Bose particle

The sum over n. can now be perfonned, but this depends on the statistics that the particles in the ideal gas obey. Fenni particles obey the Pauli exclusion principle, which allows only two possible values n. = 0, 1. For Bose particles, n. can be any integer between zero and infinity. Thus the grand partition fiinction is... 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

The fiinction N (T) is sketched in fignre A2.2.7. At zero temperature all the Bose particles occupy the ground state. This phenomenon is called the Bose-Einstein condensation and is the temperature at which the transition to the condensation occurs. 

Figure A2.2.7. Fraction of Bose particles in the ground state as a fiinction of the temperature.

Particles that obey Bose-Einstein statistics are called Bose particles or bosons. The probability density of bosons in their energy levels is represented by the Bose-Einstein function as shown in Eqn. 1-2 ... 

Further development of this analogy leads to the non-Hermitian Hamiltonian problem describing the Bose particles. Proceeding in this way, the classical diffusion problem could be related to quantum theory of multiple scattering [115-118]. 

For a symmetrical (D ) diatomic or linear polyatomic molecule with two, or any even number, of identical nuclei having the nuclear spin quantum number (see Equation 1.47) I = n + where n is zero or an integer, exchange of any two which are equidistant from the centre of the molecule results in a change of sign of i/c which is then said to be antisymmetric to nuclear exchange. In addition the nuclei are said to be Fermi particles (or fermions) and obey Fermi Dirac statistics. However, if / = , p is symmetric to nuclear exchange and the nuclei are said to be Bose particles (or bosons) and obey Bose-Einstein statistics. 

PROBLEM 5.1.3. If fermions, such as the neutrons (7=1/2) that make a hot neutron star, merge to form a massive Bose particle (cold black hole), then Bose-Einstein condensation occurs, the whole star loses energy massively, and a new minimum energy state is reached (cold black hole) (see also Problem 2.12.4). 

In a dense plasma electromagnetic waves can be quantized and behave like relativistic Bose particles with finite mass. These plasmons can decay into either / + (3+ or u u pairs. 

The strongly localized nature of low-energy polariton states should affect many processes such as light scattering and nonlinear phenomena as well as temperature-induced diffusion of polaritons. Manifestations of the localized polariton statistics (Frenkel excitons are paulions exhibiting properties intermediate between Fermi and Bose particles) in the problem of condensation also appear interesting and important. 

It may be helpful to note that the form of the Pauli principle given by Eq. (2.11) is valid not only for electrons, but for any fermions having half-integer spin. For Bose particles (bosons), however, which bear an integer spin, the Pauli principle requires the wave function to be symmetric. In that case there is no negative sign in Eqs. (2.9) and (2.10), and instead of Eq. (2.11) we have ... 

At absolute zero temperature all levels are occupied by electrons up to E. When temperature increases some of the electrons are excited and occupy free levels of energy. These electrons define the electron conductivity of metals. The average energy width of this excitation band is equal to kT, as is marked in Figures 9.9 and 9.10.The opposite case is presented by particles with integer spin, in particular photons with a spin quantum number 1. Such particles are referred to as bozons (Bose particle). They obey Bose-Einstein statistics. A distribution function/( ) for bozons is expressed as... 

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