A GEOMETRICAL THEORY OF ENERGY TRAJECTORIES IN QUANTUM MECHANICS
198312 pages
Published in:
- J.Math.Phys. 24 (1983) 324-335
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Abstract: (AIP)
Suppose f(r) is an attractive central potential of the form f(r)=∑ k i=1 g (i) ( f (i)(r)), where {f (i)} is a set of b a s i s p o t e n t i a l s (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 E n l =F n l (v) of the Hamiltonian H=−Δ+v f(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 k i n e t i c p o t e n t i a l of f and is defined by f̄(s)=inf(ψ,f,ψ) subject to ψ∈D⊆L 2(R 3), 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) =∑ k i=1 g (i)× ( f̄(i)(s)) and F *(v) =min s>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.’’- QUANTUM MECHANICS: ENERGY EIGENSTATE
- ENERGY SPECTRUM
- POTENTIAL: YUKAWA
- POTENTIAL: COULOMB
- NUMERICAL CALCULATIONS
- MODEL: BOSON
- MODEL: FERMION
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