We present a description of the quantum mechanical states of a system of n indistinguishable particles moving on a manifold M by C/S2- valued functions ψ′ defined on the configuration space Mn/Sn. These functions satisfy one of two homotopy conditions, which characterize the particles as bosons or fermions. Any closed curve of Mn/Sn which does not intersect the diagonals can be classified as even or odd according to whether its lifts to Mn have end points in Mn which are even or odd permutations of each other. For bosons, ψ′ must map any such closed curve which also avoids kerψ′ onto an even number of basic loops of (C−{0})/S2. For fermions, ψ′ must map such even loops of Mn/Sn onto an even number, and odd loops onto an odd number, of basic loops of (C−{0})/S2.
We present a description of the quantum mechanical states of a system of n indistinguishable particles moving on a manifold M by C/S2- valued functions ψ′ defined on the configuration space Mn/Sn. These functions satisfy one of two homotopy conditions, which characterize the particles as bosons or fermions. Any closed curve of Mn/Sn which does not intersect the diagonals can be classified as even or odd according to whether its lifts to Mn have end points in Mn which are even or odd permutations of each other. For bosons, ψ′ must map any such closed curve which also avoids kerψ′ onto an even number of basic loops of (C−{0})/S2. For fermions, ψ′ must map such even loops of Mn/Sn onto an even number, and odd loops onto an odd number, of basic loops of (C−{0})/S2.
In this final chapter we discuss composite systems in Quantum Mechanics. First, we lay out the necessary conditions that the state space of a composite system must satisfy by considering the trivial case of two juxtaposed systems. Then we point out the features that the interesting case of a composite system, whose parts interact with each other, must have. We then move to demonstrate how such a system can be naturally described (quantized) by employing tensor product spaces. The definition and the most relevant properties of tensor product spaces are introduced. Then the meaning of “interactions” in quantum mechanics is explored and the most nontrivial aspects of correlation and entanglement are discussed. Finally, in the last section, we introduce systems of identical particles. We describe how to represent the state space of such systems in terms of one-particle states, number operators and occupation number representations. Classification of identical particles into Bosons and Fermions is explained. Finally, introducing symmetric and antisymmetric states, the traditional quantization scheme using the symmetrization postulate is described. The permutation operator is introduced which is the main tool for dealing with identical particle systems in this language. We wind up by indicating how exchange degeneracy is removed which embodies the signature of indistinguishability.
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.
A feature of quantum mechanics is that identical particles cannot be labelled. This means that identical particles must be treated as indistinguishable and implies that if we exchange two particles in a many – particle system – e.g. the electrons in a molecule – then the wavefunction describing the system cannot change, apart from a phase factor. Besides some exotic possibilities in strictly two-dimensional systems, this phase factor has only two values: +1 and -1, which lead to symmetric and anti-symmetric wavefunctions, respectively. Particles with symmetric wavefunctions, such as photons, are called bosons, while those with anti-symmetric wavefunctions, such as electrons, are called fermions.
This chapter discusses the properties of systems of identical particles. It presents the wave functions of a system composed of N identical and independent particles. From the properties of the wave functions, the chapter shows that identical particles can be characterized by symmetric as well as antisymmetric wave functions. As a result, it distinguishes two kinds of particles: bosons and fermions. Fermions have half integer spin, whereas bosons have integer spin. Since an arbitrary number of bosons can occupy the same state, atoms with integer spin can abruptly condensate into a single ground state when the temperature of the atoms goes below a certain critical value. In atoms, a limited number of electrons can occupy the same energy level. The Pauli principle prevents the energy states being occupied by an arbitrary number of electrons. Thus, the Pauli principle prevents all atoms having the same size and the same energy.
The wave function of a system containing identical particles takes into account the relationship between a particle's intrinsic spin and its statistical property. Specifically, the exchange of two identical particles having odd-half-integer spin results in the wave function changing sign, whereas the exchange of two identical particles having integer spin is accompanied by no such sign change. This is embodied in a term (-1)(2s), which has the value +1 for integer s (bosons), and -1 for odd-half-integer s (fermions), where s is the particle spin. All of this is well-known. In the nonrelativistic limit, a detailed consideration of the exchange of two identical particles shows that exchange is accompanied by a 2pi reorientation that yields the (-1)(2s) term. The same bookkeeping is applicable to the relativistic case described by the proper orthochronous Lorentz group, because any proper orthochronous Lorentz transformation can be expressed as the product of spatial rotations and a boost along the direction of motion.
If we consider ( ) 2,kr tΨ G to be the probability density for finding, at time t, particle 1 at particle k at and particle N at ,,1 … Gr …G ,kr ,Nr G a normalization condition analogous to Eq.(29.2) holds as (35.2) 1V* =ΨΨ∫ d The integral is defined in a space with 3N dimensions called the configuration space of the system, where a configuration point is given by 3N position coordinates 1 1 1, , , , , , , , , ,k k k N N N ,x y z x y z x y z… … and the volume element is NNNkkkNk dzdydxdzdydxdzdydxdddd ………… 1111 VVVV == Physical observables of the system will be represented by Hermitian operators, obtained from the classical representation in terms of the position and momentum variables as indicated by Postulate 2. Thus the many-particle Hamiltonian results from Eq.(2.32) as ( ) (∑ ∑ = ≠ ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ++∇−= k rrUtrU m H ,, 2 ˆ GGG= ) ) (35.3) where only acts on the coordinates of the kth particle, of mass The one-particle potential energy is due to external fields where this particle moves and k∇ .km( trU kk ,G( )jkkj rrU GG , gives the mutual interaction between the particles k and j. It is assumed that the position and momentum operators of different particles commute, and so do all the operators describing single-particle observables if they refer to different particles. The evolution of a many-particle system will be then formulated as a generalization of Eq.(30.6) in terms of given by Eqs.(35.1) and (35.3) HˆandΨ Ψ=∂ Ψ∂ H t i ˆ= (35.4) If the Hamiltonian (35.3) does not depend explicitly on the time, it is usual to express the time dependence of the many-particle state function by means of the evolution operator defined as ( )tUˆ ( ) ( ) ( )NkNk rrrtUtrrr G…G…GG…G…G ,,,,ˆ,,,,, 11 αΨ=Ψ (35.5) which is formally derived from Eq.(35.4) as ( ) ( ) ( )ˆˆ ˆU ti U t i HU t t t ˆα α α ∂∂ ⎡ ⎤Ψ = Ψ =⎣ ⎦∂ ∂= = Ψ It follows that (35.6) ( ) =/ˆˆ tHietU −= which is the analogue of Eq.(30.12). The evolution operator is unitary, Eq.(II.22), since ( ) ( )†ˆ ˆU t U t Iˆ= (35.7) Substituting Eq.(35.5), one obtains from Eq.(35.4) that the stationary states of the Nparticle systems are given by the energy eigenvalue equation ( ) ( )1 1ˆ , , , , , , , ,k N k NH r r r E r r rα α αΨ = ΨG G G G G G… … … … (35.8) The solution to the N-particle problem is facilitated if we take into account the symmetries of many systems of physical interest, which allow us to define constants of motion and criteria for simplifying the form (35.3) of the Hamiltonian operator. An alternative to complex explicit calculations is to associate with specific symmetries time independent unitary operators which transform the state functions into so that Eq.(35.4) gives ˆ ,U Ψ ˆ ,U Ψ ( ) ( )†ˆ ˆ ˆ ˆ ˆ ˆ ˆi U i U UH UHU Ut t∂ ∂ΨΨ = = Ψ = Ψ∂ ∂= = ˆ This is again a Schrödinger equation for the state Uˆ Ψ with the new Hamiltonian †ˆ ˆ ˆ ˆH UHU′ = If the Hamiltonian remains unchanged by the transformation, we have (35.9) †ˆ ˆ ˆ ˆ ˆ ˆor , 0H UHU U H⎡ ⎤= ⎣ ⎦ = According to the evolution equation (29.43), this implies that so that U is a constant of motion. If the unitary transformation which leaves the Hamiltonian invariant, is a function of an observable, its constancy implies that the given observable is a constant in any state of the system.