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Journal of Physics A Mathematical and General

Three-body S-state wavefunctions: symmetry and degrees of freedom associated with normalisation of the exact wavefunction

J E Gottschalk, E. N. Maslen

Published Jul 11, 1985• 6 citations
Publisher

Abstract

The few-particle Schrodinger equation does not define the symmetry of the wavefunction, which must be chosen to match the symmetry of the particles. It is shown, by reference to the S-states of a three-particle system, that the symmetry does constrain degrees of freedom associated with normalisation of the exact wavefunction. The first particle is treated as infinitely massive, and distinguishable. The systems where the second and third particles are (i) distinguishable, (ii) indistinguishable with a symmetric wavefunction (bosons) and (iii) indistinguishable with an antisymmetric wavefunction (fermions) may be treated as special cases of a continuous description of particle symmetry. Cases (ii) and (iii) are opposite extremes in the analysis.

DOI
10.1088/0305-4470/18/10/023

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The few-particle Schrodinger equation does not define the symmetry of the wavefunction, which must be chosen to match the symmetry of the particles. It is shown, by reference to the S-states of a three-particle system, that the symmetry does constrain degrees of freedom associated with normalisation of the exact wavefunction. The first particle is treated as infinitely massive, and distinguishable. The systems where the second and third particles are (i) distinguishable, (ii) indistinguishable with a symmetric wavefunction (bosons) and (iii) indistinguishable with an antisymmetric wavefunction (fermions) may be treated as special cases of a continuous description of particle symmetry. Cases (ii) and (iii) are opposite extremes in the analysis.
For pt.II see Proc. R. Soc. London, Ser. A, vol.384, no.1786, p.89-105 (1982). A procedure for solving the few-particle Schrodinger equation exactly is applied to a model system consisting of two identical particles and a massive third particle. The type of interaction potential is not specified except that it should not diverge more rapidly than r -2 at the particle positions. Allowable interactions include the Coulomb and the harmonic oscillator potentials. The principles are illustrated by reference to the spatially symmetric states of the system. The solution has the form of a multipole expansion in spherical polar coordinates. The radial dependence for each multipole component is defined by a power series including logarithmic terms. Explicit expressions are given for the coefficients in the expansion. Exact coefficients are obtained for low-order terms in the wavefunction of a two-electron atom.
This chapter addresses the quantum mechanics of identical particles, emphasizing the role of wavefunction symmetry in determining statistical behavior. Symmetric and antisymmetric wavefunctions are defined, and the implications of particle indistinguishability are examined, particularly regarding how it influences measurable outcomes and the concept of energy exchange degeneracy. The distinguishability of identical particles is discussed and shown to affect specific calculations. For systems where particle interactions can be neglected in the zero-order approximation, the total energy eigenfunction is expressed as a product of single-particle eigenfunctions. The Pauli exclusion principle is derived as a consequence of antisymmetrization. The relationship between symmetry type (bosonic or fermionic) and quantum statistics is elaborated. As an illustration, the ground state and the first excited sss-state of the helium atom are analyzed under a simplified approximation. Metadata/Keywords: identical particles, symmetric wavefunction, antisymmetric wavefunction, Pauli exclusion principle, quantum statistics, energy exchange degeneracy, indistinguishability, helium atom, symmetry and statistics
The wave functions of boson and fermion gases are known even when the particles have harmonic interactions. Here we generalize these results by solving exactly the N–body Schrödinger equation for potentials V that can be any function of the sum of the squares of the distances of the particles from one another in three dimensions. For the harmonic case that function is linear in r2. Explicit N–body solutions are given when U(r) = –2Mh–2V(r) = 𝛇r–1 – 𝛇2r–2. Here M is the sum of the masses and r2 = 1/2 M–2ΣmImJ(xI – xJ)2. For general U(r) the solution is given in terms of the one– or two–body problem with potential U(r) in three dimensions. The degeneracies of the levels are derived for distinguishable particles, for bosons of spin zero and for spin 1/2 fermions. The latter involve significant combinatorial analysis which may have application to the shell model of atomic nuclei. For large N the fermionic ground state gives the binding energy of a degenerate white dwarf star treated as a giant atom with an N–body wave function. The N–body forces involved in these extraordinary N–body problems are not the usual sums of two–body interactions, but nor are forces between quarks or molecules. Bose–Einstein condensation of particles in three dimensions interacting via these strange potentials can be treated by this method.
For pt.I see ibid., vol.14, p.2581 (1981). If one begins with N non-interacting fermions, the Pauli principle can be easily incorporated by the use of Slater determinants. This is not the case for N interacting fermions. The authors consider exactly soluble 'Coulomb-type' quantum systems in three dimensions of N interacting identical spin- 1 / 2 -fermions. A systematic procedure for constructing Pauli antisymmetry adapted wavefunctions is given. The resulting antisymmetric wavefunctions are labelled by conserved 'good' quantum numbers. In particular, for N=2, all the physically acceptable states are obtained. For N=3, they present a class of antisymmetric states which consists of all the ground states, all the first excited states and the states obtained by the hyperradial excitations of these. For N=2 and 3, the ground states of the authors' model systems are found to be 1 S(s 2 ) and 2 P(s 2 p), respectively, in the quantum numbers of the interacting system.
Abstract We show how to characterize the indistinguishability of up to four identical, bosonic or fermionic particles, which are rendered partially distinguishable through their internal degrees of freedom prepared in mixed states. This is accomplished via their counting statistics when subjected to a highly symmetric unitary acting upon their external (i.e. dynamical) degrees of freedom. For pure internal states, we further extract information on the particles’ collective phases, which ultimately allows for an experimental reconstruction of the full many-particle density operator up to complex conjugation.
General properties of many-body systems Definition of individual system So far we have studied the properties of single-body systems in prescribed external potentials. An individual physical system comprises a wave and a particle, both evolving in three-dimensional Euclidean space. Of course, isolating a system and representing the influence of other matter by an ‘external potential’ is an approximation to the real state of affairs in that we neglect the source of the potential. As a fundamental theory of matter quantum mechanics should apply to a closed many-body system (and ultimately to the universe as a whole) and reduce to a theory of systems of a few degrees of freedom as a special case under conditions where it is legitimate to neglect the ‘rest of the universe’. The extension of the quantum theory of motion to many-body systems is straightforward, although it displays some striking features not evident in the one-body case. To begin with, we define an individual n-body system as comprising: ( a ) A wavefunction ψ = ψ( x 1 , …, x n , t ) defined in a 3 n -dimensional configuration space in which x l , …, x n provide a set of rectangular Cartesian coordinates. ( b ) A set of n point particles pursuing trajectories x i ( t ) i = 1, …, n , in three-dimensional Euclidean space. A single configuration space trajectory is equivalent to n particle trajectories in Euclidean space. When we speak of a ‘many-body system’ we mean then a single wavefunction together with a set of particles.