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

physics pp. 1–12 Probabilistic Interpretation of Resonant States

Naomichi Hatano, Tatsuro Kawamoto, Joshua Feinberg

Published Aug 5, 2013• 0 citations
OpenAlex

Abstract

Abstract. We provide probabilistic interpretation of resonant states. This we do by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of time, and therefore leads to probability conservation. This is in contrast with the conventional employment of a bi-orthogonal basis that precludes probabilistic interpretation, since wave functions of resonant states diverge exponentially in space. On the other hand, resonant states decay exponentially in time, because momentum leaks out of the central scattering area. This momentum leakage is also the reason for the spatial exponential divergence of resonant state. It is by combining the opposite temporal and spatial behaviors of resonant states that we arrive at our probabilistic interpretation of these states. The physical need to normalize resonant wave functions over an expanding spatial domain arises because particles leak out of the region which contains the potential range and escape to infinity, and one has to include them in the total count of particle number.

Related papers

physics pp. 553–564 Probabilistic interpretation of resonant states

Naomichi Hatano, Tatsuro Kawamoto, Joshua Feinberg

Abstract. We provide probabilistic interpretation of resonant states. We do this by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of time, and therefore leads to probability conservation. This is in contrast with the conventional employment of a bi-orthogonal basis that precludes probabilistic interpretation, since wave functions of resonant states diverge exponentially in space. On the other hand, resonant states decay exponentially in time, because momentum leaks out of the central scattering area. This momentum leakage is also the reason for the spatial exponential divergence of resonant state. It is by combining the opposite temporal and spatial behaviours of resonant states that we arrive at our probabilistic interpretation of these states. The physical need to normalize resonant wave functions over an expanding spatial domain arises because particles leak out of the region which contains the potential range and escape to infinity, and one has to include them in the total count of particles.

•Jun 17, 2015•0 citations

Norm of resonance states in quantum scattering and electromagnetic systems

Florian Lorenz, Jan Möseritz-Schmidt, Roland Ketzmerick

Resonance states spatially diverge and are thus not square integrable. Instead, their norm is defined by the biorthogonal scalar product of left and right states. We replace the corresponding volume integral over infinite space by a convenient boundary integral in piecewise-homogeneous systems and apply this procedure to diverse physical settings. For a quantum particle in any number of dimensions we treat hard-wall and piecewise-constant potentials. For electromagnetic systems with piecewise-homogeneous material properties, we consider three-dimensional and effectively two-dimensional cavities.

quant-pharXiv:2607.22052•Jul 27, 2026

What is the resonant state in open quantum systems?

Naomichi Hatano

Abstract The article reviews the theory of open quantum system from a perspective of the non-Hermiticity that emerges from the environment with an infinite number of degrees of freedom. The non-Hermiticity produces resonant states with complex eigenvalues, resulting in peak structures in scattering amplitudes and transport coefficients. After introducing the definition of resonant states with complex eigenvalues, we answer typical questions regarding the non-Hermiticity of open quantum systems. What is the physical meaning of the complex eigenmomenta and eigenenergies? How and why do the resonant states break the time-reversal symmetry that the system observes? Can we make the probabilistic interpretation of the resonant states with diverging wave functions? What is the physical meaning of the divergence of the wave functions? We also present an alternative way of finding resonant states, namely the Feshbach formalism, in which we eliminate the infinite number of the environmental degrees of freedom. In this formalism, we attribute the non-Hermiticity to the introduction of the retarded and advanced Green’s functions.

arXiv:2106.15068•Journal of Physics Conference Series•Oct 1, 2021•7 citations

A proposed definition of resonant states

Arturo Cisneros, Harold V. McIntosh

Abstract The widely cited definition of quantization in terms of square‐integrable wave functions does not apply to continuum wave functions, to such phenomena as metastable states, or many‐body resonances. A better philosophical foundation for quantum mechanics separates the probabilistic aspects based on square integrable Hilbert space functions from the dynamical aspects based upon the solutions of Shroedinger's (or Dirac's) equation. A Hilbert space may have a non‐Hilbert space basis, which may be described by Stieltjes integrals and a spectrum measure. This viewpoint is expounded by reference to a very detailed analysis of a simple model, through which a precise definition of a Bohr–Feshbach resonance can be given. We propose a definition of a “metastable state,” showing that it is consistent with accepted usage, and that it overcomes a series of objections which have been catalogued by Simon. Its rate of decay is given by the Fourier–Stieltjes transform of the spectral density function; it is moreover the longest‐lived initially localized state which can be formed from a small span of energy eigenfunctions near its mean energy.

International Journal of Quantum Chemistry•Jul 1, 1985•1 citation

Resonant States of Open Quantum Systems

Naomichi Hatano

We first show that quantum resonant states observe particle number conservation and hence are consistent with the probabilistic interpretation of quantum mechanics.We then present for a class of quantum open systems, a resonant-state expansion of the sum of the retarded and advanced Green's functions.The expansion is given purely in terms of all discrete eigenstates and does not contain any background integrals.Using the expansion, we argue that the Fano asymmetry of resonance peaks is interpreted as interference between discrete eigenstates.We microscopically derive the Fano parameters for several cases.

arXiv:0911.4320•Progress of Theoretical Physics Supplement•Jan 1, 2010•10 citations

Interpreting the wave function - what are electrons? And how do they move?

