Symmetry of the Hamiltonian

Abstract

The notion of symmetry is familiar from daily experience. We say that an object displays a symmetry if it is invariant under a transformation. This means that after the transformation, the object’s configuration is identical with the one it had before the transformation. Thus a sphere is symmetric because it is invariant under rotations. In quantum mechanics the concept of symmetry is particularly important. The importance conmes from the nature of the formalism and from the fact that microobjects are simpler, i.e. have less structure, than human-size objects. From a mathematical point of view, the notion of symmetry is intimately related to the algebraic structure called a group. We shall see that knowledge of a quantum system’s symmetry group reveals a number of the system’s properties, without its Hamiltonian being completely known. These properties are shared by all quantum systems whose Hamiltonian has the same symmetry group.¹