ua nt - p h / 95 05 01 3 v 1 16 M ay 1 99 5 Wavelet basis for the Schrödinger equation

The self-similar representation for the Schrödinger equation is derived. The general form Schrödinger equation ı¯ h ∂ψ ∂t = ˆ Hψ, (1) which is the generalization of the free particle wave equation ı¯ h ∂ψ ∂t = − ¯ h 2 2m ∂ 2 ψ ∂x 2 , describes the evolution of the wave function ψ(x, t) for the quantum system with a general form of HamiltonianˆH. The solution of the Schrödinger equation (1), if it exists, is not uniquely determined by the Hamiltonian; the symmetry of the problem and/or boundary conditions must be given also. For this reason, for the problems with spherical symmetry (SO 3) — say, the Hydrogen atom — we look for a solution within the class of spherical functions. For homogeneous problems, i.e. the problems invariant under translations, we suppose the solution to be the superposition of plane waves, the unitary representations of the translation group G : x ′ = x + b.