The quantum n-body problem in dimension $d\ge n-1$: ground state

TL;DR

This paper introduces a new coordinate framework for the quantum n-body problem in dimensions $d \,\geq\, n-1$, revealing a structure that simplifies the spectral problem and characterizes the ground state as a function of relative distances only.

Contribution

It develops a generalized Euler coordinate approach, decomposes the kinetic energy operator, and links it to hidden algebraic structures, providing a new perspective on the quantum n-body problem.

Findings

Kinetic energy decomposed into center-of-mass, radial, and angular parts.

Radial operator exhibits large reflection symmetry and algebraic structure.

Ground state depends solely on relative distances.

Abstract

We employ generalized Euler coordinates for the $n$ body system in $d \geq n-1$ dimensional space, which consists of the centre-of-mass vector, relative (mutual), mass-independent distances $r_{ij}$ and angles as remaining coordinates. We prove that the kinetic energy of the quantum $n$-body problem for $d \geq n-1$ can be written as the sum of three terms: (i) kinetic energy of centre-of-mass, (ii) the second order differential operator $\Delta_{rad}$ which depends on relative distances alone and (iii) the differential operator $\Omega$ which annihilates any angle-independent function. The operator $\Delta_{rad}$ has a large reflection symmetry group $Z_2^{\oplus \frac{n(n-1)}{2}}$ and in $\rho_{ij}=r_{ij}^2$ variables is an algebraic operator, which can be written in terms of generators of their {\it hidden} algebra $sl(\frac{n(n-1)}{2}+1, R)$. Thus, $\Delta_{rad}$ makes sense of the Hamiltonian of a quantum Euler-Arnold $sl(\frac{n(n-1)}{2}+1, R)$ top in a constant magnetic field. It is conjectured that for any $n$, the similarity-transformed $\Delta_{rad}$ is the Laplace-Beltrami operator plus (effective) potential; thus, it describes a $\frac{n(n-1)}{2}$-dimensional quantum particle in curved space. This was verified for $n=2,3,4$. After de-quantization the similarity-transformed $\Delta_{rad}$ becomes the Hamiltonian of the classical top with variable tensor of inertia in an external potential. This approach allows a reduction of the $dn$-dimensional spectral problem to a $\frac{n(n-1)}{2}$ -dimensional spectral problem if the eigenfunctions depend only on relative distances. We prove that the ground state function of the $n$ body problem depends on relative distances alone.