Product trial function for confined quantum systems

To obtain the energy of a confined quantum system (e.g., a quantum-well wire) using the variational method, one usually uses a product ψ(r) = F(r)G(r) form for a trial wave function. The envelope function F takes care of the boundary condition (ψ = 0) by vanishing on the boundary of the confining region R while the function G can contain variational parameters for flexibility. We show that by judicious choice for F and G, the kinetic energy decouples into the sum of two terms: one associated with F and the other with G. For the particular case of a Coulomb potential (e.g., a hydrogenic impurity) we obtain (i) an expression for [Formula: see text] and (ii) a differential equation for the associated binding energy as a function of the size of the containing region R; both of these being valid for regions of arbitrary shape and size. A prediction of the differential equation is shown to agree with a published calculation for a quantum-well wire.