It is customary, when discussing configuration spaces of identical particles in two or more dimensions, to discard the configurations where two or more particles overlap, the justification being that the configuration space ceases to be a manifold at those points, and also to allow for nonbosonic statistics. We show that there is in general a loss of physical information in discarding these points by studying the simple system of two free particles moving in the plane and requiring that the Hamiltonian be self-adjoint. We find that the Hamiltonian for fermions is unique, but that in all other cases (i.e., for particles obeying properly fractional or Bose statistics) there is a one-parameter family of possible self-adjoint extensions. We show how a plausible limiting procedure selects a unique extension from each family, the favored extension being the one for which the wave function remains finite at the points of overlap. We also test our procedure by applying it to the known case of the hydrogen atom.
It is customary, when discussing configuration spaces of identical particles in two or more dimensions, to discard the configurations where two or more particles overlap, the justification being that the configuration space ceases to be a manifold at those points, and also to allow for nonbosonic statistics. We show that there is in general a loss of physical information in discarding these points by studying the simple system of two free particles moving in the plane and requiring that the Hamiltonian be self-adjoint. We find that the Hamiltonian for fermions is unique, but that in all other cases (i.e., for particles obeying properly fractional or Bose statistics) there is a one-parameter family of possible self-adjoint extensions. We show how a plausible limiting procedure selects a unique extension from each family, the favored extension being the one for which the wave function remains finite at the points of overlap. We also test our procedure by applying it to the known case of the hydrogen atom.
A quantum-mechanical three-body problem for two identical fermions of mass m and a distinct particle of mass m 1 in the universal limit of zero-range two-body interaction is studied. For the unambiguous formulation of the problem in the interval μ r < m / m 1 ≤ μ c ( μ r ≈ 8.619 and μ c ≈ 13.607 ) an additional parameter b determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. The dependence of the bound-state energies on m/m 1 and b in the sector of angular momentum and parity L P = 1 − is calculated and analysed with the aid of a simple model.
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.
We show that there are novel generalizations of statistics for identical particles in one dimension. These arise due to possible boundary conditions on wave functions, or equivalently due to δ-function interactions, at points of coincidence of particle coordinates. Special choices of these boundary conditions describe bosons, fermions or paraparticles. The general solution for the boundary conditions involves vector-valued wave functions and statistics with non-Abelian features, even though the classical configuration space has an Abelian fundamental group. Physical models leading to such non-Abelian statistics, involving for example δ-function interactions of spins, are constructed. Properties under parity and time reversal of the new boundary conditions are studied. It is shown that the Bethe ansatz does not always give eigenstates of energy. When it does, the wave numbers in the ansatz for three or more particles must satisfy a weaker form of the Yang-Baxter equations and certain additional equations. The energy spectrum and wave functions for identical particles on R1 or S1 are discussed in some simple cases. Our work generalizes the previous work of Lieb and Liniger, Lieb, Yang and Leinaas and Myrheim.
Properties of two identical particles of mass m and a distinct particle of mass m1 in the universal low-energy limit of zero-range two-body interaction are studied in different sectors of total angular momentum L and parity P. For the unambiguous formulation of the problem in the interval μr(LP)<m/m1≤μc(LP) (μr(1−)≈8.619 and μc(1−)≈13.607, μr(2+)≈32.948 and μc(2+)≈38.630,~etc.) in each LP sector an additional parameter b determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. Within the framework of this formulation, dependence of the bound-state energies on m/m1 and b in the sector of angular momentum and parity LP is calculated for L≤5 and analysed with the aid of a simple model. A number of the bound states for each LP sector is analysed and presented in the form of `phase diagrams' in the plane of two parameters m/m1 and b.
The notion of self-adjointness as distinct from Hermiticity is usually avoided in elementary quantum mechanics courses due to a feeling that it has nothing to do with physics or else is too mathematical. This is definitely not the case, and we use the model of a free particle restricted to motion on a line over a finite interval, to demonstrate this. For such a particle, there is one-parameter infinity of self-adjoint extensions of the momentum operator. We use one of these, corresponding to ’’antiperiodic’’ boundary conditions, to show that physical phenomena different from the usual case (periodic boundary conditions) occur. In fact, in this case, we have a model displaying many of the properties of spontaneously broken symmetry.
We show that the algebraic quantization method of Heisenberg and the analytical method of Schrödinger are not necessarily equivalent when applied to systems of identical particles. Heisenberg quantization is a natural approach, but inherently more ambiguous and difficult than Schrödinger quantization. We apply the Heisenberg method to the examples of two identical particles in one and two dimensions, and relate the results to the so-called fractional statistics known from Schrödinger quantization. For two particles in d dimensions we look for linear, Hermitian representations of the symplectic Lie algebra sp(d, R). The boson and fermion representations are special cases, but there exist other representations. In one dimension there is a continuous interpolation between boson and fermion systems, different from the interpolation found in Schrödinger quantization. In two dimensions we find representations that can be realized in terms of multicomponent wave functions on a three-dimensional space, but we have no clear physical interpretation of these representations, which include extra degrees of freedom compared to the classical system.