Quadratic Hamiltonians, quadratic invariants and the symmetry group SU(n)

Abstract

We show that any 2n-dimensional quadratic Hamiltonian may be transformed by a (usually time-dependent) linear canonical transformation into any other 2n-dimensional quadratic Hamiltonian, in particular that of the isotropic harmonic oscillator. This latter Hamiltonian possesses the symmetry group SU(n) and n2−1 linearly independent quadratic invariants which provide a basis for the generators of the group. Every other quadratic Hamiltonian is shown to have a quadratic invariant possessing SU(n) symmetry. The free particle structure is given explicitly. The anisotropic oscillator is shown not to possess SU(3) symmetry based on quadratic invariants. However, its wavefunctions and energy levels may be obtained directly from those of the isotropic oscillator whether the frequencies are commensurable or not.