A geometrical theory of energy trajectories in quantum mechanics

Abstract

Suppose f(r) is an attractive central potential of the form f(r)=∑ki=1 g(i)( f(i)(r)), where {f(i)} is a set of basis potentials (powers, log, Hulthén, sech2) and {g(i)} is a set of smooth increasing transformations which, for a given f, are either all convex or all concave. Formulas are derived for bounds on the energy trajectories Enl =Fnl(v) of the Hamiltonian H=−Δ+vf(r), where v is a coupling constant. The transform Λ( f)=F is carried out in two steps: f→f̄→F, where f̄(s) is called the kinetic potential of f and is defined by f̄(s)=inf(ψ,f,ψ) subject to ψ∈𝒟⊆L2(R3), where 𝒟 is the domain of H, ∥ψ∥=1, and (ψ,−Δψ)=s. A table is presented of the basis kinetic potentials { f̄(i)(s)}; the general trajectory bounds F*(v) are then shown to be given by a Legendre transformation of the form (s, f̄*(s)) →(v, F*(v)), where f̄*(s) =∑ki=1g(i)× ( f̄(i)(s)) and F*(v) =mins>0{s+v f̄*(s)}. With the aid of this potential construction set (a kind of Schrödinger Lego), ground-state trajectory bounds are derived for a variety of translation-invariant N-boson and N-fermion problems together with some excited-state trajectory bounds in the special case N=2. This article combines into a single simplified and more general theory the earlier ‘‘potential envelope method’’ and the ‘‘method for linear combinations of elementary potentials.’’