Quantum mechanics, like relativity, is a pillar of contemporary physics, with remarkable, puzzling consequences. In these chapters some very simple ‘games’ – distributing prizes of different kinds – are used to explain the implications for ‘identity’. Classical particles have ‘Maxwell‐Boltzmann’ statistics, according to which swapping properties makes a difference; similarly for people, as it makes a difference if we swap bank accounts! Quantum particles with ‘Bose‐Einstein’ statistics (called bosons) or ‘Fermi‐Dirac’ statistics (fermions) are like dollars in bank accounts: if we swap $1 it makes no difference. Chapter 17 shows how Schrödinger used the prize games to argue that – like the dollars – bosons and fermions are not ‘identifiable things’. Chapter 18 extends the models to other possible quantum particles, arguing that quantum mechanics also restricts identity by restricting what states particles can possess together – for instance, Fermi's ‘exclusion principle’, that fermions must be in different states.
Quantum mechanics, like relativity, is a pillar of contemporary physics, with remarkable, puzzling consequences. In these chapters some very simple ‘games’ – distributing prizes of different kinds – are used to explain the implications for ‘identity’. Classical particles have ‘Maxwell‐Boltzmann’ statistics, according to which swapping properties makes a difference; similarly for people, as it makes a difference if we swap bank accounts! Quantum particles with ‘Bose‐Einstein’ statistics (called bosons) or ‘Fermi‐Dirac’ statistics (fermions) are like dollars in bank accounts: if we swap $1 it makes no difference. Chapter 17 shows how Schrödinger used the prize games to argue that – like the dollars – bosons and fermions are not ‘identifiable things’. Chapter 18 extends the models to other possible quantum particles, arguing that quantum mechanics also restricts identity by restricting what states particles can possess together – for instance, Fermi's ‘exclusion principle’, that fermions must be in different states.
Quantum mechanics, like relativity, is a pillar of contemporary physics, with remarkable, puzzling consequences. In these chapters some very simple ‘games’ – distributing prizes of different kinds – are used to explain the implications for ‘identity’. Classical particles have ‘Maxwell‐Boltzmann’ statistics, according to which swapping properties makes a difference; similarly for people, as it makes a difference if we swap bank accounts! Quantum particles with ‘Bose‐Einstein’ statistics (called bosons) or ‘Fermi‐Dirac’ statistics (fermions) are like dollars in bank accounts: if we swap $1 it makes no difference. Chapter 17 shows how Schrödinger used the prize games to argue that – like the dollars – bosons and fermions are not ‘identifiable things’. Chapter 18 extends the models to other possible quantum particles, arguing that quantum mechanics also restricts identity by restricting what states particles can possess together – for instance, Fermi's ‘exclusion principle’, that fermions must be in different states.
In the previous chapters, the focus was on systems of particles following the law of classical mechanics. Of course, this was mainly done within the framework of classical mechanics but a quantum description of individual (discretised) states was also used, while assuming distinguishability of the particles (for instance, particles located at the nodes of a lattice (spin system, quantum harmonic oscillators,) ). The present chapter as well as Chapters 7 and 8 are dedicated to the study of statistical physics in a fully quantum framework. The first obstacle encountered by a quantum approach is the impossibility to define a quantum phase space (in which the ergodic hypothesis would hold for instance). This problem is inherent to quantum mechanics: because of Heisenberg’s uncertainty principle, one cannot define both the momentum and the position of a particle. Besides, statistical physics considers systems made of a large number of particles, a priori in mutual interaction. In classical physics, model situations for which the interactions between the constituents of the system are neglected can be studied. In quantum mechanics, the indistinguishability of identical particles makes the study of the problem in the absence of interaction already difficult: even for non-interacting particles, the symmetrisation (or antisymmetrisation) of the wave function of https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_25.tif "/> identical particles induces correlations which have non-trivial thermodynamical effects such as the existence of a “Fermi sea” for fermions (see Section 8.1 ) or the Bose–Einstein condensation phenomenon for bosons (see Section 7.4 ). The present chapter addresses these two points consecutively.
The universe we live in is both strange and interesting. This strangeness comes about because, at the most fundamental level, the universe is governed by the laws of quantum mechanics. This is the most spectacularly accurate and powerful theory ever devised, one that has given us insights into many aspects of the world, from the structure of matter to the meaning of information. This textbook provides a comprehensive account of all things quantum. It starts by introducing the wavefunction and its interpretation as an ephemeral wave of complex probability, before delving into the mathematical formalism of quantum mechanics and exploring its diverse applications, from atomic physics and scattering, to quantum computing. Designed to be accessible, this volume is suitable for both students and researchers, beginning with the basics before progressing to more advanced topics.
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.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
In this work we present a classical kinetic model of intermediate statistics. In the case of Brownian particles we show that the Fermi-Dirac (FD) and Bose-Einstein (BE) distributions can be obtained, just as the Maxwell-Boltzmann (MB) distribution, as steady states of a classical kinetic equation that intrinsically takes into account an exclusion-inclusion principle. In our model the intermediate statistics are obtained as steady states of a system of coupled nonlinear kinetic equations, where the coupling constants are the transmutational potentials ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\kappa}}\mathrm{\ensuremath{\kappa}}\ensuremath{'}}$. We show that, besides the FD-BE intermediate statistics extensively studied from the quantum point of view, we can also study the MB-FD and MB-BE ones. Moreover, our model allows us to treat the three-state mixing FD-MB-BE intermediate statistics. For boson and fermion mixing in a D-dimensional space, we obtain a family of FD-BE intermediate statistics by varying the transmutational potential ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. This family contains, as a particular case, when ${\mathrm{\ensuremath{\eta}}}_{\mathrm{B}\mathrm{F}=0}$, the quantum statistics recently proposed by L. Wu, Z. Wu, and J. Sun [Phys. Lett. A 170, 280 (1992)]. When we consider the two-dimensional FD-BE statistics, we derive an analytic expression of the fraction of fermions. When the temperature T\ensuremath{\rightarrow}\ensuremath{\infty}, the system is composed by an equal number of bosons and fermions, regardless of the value of ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. On the contrary, when T\ensuremath{\rightarrow}0, ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$ becomes important and, according to its value, the system can be completely bosonic or fermionic, or composed both by bosons and fermions.