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R. de la Madrid

Publications and source records attributed to R. de la Madrid.

At least 19 recordsLinked to original sources
arXiv2018

The Gamow-state description of the decay energy spectrum of neutron-unbound $^{25}$O

We show the feasibility of calculating the decay energy spectrum of neutron emitting nuclei within the Gamow-state description of resonances by obtaining the decay energy spectrum of $^{25}$O. We model this nucleus as a valence neutron interacting with an $^{24}$O inert core, and we obtain the resulting resonant energies, widths and decay energy spectra for the ground and first excited states. We also discuss the similarities and differences between the decay energy spectrum of a Gamow state and the Breit-Wigner distribution with energy-dependent width.

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arXiv2010

Replacing the Breit-Wigner amplitude by the complex delta function to describe resonances

Whenever the Breit-Wigner amplitude appears in a calculation,there are many instances (e.g., Fermi's two-level system and the Weisskopf-Wigner approximation) where energy integrations are extended from the scattering spectrum of the Hamiltonian to the whole real line. Such extensions are performed in order to obtain a desirable, causal result. In this paper, we recall several of those instances and show that substituting the Breit-Wigner amplitude by the complex delta function allows us to recover such desirable results without having to extend energy integrations outside of the scattering spectrum.

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arXiv2008

The resonance amplitude associated with the Gamow states

The Gamow states describe the quasinormal modes of quantum systems. It is shown that the resonance amplitude associated with the Gamow states is given by the complex delta function. It is also shown that under the near-resonance approximation of neglecting the lower bound of the energy, such resonance amplitude becomes the Breit-Wigner amplitude. This result establishes the precise connection between the Gamow states, Nakanishi's complex delta function and the Breit-Wigner amplitude. In addition, this result provides another theoretical basis for the phenomenological fact that the almost-Lorentzian peaks in cross sections are produced by intermediate, unstable particles.

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arXiv2007

Reply to Comment arXiv:0704.3529v1

In this reply, we show that the author of the Comment arXiv:0704.3529v1 inadvertently provides additional arguments against the use of Hardy functions as test functions for the Gamow states.

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arXiv2007

Reply to ``Comment on `On the inconsistency of the Bohm-Gadella theory with quantum mechanics'''

In this reply, we show that when we apply standard distribution theory to the Lippmann-Schwinger equation, the resulting spaces of test functions would comply with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov, and of the theory of ultradistributions were wrong. As well, we point out several differences between the ``standard method'' of constructing rigged Hilbert spaces in quantum mechanics and the method used in Time Asymmetric Quantum Theory.

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arXiv2007

A selfadjoint variant of the time operator

We study the selfadjoint time operator recently constructed by one of the authors. We will show that this time operator must be interpreted as a ``selfadjoint variant'' of the time operator.

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arXiv2006

Description of resonances within the rigged Hilbert space

The spectrum of a quantum system has in general bound, scattering and resonant parts. The Hilbert space includes only the bound and scattering spectra, and discards the resonances. One must therefore enlarge the Hilbert space to a rigged Hilbert space, within which the physical bound, scattering and resonance spectra are included on the same footing. In these lectures, I will explain how this is done.

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arXiv2006

On the inconsistency of the Bohm-Gadella theory with quantum mechanics

The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann-Schwinger equation are functionals over spaces of Hardy functions. The preparation-registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann-Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation-registration arrow of time is flawed. Thus, Hardy functions neither appear when we solve the Lippmann-Schwinger equation nor they should appear. It is also shown that the Bohm-Gadella theory does not rest on the same physical principles as quantum mechanics, and that it does not solve any problem that quantum mechanics cannot solve. The Bohm-Gadella theory must therefore be abandoned.

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arXiv2006

The rigged Hilbert space approach to the Lippmann-Schwinger equation. Part I

We exemplify the way the rigged Hilbert space deals with the Lippmann-Schwinger equation by way of the spherical shell potential. We explicitly construct the Lippmann-Schwinger bras and kets along with their energy representation, their time evolution and the rigged Hilbert spaces to which they belong. It will be concluded that the natural setting for the solutions of the Lippmann-Schwinger equation--and therefore for scattering theory--is the rigged Hilbert space rather than just the Hilbert space.

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arXiv2006

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

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 into a semigroup time evolution. Physically, the semigroup time evolution is simply a (retarded or advanced) propagator.

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arXiv2006

Localization of non-relativistic particles

This paper is a contribution to the problem of particle localization in non-relativistic Quantum Mechanics. Our main results will be (1) to formulate the problem of localization in terms of invariant subspaces of the Hilbert space, and (2) to show that the rigged Hilbert space incorporates particle localization in a natural manner.

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arXiv2005

On the connection between the Schrodinger and the Heisenberg pictures for unbounded operators

It is well known that the unboundedness of operators in Hilbert space entails domain troubles. It is also well known that most domain troubles can be surmounted by extending the Hilbert space to a rigged Hilbert space. In this note, we point out another of such troubles, namely the correspondence between the Schrodinger and the Heisenberg pictures for unbounded operators, and sketch the solution of this problem within the rigged Hilbert space.

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arXiv2005

The role of the rigged Hilbert space in Quantum Mechanics

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.

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arXiv2004

The Rigged Hilbert Space of the Free Hamiltonian

We explicitly construct the Rigged Hilbert Space (RHS) of the free Hamiltonian $H_0$. The construction of the RHS of $H_0$ provides yet another opportunity to see that when continuous spectrum is present, the solutions of the Schrodinger equation lie in a RHS rather than just in a Hilbert space.

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arXiv2003

The Importance of Boundary Conditions in Quantum Mechanics

We discuss the role of boundary conditions in determining the physical content of the solutions of the Schrodinger equation. We study the standing-wave, the ``in,'' the ``out,'' and the purely outgoing boundary conditions. As well, we rephrase Feynman's $+i ε$ prescription as a time-asymmetric, causal boundary condition, and discuss the connection of Feynman's $+i ε$ prescription with the arrow of time of Quantum Electrodynamics. A parallel of this arrow of time with that of Classical Electrodynamics is made. We conclude that in general, the time evolution of a closed quantum system has indeed an arrow of time built into the propagators.

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