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American Journal of Physics

The quantum mechanical few-body problem

John W. Norbury

Published Mar 1, 1989• 5 citations
Publisher

Abstract

A solution to quantum mechanical problems of two, three, and four bodies is discussed from the point of view of two-body harmonic oscillator forces. An exact solution to the three- and four-body problems is found for harmonic oscillator forces, assuming equal masses. The aim of the discussion is to make students aware of the real difficulties of the few-body problem.

DOI
10.1119/1.16050

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A solution to quantum mechanical problems of two, three, and four bodies is discussed from the point of view of two-body harmonic oscillator forces. An exact solution to the three- and four-body problems is found for harmonic oscillator forces, assuming equal masses. The aim of the discussion is to make students aware of the real difficulties of the few-body problem.
In this thesis, I go through the well-known solutions to the one and two-particle systems trapped in a quantum harmonic oscillator and then continue to the three, four and many-body quantum systems. This is done by developing new analytical models and numerical methods both for the few- and many-body systems. One-dimensional systems are very interesting in a sense that particles aligned on a line can only change seats by going through each other. This property can be exploited in the strongly interacting regime, where particles are forced to sit in a specific configuration, which can be easily manipulated. The knowledge of how and where the particles are can be exploited in future quantum applications. In short, the thesis is about establishing a solid knowledge about everything that one needs to know about the one-dimensional few- and many-component interacting quantum systems trapped in harmonic oscillator potentials.
We consider an analytic way to make the interacting N -body problem tractable by using harmonic oscillators in place of the relevant two-body interactions. The two-body terms of the N -body Hamiltonian are approximated by considering the energy spectrum and radius of the relevant two-body problem which gives frequency, centre position, and zero point energy of the corresponding harmonic oscillator. Adding external harmonic one-body terms, we proceed to solve the full quantum mechanical N -body problem analytically for arbitrary masses. Energy eigenvalues, eigenmodes, and correlation functions like density matrices can then be computed analytically. As a first application of our formalism, we consider the N -boson problem in two and three dimensions where we fit the two-body interactions to agree with the well-known zero-range model for two particles in a harmonic trap. Subsequently, condensate fractions, spectra, radii, and eigenmodes are discussed as a function of dimension, boson number N , and scattering length obtained in the zero-range model. We find that energies, radii, and condensate fraction increase with scattering length as well as boson number, while radii decrease with increasing boson number. Our formalism is completely general and can also be applied to fermions, Bose–Fermi mixtures, and to more exotic geometries.
A few-body quantum mechanical system is examined. It is assumed that the scattering operator of the system is a sum of two-body potential operators. Some odd ends in the sum commute. Four-dimensional perturbation potentials and second quantized formalism is used for a solution of the problem. The scattering of two distinct particles in a force field is considered as an example. For this case, the system of the integral equations for the scattering amplitudes are obtained. These amplitudes permit us to construct channel Möller operators. It has been shown that such system has unique solution equivalent to the solution of the corresponding Schrödinger equation.