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arXiv:quant-ph/0603177 · Journal of Physics A Mathematical and General

The rigged Hilbert space approach to the Lippmann–Schwinger equation: II. The analytic continuation of the Lippmann–Schwinger bras and kets

Rafael de la Madrid

Published Mar 29, 2006• 10 citations
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Abstract

The analytic continuation of the Lippmann–Schwinger bras and kets is obtained and characterized. It is shown that the natural mathematical setting for the analytic continuation of the solutions of the Lippmann–Schwinger equation is the rigged Hilbert space rather than just the Hilbert space. It is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution, resulting in a semigroup time evolution. Physically, the semigroup time evolution is simply a (retarded or advanced) propagator.

DOI
10.1088/0305-4470/39/15/009

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The analytic continuation of the Lippmann–Schwinger bras and kets is obtained and characterized. It is shown that the natural mathematical setting for the analytic continuation of the solutions of the Lippmann–Schwinger equation is the rigged Hilbert space rather than just the Hilbert space. It is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution, resulting in a semigroup time evolution. Physically, the semigroup time evolution is simply a (retarded or advanced) propagator.
The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states ϕ+ϕ^+ϕ+ and the out-states (observables) ψ−ψ^-ψ− of the S-matrix elements (ψ−,ϕ+)=(ψout,Sϕin)(ψ^-,ϕ^+)=(ψ^{out},S ϕ^{in})(ψ−,ϕ+)=(ψout,Sϕin) any elements of the HS $\H$, {ψ−}={ϕ+}(=)˝\{ψ^-\}=\{ϕ^+\}(=\H){ψ−}={ϕ+}(=)˝​, the RHS theory chooses the boundary conditions~: ϕ+∈Φ−⊂⊂˝Φ−×ϕ^+\inΦ_-\subset\H\subsetΦ_-^\timesϕ+∈Φ−​⊂⊂˝​Φ−×​, ψ−∈Φ+⊂⊂˝Φ+×ψ^-\inΦ_+\subset \H\subset Φ_+^\timesψ−∈Φ+​⊂⊂˝​Φ+×​, where Φ−Φ_-Φ−​ (Φ+Φ_+Φ+​) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the analytically continued S-matrix. This can be phenomenologically justified by causality. The two RHS's for states ϕ+ϕ^+ϕ+ and observables ψ−ψ^-ψ− provide new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets ∣E±∈Φ∓×|E^{\pm}\inΦ_{\mp}^{\times}∣E±∈Φ∓×​ (solutions of the Lippmann-Schwinger equation with ±iε\pm iε±iε respectively) and the Gamow vectors ∣ER−iΓ/2±∈Φ∓×|E_R-iΓ/2^\pm\inΦ_{\mp}^\times∣ER​−iΓ/2±∈Φ∓×​. The Gamow vectors ∣ER−iΓ/2−|E_R-iΓ/2^-∣ER​−iΓ/2− have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.
The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential. The non-square integrable solutions of the time-independent Schrödinger equation are used to define Dirac kets, which are (generalized) eigenvectors of the Hamiltonian. These Dirac kets are antilinear functionals over the space of physical wave functions. They are also basis vectors that expand any physical wave function in a Dirac basis vector expansion. It is shown that an acceptable physical wave function must fulfill stronger conditions than just square integrability — the space of physical wave function is not the whole Hilbert space but rather a dense subspace of the Hilbert space. We construct the position and energy representations of the Rigged Hilbert Space generated by the square barrier potential Hamiltonian. We shall also construct the unitary operator that transforms from the position into the energy representation. We shall see that in the energy representation the Dirac kets act as the antilinear Schwartz delta functional. In constructing the Rigged Hilbert Space of the square barrier potential, we will find a systematic procedure to construct the Rigged Hilbert Space of a large class of spherically symmetric potentials. The example of the square barrier potential will also make apparent that the natural framework for the solutions of a Schrödinger operator with continuous spectrum is the Rigged Hilbert Space rather than just the Hilbert space.
Abstract. There is compelling evidence that, when continuous spectrum is present, the natural mathematical setting for Quantum Mechanics is the rigged Hilbert space rather than just the Hilbert space. In particular, Dirac’s bra-ket formalism is fully implemented by the rigged Hilbert space rather than just by the Hilbert space. In this paper, we provide a pedestrian introduction to the role the rigged Hilbert space plays in Quantum Mechanics, by way of a simple, exactly solvable example. The procedure will be constructive and based on a recent publication. We also provide a thorough discussion on the physical significance of the rigged Hilbert space. PACS numbers: 03.65.-w, 02.30.HqThe RHS in Quantum Mechanics 2 1.