Curvature, Dimentionality, and the Necessity of 3D Hamiltonians

Eardley, Jon

Hamiltonians

Eardley, Jon

3D Hamiltonians

R. J. Baxter, G. R. W. Quispel

M. S. Shikakhwa, Noureddine Chair

Abstract The Hamiltonian for a spin zero particle that is confined to a 1D curve embedded in the 3D space is constructed. Confinement is achieved by starting with the particle living in a small tube surrounding the curve, and assuming an infinitely strong normal force that squeezes the thickness of the tube to zero, eventually pinning the particle to the curve. We follow the new approach that we applied to confine a particle to a surface, in that we start with an expression for the 3D momentum operators whose components along and normal to the curve directions are separately Hermitian. The kinetic energy operator expressed in terms of the momentum operator in the normal direction is then a Hermitian operator in this case. When this operator is dropped and the thickness of the tube surrounding the curve is set to zero, one automatically gets the Hermitian curve Hamiltonian that contains the geometric potential term as expected. It is demonstrated that the origin of this potential lies in the ordering or symmetrization of the original 3D momentum operators in order to render them Hermitian. The Hermitian momentum operator for the particle as it is confined to the curve is also constructed and is seen to be similar to what is known as the geometric momentum of a particle confined to a surface in that it has a term proportional to the curvature that is along the normal to the curve. The force operator of the particle on the curve is also derived, and is shown to reduce, for a curve with a constant curvature and torsion, to a -apparently- single component normal to the curve that is a symmetrization of the classical expression plus a quantum term. All the above quantities are then derived for the specific case of a particle confined to a cylindrical helix embedded in 3D space.

Daniel R. Bes

C. Grosche

In this contribution, I show that it is possible to construct three-dimensional spaces of nonconstant curvature, i.e., three-dimensional Darboux spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that, in the two three-dimensional Darboux spaces which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In D 3d-I, we find seven coordinate systems which separate the Schrödinger equation. For the second space, D 3d-II, all coordinate systems of flat three-dimensional Euclidean space which separate the Schrödinger equation also separate the Schrödinger equation in D 3d-II. I solve the path integral on D 3d-I in the (u, v, w) system and on D 3d-II in the (u, v, w) system and in spherical coordinates.