For both identical and distinguishable particles and for an arbitrary polynomial approximation to the realistic two-body forces, the A-body Hamiltonian may be converted into an infinite block-three-diagonal matrix in the properly arranged, translationally invariant, oscillator basis. As a consequence, the exact Green function and all the projections of eigenstates of the microscopic Schrodinger equation become expressible in terms of the matrix continued fractions. This generalises the recent reformulation of the one-dimensional A=1 anharmonic problem by Graffi and Greechi (1975). The quick convergence of this non-perturbative method of solving the many-body bound-state problem is demonstrated for the simplest three- and four-body examples. With the core-possessing type V(r)=-r 2 +r 4 of the spin- and isospin-independent two-body interaction, only the 10 -3 error in the three-bosonic ground-state energy arises from restricting the formulae to 5*5 dimensional matrices.
For both identical and distinguishable particles and for an arbitrary polynomial approximation to the realistic two-body forces, the A-body Hamiltonian may be converted into an infinite block-three-diagonal matrix in the properly arranged, translationally invariant, oscillator basis. As a consequence, the exact Green function and all the projections of eigenstates of the microscopic Schrodinger equation become expressible in terms of the matrix continued fractions. This generalises the recent reformulation of the one-dimensional A=1 anharmonic problem by Graffi and Greechi (1975). The quick convergence of this non-perturbative method of solving the many-body bound-state problem is demonstrated for the simplest three- and four-body examples. With the core-possessing type V(r)=-r 2 +r 4 of the spin- and isospin-independent two-body interaction, only the 10 -3 error in the three-bosonic ground-state energy arises from restricting the formulae to 5*5 dimensional matrices.
Abstract The resolution of the Schrödinger equation for the translation-invariant N-body harmonic oscillator Hamiltonian in D dimensions with one-body and two-body interactions is performed by diagonalizing a matrix J of order N − 1 . It has been previously established that the diagonalization can be analytically performed in specific situations, such as for N ≤ 5 or for N identical particles. We show that the matrix J is diagonal, and thus the problem can be analytically solved, for any number of arbitrary masses provided some specific relations exist between the coupling constants and the masses. We present analytical expressions for the energies under those constraints.
We propose a new nonrelativistic Pauli-type equation where some specific small relativistic terms are retained. With the confining potentials ${V}_{\ensuremath{\infty}}(x)$ approximated by the polynomials ${V}_{m}(x)={g}_{0}{x}^{2}+\ensuremath{\cdots}+{g}_{m}{x}^{2m+2}$, gm>0, the nonzero kinematical corrections ${T}_{m}\ensuremath{-}{T}_{0}$, where ${T}_{m}={h}_{0}{p}^{2}+\ensuremath{\cdots}+{h}_{m}{p}^{2m+2}\ensuremath{\simeq}{T}_{\ensuremath{\infty}} ={({\ensuremath{\mu}}^{2}{c}^{4}+{p}^{2}{c}^{2})}^{\frac{1}{2}}\ensuremath{-}\ensuremath{\mu}{c}^{2}$, are added to the anharmonic-oscillator Schr\"odinger equation, so that the $p\ensuremath{-}x$ symmetry typical for a harmonic-oscillator Hamiltonian is restored. As a consequence of this semirelativistic regularization, the analytic diagonalization of an entirely anharmonic Hamiltonian Hmm=Tm+Vm in terms of the $m\ifmmode\times\else\texttimes\fi{}m$-matrix continued fractions is obtained. Both the auxiliary fractions and the eigenstates converge very quickly. In the cases of the bounded spectrum of Hmm (m=2q), it is proved exactly for q=1,2,and3 and conjectured for $q\ensuremath{\ge}4$.
This is an investigation of an anharmonic oscillator characterized by the potential (ω02/ 2) q2 + λ′ q4. By using the exact relations obtained by evaluating ≪i|[H, O]∓|j > where O is an arbitrary operator, H is our total Hamiltonian and |i > and |j > are exact eigenstates of H, we derive an exact recurrence formula. This formula allows us to express τ-functions with a higher power of the variables through τ-functions with a lower power of the variables and energy eigenvalues. In this way we derive several exact relations, which are, in a sense, generalizations of the virial theorem and sum rules (Eqs. (4·5), (4·6) and (5·4)). These exact relations are the central equaions of this paper. On the basis of these exact relations we propose our “nearest neighbour level” (N. N. L) approximation, which seems to provide a good approximation scheme. We can also use our exact relations to test the validity of various approximation methods, and as an example, we discuss the “New-Tamm-Dancoff” (N. T. D)-type of approximation in detail.
We construct an eigenvalue problem by confining many-body system to a bounded domain with the boundary condition that the wave function vanishes. By changing the boundary, however, the eigenvalues of the energy can be varied continuously. The D-matrix is defined for a series of bounded problems with the same value for the ground state energy. The D-matrix is related to the S-matrix, enabling us to calculate the S-matrix at a given energy. The Schrodinger equation for the system is transformed to a diffusion equation by regarding time as imaginary. Initial ensemble, representing an approximate wave function, is evolved, through Monte Carlo simulation of random walks and branching, to the ground state ensemble. The limitations of investigation are: 1. Ingoing and outgoing channels have two fragments. 2. The interaction between the fragments is negligible outside the boundary mentioned above. 3. The particles are bosons or we know the zeros of the wave function. First we consider the scattering of a particle by a potential, which is equivalent to the two-body problem, in one dimension. Here we use the Poschl-Teller potential for which the exact solution is known. We use this case to investigate a new sampling method and study of various parameters. Next we consider three particles in one dimension. Here we take interaction to be a potential well, where at least one of the interactions is attractive so that a two-body bound state is possible.
Abstract As the first stage in a project aimed at finding exact bound-state solutions for the few-particle Schrödinger equation, some restricted two-and three- body systems are treated by Rayleigh–Schrödinger perturbation theory. Formal solutions for the perturbation equations are obtained by using standard techniques for solving differential equations. Particular attention is given to justifying the various boundary conditions, since the same conditions apply to formal solutions of the eigenvalue equations. It is shown that boundary conditions designed to preserve the Hermitian character of the Hamiltonian are sufficient to determine all parameters in the formal solutions. The procedure is demonstrated by application to a hydrogen atom perturbed by stationary point charges. Results are given for the ground and some excited states.
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.