Quantum Particle Constrained to a Curved Surface

This thesis begins with an introduction of discussing the motion of P non interacting particles constrained in a curved surface of N dimensions. First the particles are described in the flat space R and Cartesian coordinates, an external potential, Vρ is considered to maintain the system constrained in a curved subspace DN. Assuming normal forces to keep particles on the curved surface, the part of Schrödinger equation which is independent of the potential Vρ and contains the curve space variables will be separated to internal and external parts, therefore a new Schrödinger equation is raised that depends on curved surface geometrical properties. In continuation, to explain this problem clearly, we study the motion of one quantum particle that is bounded at an arbitrary point p on a curved surface S in three dimensions. In order to get the true result, a potential V𝝀���� is assumed for the constrained particle so the wave function will be uniformly compressed. Since classical mechanics principal guides us to avoid tangential forces, only normal constraint forces are acceptable to keep particle on the surface. Choosing point Q as an immediate neighborhood for point p and considering differential geometry relations for these points, Schrödinger equation can be obtained. Hence it will be demonstrated that, the internal potential for bounded particle is a function of surface curvatures which cannot be found from metric tensors or its derivatives and so on classical Lagrangian. Therefore there is a strike contrast with classical mechanics where Lagrangian and Newtonian approach give same results. In addition, inconsequence, Schrödinger equation for a particle constrained on a Spherical shell, Cylindrical shell and a Toroid is determined. Finally we obtained Schrödinger equation for a particle constrained on a pseudosphere surface