In this thesis, we investigate solutions of the Schrodinger equation inside regular polygons. It is well-known that the Schrodinger equation in two dimensions is separable for rectangular and circular 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 equation acquires 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 construct solutions which satisfy Dirichlet boundary condition. These are the only polygons which can tile the flat plane besides rectangles. We found that just rectangle, regular triangle, half a square and half a regular triangle are polygons which possess exact solutions of Schrodinger equation. In the problem of dir=rtl We also consider an array of regular hexagons tilling the flat plane with periodic boundary conditions and solved the Schrodinger equation by investigating the translational, rotational and reflectional symmetries. We show that for the special case of Dirichlet or Neumann boundary condition, solutions are exactly 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
