Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Institute of Physics eBooks

Resonances and bound states

Bimalendu Deb

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Resonances manifest as the sharp peaks observed in graphs of scattering cross section versus energy. Resonances in binary collisions are closely related to twobody bound or quasi-bound states, which are the discrete quantum states whose amplitudes fall off (usually exponentially) in space as the separation r → ∞. In the case of a single channel, a bound state is formed at a specific negative energy; that is, below the channel threshold. In the multichannel case, as mentioned in the preceding chapter, a bound state occurs when all the channels are closed; that is, below the threshold of the lowest channel. Under specific circumstances, a bound state may occur at zero energy; that is, right at the lowest threshold. Quasi-bound states are also discrete quantum states that are formed temporarily at a discrete positive energy (above the lowest threshold). Both resonant and bound states (including quasibound ones) can be described by a unified approach to cold collisions. The temporary binding of a quasi-bound state at a discrete energy enhances the scattering cross section at that energy. This occurs because the quasi-bound state eventually decays into the continuum of scattering states whose central energy equals the binding energy. The resonance width is determined by the coupling between the bound and continuum states.

DOI
10.1088/978-0-7503-3959-9ch4

Related papers

Historical Notes on Feshbach and Shape Resonances

A. R. P. Rau

In his interesting Reference Frame piece (Physics Today, August 2004, page 12), Daniel Kleppner relates Herman Feshbach’s reaction to the term “Feshbach resonance” and gives a capsule account of the topical and vital role this phenomenon plays in today’s explorations of cold collisions of fermions and bosons, and of condensates of atoms and molecules. Readers may be interested in the earlier history and why atomic physicists began to use a term that Feshbach himself dismissed as jargon.In the Los Alamos experiment of Howard Bryant’s group on photo-absorption by the negative ion of hydrogen, two prominent resonances were seen at photon energies of about 10.95 eV. This energy marks the position of the first excited state of the hydrogen atom above the single bound state of the negative ion.One narrow resonance occurs just below the threshold of that excitation, whereas the second, broad one occurs just above. Their physical mechanisms are different and, following the paradigm set by that historic experiment, it has become customary to refer to the two types of resonance, both common in atomic and molecular physics, as Feshbach and shape resonance, respectively. That is, a resonant coupling just below the relevant threshold, leading to narrow profiles, is distinguished from those lying just above the threshold to which they are most strongly coupled, and typically broad.The two types of resonances are most naturally viewed in state and coordinate space, respectively. Feshbach resonances are thought of in terms of the superposition of two quantum states, one with discrete and one with continuum character, leading to a temporary quasi-bound state that manifests itself as a resonance in that continuum channel. The independent and almost simultaneous work in nuclear physics by Feshbach using the language of projection operators and in atomic physics by Ugo Fano using the language of superposition of wave functions serves for their technical description.Shape resonances, on the other hand, may be associated with the shape of a trapping potential in real, coordinate space. Most often, the trapping potential is an angular momentum potential that combines with the internal one to create intervening barriers to low-energy particles. The resulting pictures often display “two-valley” potentials. In the case of the negative ion of hydrogen, such a barrier has been described in terms of the hyperspherical coordinates of that three-body system. Examples of such shape resonances abound in low-energy scattering of electrons from molecules.For more discussion and references, including to recent applications in mesoscopic condensed matter systems, see my pedagogical mini-review in Physica Scripta, volume 69, page C10, 2004.© 2005 American Institute of Physics.

Physics Today•Feb 1, 2005•22 citations

Feshbach resonance in atomic binary collisions in the Wigner threshold law regime

Maurice Raoult, Frederick H. Mies

We analyze, within the generalized multichannel quantum defect theory framework, atomic binary collision cross sections over the $1\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{K}$ to $10\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$ energy range just above the entrance channel threshold, a domain where the Wigner threshold law should apply. By adjusting the strength of a constant external magnetic field a Feshbach resonance is tuned at will over this energy range. In the threshold regime, the quasibound state interacts with an opening continuum whose wave function presents a strong energy dependence which reflects the breakdown of the WKB approximation. The effective discrete-continuum interactions become very sensitive to the energy. The consequences of these variations have been investigated by including threshold effects in the analysis of the Fano configuration-interaction theory [Phys. Rev. 124, 1866 (1961)] in terms of quantum defect theory quantities proposed by [Lecomte J. Phys. B 20, 3645 (1987)]. This analysis shows that the energy variations are in general so important that it becomes meaningless to associate a width with a Feshbach resonance. However, it is still possible to define the resonance energy, as long as the energy variation of the shift of the resonance, induced by the effective discrete-continuum interaction, remains linear over an energy range corresponding to the magnitude of the shift.

