Statistical physics (2005)

mical properties, determined by the number of electrons that can participate in chemical bonds, are the same.

5.4Case of Two Particles of Spin 1/2

Here the properties of the wave functions of two indistinguishable particles of spin 1/2 are recalled, this is in particular the case for two electrons. From the theorem of spin-statistics connection, this wave function must be antisymmetrical with respect to the transposition of these two particles. Now, in addition to their spin degree of freedom, these two particles also have an orbital degree of freedom concerning the space coordinates; from Quantum Mechanics courses, the wave function for a single particle of this type can be decomposed into tensor products concerning the space and spin variables. The most general form of decomposition is :

5.4.1Triplet and Singlet Spin States

Let us first assume, for simplification, that this decomposition is reduced to a single term, i.e.,

Since we are dealing with fermions, the two-particle wave function changes sign when particles 1 and 2 are exchanged. This sign change can occur in two di erent ways :

–either it is the space part ψ(r1, r2) of the wave function which is antisymmetrical in this exchange, the spin function remaining unchanged;

–or it is |σ1, σ2 which changes its sign in this exchange, the space function remaining unchanged.

Consider the spin states. Each particle has two possible spin states, noted | ↑ or | ↓ . There are thus four basis states for the set of two spins, which can be chosen as

(the order of notation is : particle 1, then particle 2).

From these states one deduces four new independent states, classified according to their symmetry, that is, their behavior when exchanging the spins of the two particles.

There are three symmetrical states, unchanged when the role of particles 1 and 2 is exchanged :

on the quantization axis Oz described by the respective quantum numbers Sz = +1, 0, −1 : the eigenvalue of the squared length of this spin is equal to S(S + 1) 2, i.e., 2 2 when the eigenvalue of the projection along Oz is Sz , i.e., , 0 or − .

There remains an antisymmetrical state, which changes its sign in the exchange of the two spins, this is the singlet state

in which S = Sz = 0.

Even when the two-particle hamiltonian does not explicitly depend on spin, orbital wave functions with di erent symmetries and possibly di erent energy eigenvalues are associated with the triplet and singlet states : for example, in the special case treated in §5.2.2 the fundamental state, with a symmetrical orbital wave function [see Eq. (5.14)], must be a spin singlet state, whereas in the first excited state one can have a spin triplet. This energy splitting between the symmetrical and antisymmetrical orbital solutions is called the exchange energy or exchange interaction.

The exchange interaction is responsible of ferromagnetism, as understood by Werner Heisenberg (1901-1976) as early as 1926 : indeed the dipolar interaction between two magnetic moments a fraction of a nm apart, each of the order of the Bohr magneton, is much too weak to allow the existence of an ordered phase of magnetization at room temperature. On the contrary, owing to the Pauli principle, in ferromagnetic systems the energy splittings between states with di erent spins correspond to separations between di erent orbital levels. These are of a fraction of an electron-volt, which is equivalent to temperatures of the order of 1000 K, since these states are the ones responsible for valence bindings.

In a ferromagnetic solid, magnetism can originate from mobile electrons in metals like in Fe, Ni, Co, or from ions in insulating solids, as in the magnetite iron oxide Fe3O4, and the spin can di er from 1/2. However this interaction between the spins, which relies on the Pauli principle, is responsible for the energy splitting between di erent orbital states : one state in which the total magnetic moment is di erent from zero, corresponding to the lower temperature state, and the other one in which it is zero.

5.4.2General Properties of the Wave Function of Two Spin 1/2 Particles

It has just been shown that, because of the Pauli principle, space wave functions are associated with the triplet states, which are changed into their opposite in the exchange of the two particles (antisymmetrical wave function). To the singlet state is associated a symmetrical wave function. The most general wave function for two spin 1/2 particles will be written, for a decomposition onto the triplet and singlet states :

ψ11A |11 + ψ10A |10 + ψ1A−1|1 − 1 + ψ00S |00

The wave functions of the type ψ11A are spatial functions of r1 and r2, the upper index recalls the symmetry of each component.

