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Journal of Hubei Institute for Nationalities

The Quantum Effect Analysis of Identical Particles Scattering

Xing Li

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Scattering is an important phenomenon of microscopic particle's interaction. The identical particles are the particles with inherent identical character in mass,electric charge,spin,etc. They show some special characteristic in their scattering. In this paper,the identical particles' scattering has been analysed from the symmetry of wave function, scattering cross - section and probabiliy. A comparison is made betwwen scattering of the identical particles and that of the non - identical particles.

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Scattering is an important phenomenon of microscopic particle's interaction. The identical particles are the particles with inherent identical character in mass,electric charge,spin,etc. They show some special characteristic in their scattering. In this paper,the identical particles' scattering has been analysed from the symmetry of wave function, scattering cross - section and probabiliy. A comparison is made betwwen scattering of the identical particles and that of the non - identical particles.
In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.
Particles that must have symmetric states are called bosons; pions, photons, and He 4+ nuclei are such particles. Particles whose states must be ant symmetric under interchange of the two particles are called fermions; electrons, protons, and neutrons are examples of fermions. Particles with integer spin are always bosons while particles with half-integer spin are always fermions. It is not at all obvious from the point of view of non-relativistic quantum mechanics why this relation, called the connection between spin and statistics, is true, and one must look within the framework of relativistic quantum field theory to understand it fully theoretically. Effects of identity can also show up in the scattering of seemingly non-identical particles. The elastic scattering of a He 4+ ion from a He 4+ atom, both are in their ground states. These are certainly distinguishable particles, since they have different charges. Particle identity also plays an important role in determining the properties of molecules made of identical atoms.
Quantum scattering, a fundamental phenomenon in quantum mechanics, is very important in understanding the interactions between particles at the microscopic level.1,2 Quantum scattering plays a pivotal role in various fields of physics, chemistry, and even beyond, influencing areas such as materials science, quantum computing, and nuclear physics among others.3,4 Theoretical concepts such as the Schrödinger equation, scattering amplitude, and scattering cross-section are highlighted in this, along with their significance in describing particle interactions with potential energy fields.5,6
An analysis of the Wigner function for identical particles is presented. Four situations have been considered. (i) The first is scattering process between two indistinguishable particles described by a minimum uncertainty wave packets showing the exchange and correlation effects in Wigner phase space. (ii) An equilibrium ensemble of NNN particles in a one-dimensional box and in a one-dimensional harmonic potential is considered second, showing that the reduced one-particle Wigner function, as a function of the energy defined in the Wigner phase space, tends to the Fermi-Dirac or to the Bose-Einstein distribution function, depending on the considered statistics. (iii) The third situation is reduced one-particle transport equation for the Wigner function, in the case of interacting particles, showing the need for the two-particle reduced Wigner function within the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy scheme. (iv) Finally, the electron-phonon interaction in the two-particle case is considered, showing coparticipation of two electrons in the interaction with the phonon bath.