Physical Review A•Jul 27, 2004•41 citations

Ultracold molecules from ultracold atoms: a case study with the KRb molecule

Paul S. Julienne

Ultracold collisions of cold atoms or molecules make the bound states of the collision complex formed from the two colliding species accessible for control and manipulation of the cold species or the complex. Such resonances are best treated by a resonant scattering theory, which in the ultracold domain can take advantage of the properties of the long-range potential and the methods of multichannel quantum defect theory. Coupled channels calculations on the threshold scattering states and bound states of the 40 K 87 Rb molecule illustrate the ideas and methodology of quantum defect theory using the long-range potential and also demonstrate the spin properties of the bound states throughout the spectrum. 1

•Jan 1, 2009•33 citations

Feshbach resonances in atomic structure and dynamics

J. H. Macek, Jacob Schach Møller, Dmitri Fedorov

Feshbach resonances are quasi‐bound states that are represented, in first approximation, as bound states in energetically closed channels. Owing to coupling with open channels they can spontaneously break up into two or more fragments. I will review those states that separate into two fragments within the framework of multichannel effective range theory. Magnetic tuning of scattering lengths is described within this context. Such tuning has opened up many opportunities for fundamental studies of few‐body systems. By using multichannel zero‐range potentials, conventionally called pseudo‐potentials, it is possible to employ the model when the few‐body systems involve three or more fragments. The emergence of Efimov states for critical values of the tuning parameters is discussed.

AIP conference proceedings•Jan 1, 2008•0 citations

Threshold Resonance Effects and the Phase-Amplitude Approach

Di Shu

A fundamental aspect of scattering is the appearance of resonances, which appear in many systems. Although their effect is often lost due to averaging at room temperatures or higher, they can become dominant features at low or ultralow temperatures, where only a few partial waves contribute to the scattering process. Our ability to form and manipulate ultracold molecules provides the seed to study in a precise and controlled fashion the role of single partial waves, and state-to-state processes in chemical systems. We explore resonances occurring due to the existence of a quasi-bound state in the entrance channel of a scattering system and explain the energy scaling due to these near threshold resonances (NTR) based on the properties of the Jost functions. We also investigate the threshold resonance effects in the Efimov systems which have been studied in a variety of context, such as three-body Coulomb systems and nuclear three-body systems. Numerical schemes based on Milne's phase-amplitude approach are formulated to analyze NTR effects. We present a simple and practical approach to solve the long-standing problem of finding the non-oscillatory solution to Milne's equation. Our numerical approach also gives an integral representation of scattering phase shift and allows us to compute ultra-narrow shape resonances.

•Jan 1, 2018•0 citations

Ultracold molecules from ultracold atoms: a case study with the KRb molecule

Paul S. Julienne

Ultracold collisions of cold atoms or molecules make the bound states of the collision complex formed from the two colliding species accessible for control and manipulation of the cold species or the complex. Such resonances are best treated by a resonant scattering theory, which in the ultracold domain can take advantage of the properties of the long-range potential and the methods of multichannel quantum defect theory. Coupled channels calculations on the threshold scattering states and bound states of the 40K87Rb molecule illustrate the ideas and methodology of quantum defect theory using the long-range potential and also demonstrate the spin properties of the bound states throughout the spectrum.

