Path integral approach for spaces of nonconstant curvature in three dimensions
- PDF / 639,841 Bytes
- 8 Pages / 612 x 792 pts (letter) Page_size
- 85 Downloads / 150 Views
SECOND INTERNATIONAL WORKSHOP ON SUPERINTEGRABLE SYSTEMS IN CLASSICAL AND QUANTUM MECHANICS Theory
Path Integral Approach for Spaces of Nonconstant Curvature in Three Dimensions* C. Grosche** II. Institut fur ¨ theoretische Physik, Universitat ¨ Hamburg, Hamburg, Germany Received May 16, 2006
Abstract—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 threedimensional Darboux spaces which are discussed in this paper, the quantum motion is similar to the two¨ dimensional case. In D3d-I , we find seven coordinate systems which separate the Schrodinger equation. For the second space, D3d-II , all coordinate systems of flat three-dimensional Euclidean space which separate ¨ ¨ the Schrodinger equation also separate the Schrodinger equation in D3d-II . I solve the path integral on D3d-I in the (u, v, w) system and on D3d-II in the (u, v, w) system and in spherical coordinates. PACS numbers: 03.65.Db DOI: 10.1134/S1063778807030131
1. INTRODUCTION
(II) ds2 =
In this paper, the quantum motion on threedimensional spaces of nonconstant curvature is studied. In [1, 2], two-dimensional spaces of nonconstant curvature, called Darboux spaces, were introduced. Particular emphasis was put on separation of variables and finding all coordinate systems which ¨ separate the Schrodinger equation (respectively, the Helmholtz equation) and the path integral. Another important issue was to find all potentials in these spaces which are superintegrable. These potentials have the property that there are additional constants ¨ of motion and that the corresponding Schrodinger equation separates in more than one coordinate system. Actually, in two dimensions, these systems have three constants of motion. In [3], the path integral method [4–7] was applied to study the free motion on the four Darboux spaces DI −DIV , and a study of superintegrable potentials was completed in [8]. In [1, 2], the two-dimensional Darboux spaces were introduced as follows (we also insert for the light-cone coordinates (x, y) the coordinates (u, v), which will be called the (u, v) system): (I) ds2 = (x + y)dxdy = 2u(du2 + dv 2 ), ∗ **
a + b dxdy (x − y)2
bu2 − a (du2 + dv 2 ), u2 ds2 = (ae−(x+y)/2 + be−x−y )dxdy
(1.2)
=
(III)
−2u
=e
u
2
(1.3)
2
(b + ae )(du + dv ),
a(e(x−y)/2 + e(y−x)/2 ) + b dxdy (e(x−y)/2 − e(y−x)/2 )2 (1.4) a a− + 2 2 + (du + dv ), = sin2 u cos2 u where a and b are additional (real) parameters (a± = (a ± 2b)/4). DII has the property that, for a = 0, b = 1, we recover the two-dimensional Euclidean plane, and all four coordinate systems on the twodimensional Euclidean plane are also separable coo
Data Loading...
