We present two examples where the Schrödinger equation admits R-separable solutions. In one of them (a particle in a uniform force field) the Schrödinger equation admits separable and R-separable solutions.
We discuss separability of solutions to a Schrödinger equation that describes a composite quantum system and give some kinds of Hamiltonians H(t) such that the solution to Schrödinger equation induced by H(t) is separable at any time provided that it is separable at t = 0. For example, we prove that if the Hamiltonian H is time-independent and equals to the product P A ⊗ P B of two projections on the subsystems K A and K B , respectively, then the state |ψ(t)〉 of the composite system starting from a separable initial |ψ(0)〉 = |ψ A 〉 ⊗ |ψ B 〉 is separable for all t ∈ [0, T] if and only if either |ψ A 〉 is an eigenstate of P A , or |ψ B 〉 is an eigenstate of P B .
In recent times attention has been paid to the fact that (linear) wave equations admit of “soliton-like” solutions, known as localized waves or non-diffracting waves, which propagate without distortion in one direction. Such localized solutions (existing also for K-G or Dirac equations) are a priori suitable, more than gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, localized solutions exist even for the ordinary (linear) Schrödinger equation within standard quantum mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions (even if localized and “decaying”) are not square-integrable, as well as plane or spherical waves: we show therefore how to obtain finite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential.