Momentum And Kinetic Energy Operators For Particles Moving On Curved Surfaces

The quantum motion constrained on quantum dots is currently a hot research topic in condensed matter physics. A very interesting issue is the geometrical shapes of quantum dots. There are toroidal surface and M?bius surface etc besides the typical sphere and spheroid. In classical mechanics, for a particle moving on the curved surface embedded in three dimension Cartesian coordinates, the local curved coordinates on the surface and the Cartesian coordinates play equal roles in the description of its classical motion. In the differential geometry, we can use two complementary and fundamental methods such as intrinsic geometry and extrinsic geometry as well. However, traditional quantum mechanics, also the tradition of the modern physics research such as general relativity and gauge field, uses the local coordinate system only. Then can we use the three dimension Cartesian coordinates to describe motion constrained on the two dimensional curved surfaces?Expressing this problem in mathematical language, we have that two variables are needed to describe a two dimensional regular curved surface, for example (u , v ); and the surface equation is r =( x (u , v ), y (u , v ), z (u , v)), where the Hermitian momentum operators p_x , p_y ,p_x are the so called Cartesian momentum operators. They are entirely different from the canonical momentum operators on the curved surface. Generally speaking, these canonical momentum operators are not bounded operators, for example radial momentum operators , and their forms change with the coordinates transformations. This problem has been criticized by the mathematics and the physics circles.The research of the Cartesian momentum operators began in 2003. This present work proves that the Cartesian momentum operators take the following forms: where Hn is a geometrical invariant, which is called mean curvature vector field. This is an invariant under the transformations of coordinates.Furthermore, the Hamilton of the system seems to be but it is not the case. The correct Hamilton should be, where the f_x , f_y ,f_z are the nontrivial functions of u ,v . This is a new kind operator ordering problem. This paper also gives the differential equations determining these functions. Moreover, some explicit solutions are given for some concrete examples.The physical picture with Cartesian operators is much clear and classical correspondence is straightforward. Our research casts a new insight into the understanding of the classical correspondence of quantum mechanics.