The same wavefunction can be expressed on the spin basis |σ1, σ2 , accounting for expressions (5.24) and (5.25) of the triplet and singlet wave functions. One thus obtains :

It thus appears, by comparing to the general expression (5.21), that the Pauli principle dictates symmetry conditions on the spin 1/2 components :

•ψ++(r1, r2) and ψ−−(r1, r2) must be antisymmetrical in the exchange of particles 1 and 2

•ψ+−(r1, r2) and ψ−+(r1, r2) do not have a specific symmetry, but their sum is antisymmetrical and their di erence symmetrical.

5.5.2Occupation Numbers

The N -fermion (or N -boson) wavefunctions are complicated to write because, as soon as a particle is put in a state of given energy, this induces conditions on the available states for the other particles; then the antisymmetrization of the total wave function has to be realized for the fermions (or the symmetrization for the bosons). Since these particles are indistinguishable, the only interesting datum in Statistical Physics is the number of particles in a defined quantum state.

Instead of writing the N -particle wavefunction, which will be of no use in what follows, we will described the system as a whole by the occupation numbers of its di erent states, this is the so-called “Fock basis” : these one-particle states correspond to distinct εk , which can have the same energy but di erent indexes if the state is degenerate (for example two states of the same energy and di erent spins will have di erent indexes k and k ); the states are ordered by increasing energies (Fig. 5.4).

εk nk

ε3 n3

ε2 n2

ε1 n1

Fig. 5.4 : Definition of the occupation numbers of the one-particle states.

The notation |n1,2 , . . . nk , . . . will express the N -particle state in which n1 particles occupy the state of energy ε1, n2 that of energy ε2, . . . nk the state of energy εk . One must satisfy

the total number of particles in the system. Under these conditions, the total energy of the N -particle system is given by

This description will be used in the remainder of the course. The whole physical information is included in the Fock notation. Indeed one cannot tell which particle is in which state, this would be meaningless since the particles are indistinguishable. The only thing that can be specified is how many particles are in each one-particle state.

5.6Return to the Introduction Examples

Now we are able to propose a preliminary interpretation of the physical phenomena described in the introduction, §5.1, some of which will be analyzed in more detail in the following chapters :

5.6.1Fermions Properties

1.In the periodic classification of the elements, the atomic levels are filled with electrons which are fermions. In a given orbital level only two electrons of distinct spin states can be placed, i.e., according to the Pauli principle they are in the singlet spin state.

2.When a chemical bond is created, the electrons coming from the two atoms of the bond go into the atomic levels originating from the coupling between the levels of the separated atoms. The resulting bonding and antibonding states must also be populated by electrons in di erent spin states, thus in the singlet spin state.

3.The ferromagnetism arises, as already mentioned, from the energy difference between the singlet and the triplet spin states for a pair of electrons. If the fundamental state presents a macroscopic magnetization, it must be made from microscopic states with a nonzero spin. In fact, in a ferromagnetic metal such as iron or nickel, the spins of two iron nuclei couple through conduction electrons which travel from nucleus to nucleus.

5.6.2Bosons Properties

4.In superconductivity the electrons of a solid couple by pairs due to their interaction with the solid vibrations. These pairs, called “Cooper pairs,” are bosons which gather into the same fundamental state at low temperature, thus producing the phenomenon of superconductivity.

5.In the same way the 4He atoms are bosons, which can occupy the same quantum state below a critical temperature (see chapter 9). Their ma-

croscopic behavior expresses this collective property, whence the absence of viscosity.

6.Photons are also bosons and the stimulated emission, a specific property of bosons in nonconserved number, will be studied in §9.2.4.

levels, rather than by its wave function. A given state can be occupied at most by one fermion, whereas the number of bosons that can be located in a given state is arbitrary.