arXiv:1002.3051 · Journal of Physics A Mathematical and Theoretical

Inner products of resonance solutions in 1D quantum barriers

J. Julve, Fernando Jordán de Urries

Published Apr 13, 2010• 10 citations

Abstract

The properties of a prescription for the inner products of resonance (Gamow states), scattering (Dirac kets) and bound states for one-dimensional quantum barriers are worked out. The divergent asypmtotic behaviour of the Gamow states is regularized using a Gaussian convergence factor first introduced by Zel'dovich. With this prescription, most of these states (with discrete complex energies) are found to be orthogonal to each other and to the Dirac kets, except when they are neighbours, in which case the inner product is divergent. Therefore, as it happens for the continuum scattering states, the norm of the resonant ones remains non-calculable. Thus, they exhibit properties halfway between the (continuum real) Dirac-δ orthogonality and the (discrete real) Kronecker-δ orthogonality of the bound states.

DOI: 10.1088/1751-8113/43/17/175301

Related papers

General inner products for energy eigenstates

J. Julve, Sonia Turrini, Fernando Jordán de Urries

The features of the inner products between all the types of real and complex-energy solutions of the Schrödinger equation for 1-dimensional cut-off quantum potentials are worked out using a Gaussian regularization. A general Master Solution is introduced which describes any of the above solutions as particular cases. From it, a Master Inner Product is obtained which yields all the particular products. We show that the Outgoing and the Incoming Boundary Conditions fully determine the location of the momenta respectively in the lower and upper half complex plane even for purely imaginary momenta (anti-bound and bound solutions).

arXiv:1302.0630Feb 4, 20131 citation

Scattering by singular potentials in coupled Schrödinger equations

Peter Leth Christiansen, Jens Juul Rasmussen, Mads Peter Sørensen

Scattering in pairs of one-dimensional time-independent Schrödinger equations coupled via singular potentials are considered. For potentials given by Dirac’s delta function reflected and transmitted waves of the two modes as well as a bound state is found by integration. For potentials given by the derivative of Dirac’s delta function, resonant reflection and transmission occur for a discrete set of amplitudes of the scattering potential. These results depend on the applied regularization.

Low Temperature PhysicsDec 1, 20220 citations

Square integrable resonance wavefunctions

Nimrod Moiseyev

As discussed in the previous chapter, the poles of the S-matrix are identified with discrete eigenvalues of the time-independent Schrödinger equation, where the asymptotes of the corresponding eigenfunctions are either purely outgoing waves or purely incoming waves. More specifically, the bound and decay resonance poles are obtained by imposing the outgoing boundary conditions on the solutions of the time-independent Schrödinger equation, while the anti-bound and virtual states (sometimes referred to as capture resonances) are associated with the solutions obtained under the requirement of the incoming boundary conditions. Except for the bound states, all other poles of the S-matrix are associated with exponentially divergent wavefunctions which by definition do not belong to the Hilbert space of conventional Hermitian quantum mechanics. This fact represents a major difficulty for the development of a non-Hermitian quantum mechanical formalism. Consequently, one may wonder, for example, how to properly define an inner product in non-Hermitian quantum mechanics (NHQM) if the wavefunctions diverge asymptotically. We recall in this context that the concept of an inner product constitutes a fundamental building block of standard (Hermitian) quantum mechanics (QM), by means of which one defines the quantum mechanical expectation values of physically observable quantities over the quantum states under consideration. An inner product for NHQM is necessary in order to accommodate the tools of conventional QM. Furthermore, one might anticipate that an appropriate NHQM inner product would also facilitate practical numerical calculation of the S-matrix poles for those cases where the eigenvalues of the Hamiltonian do not possess an analytical closed form expression (unlike the cases studied in the previous chapter).

Cambridge University Press eBooksFeb 17, 20110 citations

On scattering from the one-dimensional multiple Dirac delta potentials

Fatih Erman, Manuel Gadella, Haydar Uncu

Abstract In this paper, we propose a pedagogical presentation of the Lippmann–Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann–Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for an arbitrary number of centres and study the threshold anomaly for the N = 2 and N = 4 cases. We also study further features like the quantum metastable states and resonances, including their corresponding Gamow functions and virtual or antibound states. The use of the Lippmann–Schwinger equation simplifies our analysis enormously and gives exact results for an arbitrary number of Dirac delta potentials.

European Journal of PhysicsJan 18, 201817 citations

Scattering, bound states, and resonances in the one-dimensional Dirac equation via supersymmetric quantum mechanics

Camila C. Soares, Luis B. Castro, Antonio S. de Castro

We develop a unified treatment of scattering and discrete spectra for the one-dimensional Dirac equation with scalar and vector interactions. Under the spin-symmetry condition, the coupled first-order Dirac system maps exactly onto an effective Sturm--Liouville (Schrö\-din\-ger-like) problem for a single spinor component. This mapping provides a convenient framework for analyzing transmission, reflection, and analytic continuation. As an explicit application, we consider effective interactions of hyperbolic Pöschl--Teller type and exploit supersymmetric quantum mechanics and shape invariance to obtain a closed-form expression for the transmission probability. The bound-state spectrum is then recovered from the poles of the analytically continued transmission amplitude, reproducing known results and offering a unified description of scattering and bound states. For the barrier configuration, we briefly comment on the resulting pole pattern in the complex momentum plane and its connection with resonance and quasi-normal-mode behavior. Moreover, we use the chiral transformation to relate the spin- and pseudospin-symmetry sectors and translate results between them without repeating the full derivation.

quant-pharXiv:2607.00306Jul 2, 2026

The Time-Independent Schrödinger Equation

Reinhold A. Bertlmann, Nicolai Friis

Abstract We discuss the procedure of solving the time-dependent Schrödinger equation via a separation ansatz that leads to the time-independent Schrödinger equation. The resulting equation is in the form of an eigenvalue problem for the Hamiltonian, whose solutions are the stationary states. We derive and discuss the bound states and corresponding energy levels for several one-dimensional potentials of interest, including the finite and infinite potential well, the Dirac-delta potential, and the square double-well modelling the spatial configuration of the ammonia molecule. We then turn to the scattered solutions and discuss reflection from and transmission through a finite potential barrier, thus encountering the tunnel effect, which we in turn formulate more generally using the Gamow factor. This leads us to the interesting phenomenon of tunnelling resonances

Sep 7, 20230 citations