The quantum n-body problem in dimension d ⩾ n – 1: ground state (original) (raw)
Related papers
Journal of Mathematical Physics, 2000
This selective review is written as an introduction to the mathematical theory of the Schro ¤ d inger ¥ equation for N ¦ particles. § Characteristic for these systems are the cluster properties of the potential in configuration space, which are expressed in a simple © geometric language. The methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results. The central theme is the interplay between the spectral theorÿ of N ¦-body Hamiltonians and the space-time and phase-space analysis of bound states and scattering states.
N -body gravity and the Schrödinger equation
Classical and Quantum Gravity, 2007
We consider the problem of the motion of N bodies in a self-gravitating system in two spacetime dimensions. We point out that this system can be mapped onto the quantum-mechanical problem of an N-body generalization of the problem of the H + 2 molecular ion in one dimension. The canonical gravitational N-body formalism can be extended to include electromagnetic charges. We derive a general algorithm for solving this problem, and show how it reduces to known results for the 2-body and 3-body systems.
Four-body problem in d-dimensional space: Ground state, (quasi)-exact-solvability. IV
Journal of Mathematical Physics
Due to its great importance for applications, we generalize and extend the approach of our previous papers to study aspects of the quantum and classical dynamics of a 4-body system with equal masses in d-dimensional space with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. The ground state (and some other states) in the quantum case and some trajectories in the classical case are of this type. We construct the quantum Hamiltonian for which these states are eigenstates. For d ≥ 3, this describes a six-dimensional quantum particle moving in a curved space with special d-independent metric in a certain d-dependent singular potential, while for d = 1 it corresponds to a three-dimensional particle and coincides with the A 3 (4-body) rational Calogero model; the case d = 2 is exceptional and is discussed separately. The kinetic energy of the system has a hidden sl(7, R) Lie (Poisson) algebra structure, but for the special case d = 1 it becomes degenerate with hidden algebra sl(4, R). We find an exactlysolvable four-body S 4-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable four-body sextic polynomial type potential with singular terms. Naturally, the tetrahedron whose vertices correspond to the positions of the particles provides pure geometrical variables, volume variables, that lead to exactly solvable models. Their generalization to the n-body system as well as the case of non-equal masses is briefly discussed.
Some specific solutions to the translation-invariant N-body harmonic oscillator Hamiltonian
Journal of Physics Communications
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.
The Relativistic Many Body Problem in Quantum Mechanics
Symmetries in Science XI, 2005
We discusse a relativistic Hamiltonian for an n-body problem in which all the masses are equal and all spins take value 1/2. In the frame of reference in which the total momentum P = 0, the Foldy-Wouthuysen transformation is applies and the positive energy part of the Hamiltonian is separated. The Hamiltonian with unharmonic oscillator potential is applied to describe mass differences for charmonium and bottonium states. * Member of El Colegio Nacional and Sistema Nacional de Investigadores
Approximate solutions for N-body Hamiltonians with identical particles in D dimensions
Results in Physics, 2013
A method based on the envelope theory is presented to compute approximate solutions for N-body Hamiltonians with identical particles in D dimensions (D P 2). In some favorable cases, the approximate eigenvalues can be analytically determined and can be lower or upper bounds. The accuracy of the method is tested with several examples, and an application to a N-body system with a minimal length is studied. Finally, a semiclassical interpretation is given for the generic formula of the eigenvalues.
Three-body problem in d-dimensional space: Ground state, (quasi)-exact-solvability
Journal of Mathematical Physics
As a straightforward generalization and extension of our previous paper, J. Phys. A50 (2017) 215201 [1] we study aspects of the quantum and classical dynamics of a 3-body system with equal masses, each body with d degrees of freedom, with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories (which are in the interaction plane) in the classical case are of this type. The quantum (and classical) Hamiltonian for which the states are defined by this type eigenfunctions is derived. It corresponds to a three-dimensional quantum particle moving in a curved space with special d-dimension-independent metric in a certain d-dependent singular potential, while at d = 1 it elegantly degenerates to a two-dimensional particle moving in flat space. It admits a description in terms of pure geometrical characteristics of the interaction triangle which is defined by the three relative distances. The kinetic energy of the system is dindependent, it has a hidden sl(4, R) Lie (Poisson) algebra structure, alternatively, the hidden algebra h (3) typical for the H 3 Calogero model as in the d = 3 case. We find an exactly-solvable three-body S 3-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential with singular terms. For both models an extra first order integral exists. For d = 1 the whole family of 3-body (twodimensional) Calogero-Moser-Sutherland systems as well as the TTW model are reproduced. It is shown that a straightforward generalization of the 3-body (rational) Calogero model to d > 1 leads to two primitive quasi-exactly-solvable problems. The extension to the case of non-equal masses is straightforward and is briefly discussed. 12 r 13 r 23 ∂ r 23 ∂ r 13 + 2(d − 1)(r 2 13 r 2 23) + γ (6r 2 13 r 2 23 + r 2 12 (r 2 13 + r 2 23) − r 4 13 − r 4 23) − 6 ω r 2 12 r 2 13 r 2 23 r 12 r 2 13 r 2 23 ∂ r 12 + 2(d − 1)(r 2 13 r 2 12) + γ (6r 2 13 r 2 12 + r 2 23 (r 2 13 + r 2 12) − r 4 13 − r 4 12) − 6 ω r 2 12 r 2 13 r 2 23 r 23 r 2 13 r 2 12 ∂ r 23 + 2(d − 1)(r 2 12 r 2 23) + γ (6r 2 12 r 2 23 + r 2 13 (r 2 12 + r 2 23) − r 4 12 − r 4 23) − 6 ω r 2 12 r 2 13 r 2 23 r 13 r 2 12 r 2 23 ∂ r 13 .
Relativistic N-body problem in a separable two-body basis
Physical Review C, 2001
We use Dirac's constraint dynamics to obtain a Hamiltonian formulation of the relativistic N -body problem in a separable two-body basis in which the particles interact pair-wise through scalar and vector interactions. The resultant N -body Hamiltonian is relativistically covariant. It can be easily separated in terms of the center-of-mass and the relative motion of any twobody subsystem. It can also be separated into an unperturbed Hamiltonian with a residual interaction. In a system of two-body composite particles, the solutions of the unperturbed Hamiltonian are relativistic two-body internal states, each of which can be obtained by solving a relativistic Schrödingerlike equation. The resultant two-body wave functions can be used as basis states to evaluate reaction matrix elements in the general N -body problem.
Journal of Statistical Physics, 2014
In the context of an ambient space with an arbitrary number d of dimensions, the many-body problem consisting of an arbitrary number N of particles confined by a common, external harmonic potential (realizing a container with soft walls) and interacting among themselves and with the environment with arbitrary conservative repulsive forces scaling as the inverse cube of distances, displays a peculiar behaviour: its effective volume oscillates isochronously without damping. We recently discovered this remarkable phenomenon (valid in the context of both classical and quantum mechanics) and discussed its implications in the context of statistical mechanics and thermodynamics; but after publishing these findings we were informed that essentially analogous results had been previously obtained by Lyndell-Bell and Lyndell-Bell. In the present paper, motivated by the need we felt to acknowledge this fact, we also offer some retrospective remarks on the N-body problem with quadratic and/or inversely-quadratic potentials in one-and more-dimensional space.