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چكيده انگليسي :

Solutions of the Schrodinger Equation in Regular Polygon Billiards Using Group Theoretical Methods Azadeh Nemati a nemati@ph iut ac ir Date of Submission September 12 2011 Department of Physics Isfahan University of Technology Isfahan 84156 83111 IranDegree M Sc Language FarsiSupervisor Ahmad Shirzad shirzad@ipm ir AbstractIn this thesis we investigate solutions of the Schrodinger equation inside regular polygons It iswell known that the Schrodinger equation in two dimensions is separable for rectangular andcircular billiards Moreover the solutions constitute complete orthonormal sets in each case A basic question arises as Is there any other polygon in which the Schrodinger equationacquires exact solutions and do these solutions constitute complete sets of orthogonal functions We consider the symmetry group of an equilateral triangle and regular hexagon to constructsolutions which satisfy Dirichlet boundary condition These are the only polygons which can tilethe flat plane besides rectangles We found that just rectangle regular triangle half a square andhalf a regular triangle are polygons which possess exact solutions of Schrodinger equation In theproblem of classic billiard these polygons are recognized as exactly solvable models in which thetrajectories in the phase space are restricted to a torus Going back to quantum billiards anyother two dimensional polygon can acquire exact solution of Schrodinger equation with Dirichletor Neumann boundary condition only if it is tiled with the above polygons We propose twokinds of solutions for Schrodinger equation Which we call them primitive or evolved ones An evolved solution is suggested as a solution in which the considered polygon is tiled withsmaller polygons possessing primitive solutions In other words an evolved solution is tiled withprimitive ones while primitive solutions can not be tiled with solutions within smaller polygons We observed that regular hexagon has not primitive solutions instead it acquires solutions whichare derived from triangular solutions belonging to triangles which tile it i e evolved solutions We showed that some approximate methods for the hexagon problem either violate theSchrodinger equation or violate the demanded boundary conditions For example in a rescalingapproach the solutions of Schrodinger equation in a circle is deformed to match the boundaryconditions of any desired billiard However we show that such functions are no longer thesolutions of such equations There are also some approaches in which a finite number ofsolutions of Schrodinger equation are combined so that the Dirichlet or Neumann boundaryconditions are satisfied in a finite set of points In this approach the resulting function can notsatisfy the boundary condition at any arbitrary point specially near the vertices We also consider an array of regular hexagons tilling the flat plane with periodic boundaryconditions and solved the Schrodinger equation by investigating the translational rotational andreflectional symmetries We show that for the special case of Dirichlet or Neumann boundary condition solutions areexactly the solutions of smaller triangles which tile the hexagons Keywords Schrodinger equation Regular Polygon Potential Well Symmetry Group Irreducible Representation Regular Hexagon Dirichlet Boundary Condition Primitive Solution Evolved Solution

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