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The Rigged Hilbert space formalism in quantum mechanics with applications to higher-order pole resonances and to potential barriers

C. Püntmann

Published Jan 1, 1999• 0 citations
OpenAlex

Abstract

In the first part, Gamow vectors from S-matrix poles of higher multiplicity are derived in analogy to the Gamow vectors describing resonances from first-order poles. With these vectors a density operator is constructed that describes resonances associated with higher-order poles that obey an exponential decay law. It turns out that this operator formed by higher-order Gamow vectors has a unique structure. In the second part, the Gamow vectors are used in order to describe decaying and resonance states at a potential barrier of finite width. The exact numerical solutions of the S-matrix poles are presented and the out-going boundary condition for the Gamow vector description and consequences from that are discussed.

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arXiv:quant-ph/9909081•WORLD SCIENTIFIC eBooks•Nov 1, 2000•5 citations
In the first part, Gamow vectors from S-matrix poles of higher multiplicity are derived in analogy to the Gamow vectors describing resonances from first-order poles. With these vectors a density operator is constructed that describes resonances associated with higher-order poles that obey an exponential decay law. It turns out that this operator formed by higher-order Gamow vectors has a unique structure. In the second part, the Gamow vectors are used in order to describe decaying and resonance states at a potential barrier of finite width. The exact numerical solutions of the S-matrix poles are presented and the out-going boundary condition for the Gamow vector description and consequences from that are discussed.
Resonances and decaying states have been associated with first-order poles of the S-matrix and represented by exponentially decaying Gamow kets using the Rigged Hilbert Space formulation of quantum mechanics. In analogy, higher-order S- matrix poles, represented by a mixture of Gamow-Jordan vectors, are associated with higher-order decaying states. The exact Golden Rule for decaying states associated to a higher-order pole of the S-matrix is calculated. These higher-order decaying states operators also obey the exponential decay law, but they have a more complicated linewidth than the Breit-Wigner energy distribution. This linewidth depends on the interaction V. To test whether these higher-order decaying state operators represent possible physical states, we discuss their linewidth for various assumptions about the matrix elements of V.
After a summary of the Rigged Hilbert space formulation of quantum mechanics and a brief statement of its advantages over von Neumann’s formulation, a mathematically correct definition of Gamow’s exponentially decaying vectors as generalized energy eigenvectors is suggested. It is shown that exponentially decaying vectors are obtained from the S-matrix poles in the lower half of the second sheet and exponentially growing vectors from the S-matrix poles in the upper half of the second sheet. Decaying ’’state’’ vectors are defined as functionals over half of the space of physical states and growing ’’state’’ vectors are defined as functionals over the other half. On functionals over these subspaces, the dynamical group of time development splits into two semigroups, one for t ≳ 0 and the other for t < 0. The generalized basis system connected with the spectrum of the Hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated.
In this paper we construct Gamow vectors for resonances given by poles of the analytic continuation of the S matrix of any finite order. We study their modes of decay (or growth). We obtain Jordan block structures for the extended Hamiltonians and evolution operators on the subspaces spanned by these Gamow vectors. We perform this study within the context of the rigged Hilbert space extension of quantum theory. We construct an explicit Friedrichs model with a double pole resonance to illustrate the general formulation.
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.