In this paper we study the system of non-interacting spins like electrons by using notions probability, soft sets and probabilistic soft sets. Electrons are indistinguishable identical particles whose spin is one of their features like mass of electron, charge of electron etc. We also compare two systems of non-interacting spins by defining two soft spin sets and calculating their similarity measure.
In this paper we study the system of non-interacting spins like electrons by using notions probability, soft sets and probabilistic soft sets. Electrons are indistinguishable identical particles whose spin is one of their features like mass of electron, charge of electron etc. We also compare two systems of non-interacting spins by defining two soft spin sets and calculating their similarity measure.
The electron spin correlation is shown to be expressible in terms of a bona fide probability distribution function with an associated geometric representation. With this aim, an analysis is presented of the probabilistic features of the spin correlation within the quantum formalism, which helps clarify the concepts of contextuality and measurement dependence. The dependence of the spin correlation on conditional probabilities allows for a clear separation between system state and measurement context; the latter determines how the probability space should be partitioned in calculating the correlation. A probability distribution function ρ(ϕ) is then proposed, which reproduces the quantum correlation for a pair of single-particle spin projections and is amenable to a simple geometric representation that gives meaning to the variable ϕ. The same procedure is shown to be applicable to the bipartite system in the singlet spin state. This endows the spin correlation with a clear probabilistic meaning and leaves the door open for a possible physical picture of the electron spin, as discussed at the end of the paper.
In non-relativistic quantum mechanics, each particle is characterized by a set of parameters. These are intrinsic to a given particle, and they are the same for a given particle species. Think, for instance, of the mass, the charge, or the spin of an electron. All electrons have the same mass, charge and spin and, from these properties alone, we cannot distinguish two different electrons. Based on these properties, we can divide particles into classes depending on their intrinsic parameters. Particles belonging to the same class are said to be indistinguishable .
An analysis is made within the quantum formalism of the probabilistic features of the electron spin correlation, with the purpose of clarifying the concepts of contextuality and measurement dependence. The quantum formulas for the spin correlation are then derived on the basis of a probability distribution function and its associated geometric representation, both for a pair of projections of a single-particle spin and for the bipartite system in singlet spin state. This endows the spin correlation with a clear probabilistic meaning and leaves the door open for a possible physical image of the electron spin, as discussed at the end of the paper.
Although many of the principles of quantum mechanics can be adequately illustrated by considering systems that consist of only one particle subject to external forces, there are a number of important phenomena that are manifest only in systems containing two or more particles, and these will be the subject of the present chapter. We shall begin with some general statements and go on to consider the case of two interacting particles subject to no external forces, when we shall find that, in this case, the problem can be separated into one describing the behaviour of the centre of mass of the system and another describing the relative motion. We shall then consider the case of two noninteracting particles and show that these can often be treated independently as would be expected. In the case of indistinguishable particles, however, we shall find that a symmetry is imposed on the wave function that ensures that the behaviour of such particles is coupled even when they do not interact. We shall introduce the problem of interacting systems containing more than two identical particles, explaining what is meant by “fermions” and “bosons” and how their properties compare. These results will be illustrated by discussing the energy levels and optical spectra of the helium atom, starting with the approximation that the two electrons do not interact and extending the discussion to the realistic case using perturbation theory. The chapter closes with a consideration of the effects of particle indistinguishability on scattering theory.
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.
Details of the many-particle system can also be read from my book titled “Quantum Field Theory: Feynman Path Integrals and Diagrammatic Techniques in Condensed Matter” [ 23 ]. The fundamental difference between classical and quantum mechanics relates the concept of indistinguishability of identical particles where each particle can be equipped with an identifying marker without influencing its behavior in classical mechanics. Besides in classical mechanics, the particle follows its own continuous path in phase space and so it is possible to determine the particular process taking place. So, principally, each particle in a group of identical particles can be identified which is not so in quantum mechanics. It is not possible to mark a particle without influencing its physical state, and worse, if a number of identical particles are brought to the same region in space, their wave functions will rapidly spread out and overlap with one another, thereby soon rendering it impossible to say which particle is where. One of the fundamental assumptions for many-particle systems is therefore that identical particles (i.e., particles characterized by the same quantum numbers such as mass, charge and spin) are, in principle, indistinguishable.