

i i i i

represents the probability for the system to be in the phase space

volume between hyperplanes q,p

i i

and q + dq ,p + dp

i i i i

. Normalization coefficients Z and

A are proportional.

One can show that

being a sort of quantum state volume.

Remark:

This quantum state volume corresponds to the minimal precision allowed in the phase

space from the Heisenberg uncertainty principle: {Heisenberg uncertainty principle}

Partition function provided by a classical approach becomes thus:

But this passage technique from quantum description to classical description creates

some compatibility problems. For instance, in quantum mechanics, there exist a postulate

allowing to treat the case of a set of identical particles. Direct application of formula of

equation eqdensiprobaclas leads to wrong results (Gibbs paradox). In a classical

treatment of set of identical particles, a postulate has to be artificially added to the other

statistical mechanics postulates:

Postulate:

Two states that does not differ by permutations are not considered as different.

This leads to the classical partition function for a system of N identical particles:

Constraint relaxing

We have defined at section secmaxient external variables, fixed by the exterior, and

internal variables free to fluctuate around a fixed mean. Consider a system L being

described by N + N' internal variables {constraint} . This

system has a partition function Z

. Consider now a system F, such that variables n

are this

time considered as external variables having value N

. This system F has (another)

partition function we call Z

. System L is obtained from system F by constraint relaxing.

Here is theorem that binds internal variables n

of system L to partition function Z

system F :

Theorem:

Values n

the most probable in system L where n

are free to fluctuate are the values that

make zero the differential of partition function Z

where the n

's are fixed.

Proof:

Consider the description where the n

's are free to fluctuate. Probability for event n =

, ,n = N

N N

occurs is:

Values the most probable make zero differential

(this

corresponds to the maximum of a (differentiable) function P.). So

Remark:

This relation is used in chemistry: it is the fundamental relation of chemical reaction. In

this case, the N variables n

represent the numbers of particles of the N species i and N' =

2 with X = E

(the energy of the system) and X = V

(the volume of the system). Chemical

reaction equation gives a binding on variables n

that involves stoechiometric

coefficients.

Let us write a Gibbs-Duheim type relation {Gibbs-Duheim relation}:

At thermodynamical equilibrium S = S

F L

, so:

Example:

This last equality provides a way to calculate the chemical potential of the

system.{chemical potential}

In general one notes − ln(Z ) = G

Example:

Consider the case where variables n

are the numbers of particles of species i. If the

particles are independent, energy associated to a state describing the N particles (the set

of particles of type i being in state l

) is the sum of the N energies associated to states l

Thus:

where represents the partition function of the system constituted

only by particles of type i, for which the value of variable n

is fixed. So:

Remark:

Setting λ = β

,λ = βp

and with G = − k TlnZ

, we have G(T,p,n) = E + pV

− TS and G'(T,p,µ) = E + pV − TS − µ − n

. This is a Gibbs-Duheim relation.

Example:

We propose here to prove the Nernst formula{Nernst formula} describing an oxydo-

reduction reaction.{oxydo-reduction} This type of chemical reaction can be tackled using

previous formalism. Let us precise notations in a particular case. Nernst formula

demonstration that we present here is different form those classically presented in

chemistry books. Electrons undergo a potential energy variation going from solution

potential to metal potential. This energy variation can be seen as the work got by the

system or as the internal energy variation of the system, depending on the considered

system is the set of the electrons or the set of the electrons as well as the solution and the

metal. The chosen system is here the second. Consider the free enthalpy function

G(T,p,n ,E )

i p

. Variables n

and E

are free to fluctuate. They have values such that G is

minimum. let us calculate the differential of G:

Using definition\footnote{ the internal energy U is the sum of the kinetic energy and the

potential energy, so as G can be written itself as a sum:

G = U + pV − TS

}

of G:

one gets:

0 =

∑

+ dE

If we consider reaction equation:

0 =

∑

dξ + nq(V

red

− V

)

So:

dG can only decrease. Spontaneous movement of electrons is done in the sense that

implies dE < 0

. As we chose as definition of electrical potential:

V = − V

ext int

Nernst formula deals with the electrical potential seen by the exterior.

Some numerical computation in statistical

physics

In statistical physics, mean quantities evaluation can be done using by Monte--Carlo

methods. in this section, a simple example is presented.

Example:

Let us consider a Ising model. In this spin system, energy can be written:

The following Metropolis algorithm

Exercise

Exercice:

Consider a chain of N nonlinear coupled oscillators. Assuming known the (deterministic)

equation governing the dynamics of the particles, is that possible to define a system's

temperature? Can one be sure that a system of coupled nonlinear oscillators will evolve to

a state well described by statistical physics? Do a bibliographic research on this subject,

in particular on Fermi--Ulam--Pasta model.