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Journal of Physics A Mathematical and General

Boundary conditions on wavefunctions for three bodies at singular configurations

Toshihiro Iwai, Toru Hirose

Published Jan 6, 2004• 1 citation
Publisher

Abstract

Singular configurations have long been out of consideration in the study of many-particle systems. In this paper we show that a group theoretical method can provide boundary conditions on wavefunctions at singular configurations of many particles. The consequences of this method are qualitatively stated as follows. (i) If the projection of angular momentum on an axis does not vanish, the particles will not be aligned on the axis, i.e. the wavefunction vanishes at collinear configurations along the axis. (ii) If the total angular momentum does not vanish, the simultaneous collision at a point will not take place, i.e. the wavefunction vanishes at the configuration of simultaneous collision. Furthermore, on the assumption of analyticity of the wavefunction at singular configurations, the behaviour of the wavefunction around the singular configurations shows that (a) the larger the projection of angular momentum on an axis becomes, the less the particles are likely to be aligned on the line, and that (b) the more the total angular momentum grows, the less likely the particles are to collide at a point. The proof is carried out for three-body systems by transforming the power series expansion of a wavefunction into a Fourier series expansion in terms of the angular momentum eigenfunctions, in both cases of collinear configurations and of the triple collision, without reference to the Hamiltonian operator. These results are described quantitatively in terms of the angular momentum quantum numbers in propositions 1 and 3 in the text.