arXiv:0812.1233•Faraday Discussions•Jan 1, 2009•54 citations
Resonances manifest as the sharp peaks observed in graphs of scattering cross section versus energy. Resonances in binary collisions are closely related to twobody bound or quasi-bound states, which are the discrete quantum states whose amplitudes fall off (usually exponentially) in space as the separation r → ∞. In the case of a single channel, a bound state is formed at a specific negative energy; that is, below the channel threshold. In the multichannel case, as mentioned in the preceding chapter, a bound state occurs when all the channels are closed; that is, below the threshold of the lowest channel. Under specific circumstances, a bound state may occur at zero energy; that is, right at the lowest threshold. Quasi-bound states are also discrete quantum states that are formed temporarily at a discrete positive energy (above the lowest threshold). Both resonant and bound states (including quasibound ones) can be described by a unified approach to cold collisions. The temporary binding of a quasi-bound state at a discrete energy enhances the scattering cross section at that energy. This occurs because the quasi-bound state eventually decays into the continuum of scattering states whose central energy equals the binding energy. The resonance width is determined by the coupling between the bound and continuum states.
In his interesting Reference Frame piece (Physics Today, August 2004, page 12), Daniel Kleppner relates Herman Feshbach’s reaction to the term “Feshbach resonance” and gives a capsule account of the topical and vital role this phenomenon plays in today’s explorations of cold collisions of fermions and bosons, and of condensates of atoms and molecules. Readers may be interested in the earlier history and why atomic physicists began to use a term that Feshbach himself dismissed as jargon.In the Los Alamos experiment of Howard Bryant’s group on photo-absorption by the negative ion of hydrogen, two prominent resonances were seen at photon energies of about 10.95 eV. This energy marks the position of the first excited state of the hydrogen atom above the single bound state of the negative ion.One narrow resonance occurs just below the threshold of that excitation, whereas the second, broad one occurs just above. Their physical mechanisms are different and, following the paradigm set by that historic experiment, it has become customary to refer to the two types of resonance, both common in atomic and molecular physics, as Feshbach and shape resonance, respectively. That is, a resonant coupling just below the relevant threshold, leading to narrow profiles, is distinguished from those lying just above the threshold to which they are most strongly coupled, and typically broad.The two types of resonances are most naturally viewed in state and coordinate space, respectively. Feshbach resonances are thought of in terms of the superposition of two quantum states, one with discrete and one with continuum character, leading to a temporary quasi-bound state that manifests itself as a resonance in that continuum channel. The independent and almost simultaneous work in nuclear physics by Feshbach using the language of projection operators and in atomic physics by Ugo Fano using the language of superposition of wave functions serves for their technical description.Shape resonances, on the other hand, may be associated with the shape of a trapping potential in real, coordinate space. Most often, the trapping potential is an angular momentum potential that combines with the internal one to create intervening barriers to low-energy particles. The resulting pictures often display “two-valley” potentials. In the case of the negative ion of hydrogen, such a barrier has been described in terms of the hyperspherical coordinates of that three-body system. Examples of such shape resonances abound in low-energy scattering of electrons from molecules.For more discussion and references, including to recent applications in mesoscopic condensed matter systems, see my pedagogical mini-review in Physica Scripta, volume 69, page C10, 2004.© 2005 American Institute of Physics.
We analyze, within the generalized multichannel quantum defect theory framework, atomic binary collision cross sections over the $1\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{K}$ to 10mK10\phantom{\rule{0.3em}{0ex}}\mathrm{mK}10mK energy range just above the entrance channel threshold, a domain where the Wigner threshold law should apply. By adjusting the strength of a constant external magnetic field a Feshbach resonance is tuned at will over this energy range. In the threshold regime, the quasibound state interacts with an opening continuum whose wave function presents a strong energy dependence which reflects the breakdown of the WKB approximation. The effective discrete-continuum interactions become very sensitive to the energy. The consequences of these variations have been investigated by including threshold effects in the analysis of the Fano configuration-interaction theory [Phys. Rev. 124, 1866 (1961)] in terms of quantum defect theory quantities proposed by [Lecomte J. Phys. B 20, 3645 (1987)]. This analysis shows that the energy variations are in general so important that it becomes meaningless to associate a width with a Feshbach resonance. However, it is still possible to define the resonance energy, as long as the energy variation of the shift of the resonance, induced by the effective discrete-continuum interaction, remains linear over an energy range corresponding to the magnitude of the shift.
Ultracold collisions of cold atoms or molecules make the bound states of the collision complex formed from the two colliding species accessible for control and manipulation of the cold species or the complex. Such resonances are best treated by a resonant scattering theory, which in the ultracold domain can take advantage of the properties of the long-range potential and the methods of multichannel quantum defect theory. Coupled channels calculations on the threshold scattering states and bound states of the 40 K 87 Rb molecule illustrate the ideas and methodology of quantum defect theory using the long-range potential and also demonstrate the spin properties of the bound states throughout the spectrum. 1
Feshbach resonances are quasi‐bound states that are represented, in first approximation, as bound states in energetically closed channels. Owing to coupling with open channels they can spontaneously break up into two or more fragments. I will review those states that separate into two fragments within the framework of multichannel effective range theory. Magnetic tuning of scattering lengths is described within this context. Such tuning has opened up many opportunities for fundamental studies of few‐body systems. By using multichannel zero‐range potentials, conventionally called pseudo‐potentials, it is possible to employ the model when the few‐body systems involve three or more fragments. The emergence of Efimov states for critical values of the tuning parameters is discussed.
A fundamental aspect of scattering is the appearance of resonances, which appear in many systems. Although their effect is often lost due to averaging at room temperatures or higher, they can become dominant features at low or ultralow temperatures, where only a few partial waves contribute to the scattering process. Our ability to form and manipulate ultracold molecules provides the seed to study in a precise and controlled fashion the role of single partial waves, and state-to-state processes in chemical systems. We explore resonances occurring due to the existence of a quasi-bound state in the entrance channel of a scattering system and explain the energy scaling due to these near threshold resonances (NTR) based on the properties of the Jost functions. We also investigate the threshold resonance effects in the Efimov systems which have been studied in a variety of context, such as three-body Coulomb systems and nuclear three-body systems. Numerical schemes based on Milne's phase-amplitude approach are formulated to analyze NTR effects. We present a simple and practical approach to solve the long-standing problem of finding the non-oscillatory solution to Milne's equation. Our numerical approach also gives an integral representation of scattering phase shift and allows us to compute ultra-narrow shape resonances.
Ultracold collisions of cold atoms or molecules make the bound states of the collision complex formed from the two colliding species accessible for control and manipulation of the cold species or the complex. Such resonances are best treated by a resonant scattering theory, which in the ultracold domain can take advantage of the properties of the long-range potential and the methods of multichannel quantum defect theory. Coupled channels calculations on the threshold scattering states and bound states of the 40K87Rb molecule illustrate the ideas and methodology of quantum defect theory using the long-range potential and also demonstrate the spin properties of the bound states throughout the spectrum.