DOI
10.1088/0305-4470/37/3/013

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Singular configurations have long been out of consideration in the study of many-particle systems. In this paper we show that a group theoretical method can provide boundary conditions on wavefunctions at singular configurations of many particles. The consequences of this method are qualitatively stated as follows. (i) If the projection of angular momentum on an axis does not vanish, the particles will not be aligned on the axis, i.e. the wavefunction vanishes at collinear configurations along the axis. (ii) If the total angular momentum does not vanish, the simultaneous collision at a point will not take place, i.e. the wavefunction vanishes at the configuration of simultaneous collision. Furthermore, on the assumption of analyticity of the wavefunction at singular configurations, the behaviour of the wavefunction around the singular configurations shows that (a) the larger the projection of angular momentum on an axis becomes, the less the particles are likely to be aligned on the line, and that (b) the more the total angular momentum grows, the less likely the particles are to collide at a point. The proof is carried out for three-body systems by transforming the power series expansion of a wavefunction into a Fourier series expansion in terms of the angular momentum eigenfunctions, in both cases of collinear configurations and of the triple collision, without reference to the Hamiltonian operator. These results are described quantitatively in terms of the angular momentum quantum numbers in propositions 1 and 3 in the text.
Quantum systems of three identical particles on a plane are analyzed from the viewpoint of symmetry. Upon reduction by rotation, such systems are described in the space of sections of a line bundle over a three-dimensional shape space whose origin represents triple collision. It is shown that if the total angular momentum is nonzero, then the wave section must vanish at the origin, while if it is zero, then the wave section can be finite at the origin. Since the particles are assumed to be identical, the quantum system admits the action of the symmetric group S3 as well, which stands for the group of particle exchanges and is commutative with rotation. Hence the reduced system still admits the S3 action, so that Bose and Fermi states can be discussed in the space of sections of the line bundle. A detailed analysis of a system of three free particles on a plane is presented in the latter part of the article.
The present paper is devoted to the study of the few–body quantum mechanical problem with singular finite rank cluster interactions. The corresponding Hamiltonians play an important role in mathematical physics, since few body Hamiltonians with more regular interactions present considerable difficulties which make impossible a detailed analytic study [ 13 ]. Also a numerical study of such Hamiltonians using Faddeev equations present very hard problems, since the interaction between the particles does not vanish at large distances. On the other hand few-body problems with singular cluster interaction are useful and intensively studied in statistical physics, since the corresponding Hamiltonians can be analyzed in detail even if the number of particles is very large. Models describing one dimensional particles are of particular interest, since the eigenfunctions of the many body Hamiltonians can often be calculated using Bethe Ansatz [ 14 ]. In fact the well-known Yang-Baxter equation was first written in connection with the study of system of several one-dimensional particles with pairwise delta interactions. Similar methods were used in atomic physics to study collisions of several particles [ 12 ] (in three dimensions). Few-body systems in applications are usually investigated by combining the classical description of the dynamics of heavy atoms with the quantum description of the dynamics of electrons and other light particles. The first attempt to construct a three body Hamiltonian describing three quantum particles in R 3 interacting through pairwise delta functional potentials is due to G.V. Skorniakov and K.A.Ter-Martirosian [ 29 ]. R.A.Minlos and L.D.Faddeev proved that the corresponding Hamiltonian is not bounded from below and therefore cannot be used in 278 the originally intended physical applications [ 24 , 25 ]. The operator was defined using the method of self–adjoint perturbations used by F.A.Berezin and L.D.Faddeev for investigating Schödinger operators with delta potential in R 3 [ 10 ]. Almost three decades later semibounded three-body operators in dimension three with generalized two-body interactions were constructed by extending the standard Hilbert space of square integrable functions in R 9 [ 17 , 18 , 19 , 27 , 30 ]. The most interesting model considered used the theory of generalized extensions suggested by B.Pavlov [ 7 , 26 ]. Different aspects of these models were analyzed recently [ 8 , 23 , 20 ]. A general approach to these operators is described in [ 7 , 22 ]. Let us mention that a realization of a many-body lower bounded Hamiltonian with point interactions for particles in R 3 has been obtained in the original Hilbert space by using the theory of Dirichlet forms [ 2 ]. The few-body systems of two dimensional particles with two-body interactions was studied in [ 11 ], where it was proven that the corresponding Hamiltonian is semibounded.
The conventional method of constructing wave functions of pure angular momentum for a composite system is based on the idea of coupling the angular momenta of the constituents. Instead of this synthetic method, an analytical approach is here introduced. It is pointed out that an arbitrary trial wave function for the total system in a unique way must be resolvable into orthogonal components of pure angular momentum associated with different quantum numbers. A particular component can be selected by means of certain 'projection operators', which annihilate all components except the one desired. Such a projection operator is simply a product of commuting factors, each one of which annihilates a specific eigenfunction to the angular momentum under consideration. Physically this idea is of importance, since one can now start out from a rough model wave function based essentially on qualitative arguments, for instance of the independent-particle type, and then obtain a mathematically and physically much better trial function by selecting the particular component of the original function which has the correct symmetry type desired. The projection operators are studied in some detail both as products and in expanded form. The case of degenerate subspaces of the same angular momentum quantum numbers is discussed, and the problem of constructing an orthogonal subset of functions is solved by a simple elimination procedure; the connection with the conventional seniority idea is briefly discussed.
This chapter presents a detailed examination of a number of relatively simple problems pertaining to the scattering and bound states of two structureless particles interacting through a central field. It describes the spectrum of the angular momentum operators by a method that does not refer at all to the structure of the system. The spherical harmonics are the eigenfunctions appropriate to a single particle that can be described by the three classical degrees of freedom, or for a pair of such particles in their center-of-mass frame. The formulation of collision theory that is based on the one-body Schrodinger equation with a Hermitian interaction. The chapter introduces and discusses the observables pertaining to the two-body problem and explores the eigenvalues and eigenfunctions of these observables. Since the degeneracies of the energy spectrum are a consequence of the symmetries of the Hamiltonian, they can be determined experimentally by destroying this symmetry.
Properties of two identical particles of mass mmm and a distinct particle of mass m1m_1m1​ in the universal low-energy limit of zero-range two-body interaction are studied in different sectors of total angular momentum LLL and parity PPP. For the unambiguous formulation of the problem in the interval μr(LP)<m/m1≤μc(LP)μ_r(L^P) < m/m_1 \le μ_c(L^P)μr​(LP)<m/m1​≤μc​(LP) (μr(1−)≈8.619μ_r(1^-) \approx 8.619μr​(1−)≈8.619 and μc(1−)≈13.607μ_c(1^-) \approx 13.607μc​(1−)≈13.607, μr(2+)≈32.948μ_r(2^+) \approx 32.948μr​(2+)≈32.948 and μc(2+)≈38.630μ_c(2^+) \approx 38.630μc​(2+)≈38.630,~etc.) in each LPL^PLP sector an additional parameter bbb 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/m1m/m_1m/m1​ and bbb in the sector of angular momentum and parity LPL^PLP is calculated for L≤5L \le 5L≤5 and analysed with the aid of a simple model. A number of the bound states for each LPL^PLP sector is analysed and presented in the form of `phase diagrams' in the plane of two parameters m/m1m/m_1m/m1​ and bbb.
The center-of-mass system of many bodies admits a natural action of the rotation group SO(3). According to the orbit types for the SO(3) action, the center-of-mass system is stratified into three types of strata. The principal stratum consists of nonsingular configurations for which the isotropy subgroup is trivial, and the other two types of strata consist of singular configurations for which the isotropy subgroup is isomorphic with either SO(2) or SO(3). Depending on whether the isotropy subgroup is isomorphic with SO(2) or SO(3), the stratum in question consists of collinear configurations or of a single configuration of the multiple collision. It is shown that the kinetic energy operator is expressed as the sum of rotational and vibrational energy operators on each stratum except for the stratum of multiple collision. The energy operator for nonsingular configurations has singularity at singular configurations. However, the singularity is not essential in the sense that both of the rotational and vibrational energy integrals have a finite value. This can be proved by using the boundary conditions of wave functions at singular configurations for three-body systems, for simplicity. It is shown, in addition, that the energy operator for collinear configurations has also singularity at the multiple collision, but the singularity is not essential either in the sense that the kinetic energy integral is not divergent at the multiple collision. Reduction procedure is applied to the respective energy operators for the nonsingular and the collinear configurations to obtain respective reduced operators, both of which are expressed in terms of internal coordinates.