Shan Gao

In quantum mechanics, the physical state of an electron is described by a wave function. According to the standard probability interpretation, the wave function of an electron is probability amplitude, and its modulus square gives the probability density of finding the electron in a certain position in space. In this article, we show that this central assumption of quantum mechanics may have an ontological extension. It is argued that microscopic particles such as electrons are indeed particles, but their motion is not continuous but discontinuous and random. On this view, the modulus square of the wave function not only gives the probability density of the particles being found in certain locations, but also gives the probability density of the particles being there. In other words, the wave function in quantum mechanics can be regarded as a representation of the state of random discontinuous motion of particles, and at a deeper level, it may represent the dispositional property of the particles that determines their random discontinuous motion.

Discusiones Filosóficas•Jan 6, 2013•2 citations
Abstract. We provide probabilistic interpretation of resonant states. This we do by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of time, and therefore leads to probability conservation. This is in contrast with the conventional employment of a bi-orthogonal basis that precludes probabilistic interpretation, since wave functions of resonant states diverge exponentially in space. On the other hand, resonant states decay exponentially in time, because momentum leaks out of the central scattering area. This momentum leakage is also the reason for the spatial exponential divergence of resonant state. It is by combining the opposite temporal and spatial behaviors of resonant states that we arrive at our probabilistic interpretation of these states. The physical need to normalize resonant wave functions over an expanding spatial domain arises because particles leak out of the region which contains the potential range and escape to infinity, and one has to include them in the total count of particle number.
Abstract. We provide probabilistic interpretation of resonant states. We do this by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of time, and therefore leads to probability conservation. This is in contrast with the conventional employment of a bi-orthogonal basis that precludes probabilistic interpretation, since wave functions of resonant states diverge exponentially in space. On the other hand, resonant states decay exponentially in time, because momentum leaks out of the central scattering area. This momentum leakage is also the reason for the spatial exponential divergence of resonant state. It is by combining the opposite temporal and spatial behaviours of resonant states that we arrive at our probabilistic interpretation of these states. The physical need to normalize resonant wave functions over an expanding spatial domain arises because particles leak out of the region which contains the potential range and escape to infinity, and one has to include them in the total count of particles.
Resonance states spatially diverge and are thus not square integrable. Instead, their norm is defined by the biorthogonal scalar product of left and right states. We replace the corresponding volume integral over infinite space by a convenient boundary integral in piecewise-homogeneous systems and apply this procedure to diverse physical settings. For a quantum particle in any number of dimensions we treat hard-wall and piecewise-constant potentials. For electromagnetic systems with piecewise-homogeneous material properties, we consider three-dimensional and effectively two-dimensional cavities.
Abstract The article reviews the theory of open quantum system from a perspective of the non-Hermiticity that emerges from the environment with an infinite number of degrees of freedom. The non-Hermiticity produces resonant states with complex eigenvalues, resulting in peak structures in scattering amplitudes and transport coefficients. After introducing the definition of resonant states with complex eigenvalues, we answer typical questions regarding the non-Hermiticity of open quantum systems. What is the physical meaning of the complex eigenmomenta and eigenenergies? How and why do the resonant states break the time-reversal symmetry that the system observes? Can we make the probabilistic interpretation of the resonant states with diverging wave functions? What is the physical meaning of the divergence of the wave functions? We also present an alternative way of finding resonant states, namely the Feshbach formalism, in which we eliminate the infinite number of the environmental degrees of freedom. In this formalism, we attribute the non-Hermiticity to the introduction of the retarded and advanced Green’s functions.
Abstract The widely cited definition of quantization in terms of square‐integrable wave functions does not apply to continuum wave functions, to such phenomena as metastable states, or many‐body resonances. A better philosophical foundation for quantum mechanics separates the probabilistic aspects based on square integrable Hilbert space functions from the dynamical aspects based upon the solutions of Shroedinger's (or Dirac's) equation. A Hilbert space may have a non‐Hilbert space basis, which may be described by Stieltjes integrals and a spectrum measure. This viewpoint is expounded by reference to a very detailed analysis of a simple model, through which a precise definition of a Bohr–Feshbach resonance can be given. We propose a definition of a “metastable state,” showing that it is consistent with accepted usage, and that it overcomes a series of objections which have been catalogued by Simon. Its rate of decay is given by the Fourier–Stieltjes transform of the spectral density function; it is moreover the longest‐lived initially localized state which can be formed from a small span of energy eigenfunctions near its mean energy.
We first show that quantum resonant states observe particle number conservation and hence are consistent with the probabilistic interpretation of quantum mechanics.We then present for a class of quantum open systems, a resonant-state expansion of the sum of the retarded and advanced Green's functions.The expansion is given purely in terms of all discrete eigenstates and does not contain any background integrals.Using the expansion, we argue that the Fano asymmetry of resonance peaks is interpreted as interference between discrete eigenstates.We microscopically derive the Fano parameters for several cases.
In quantum mechanics, the physical state of an electron is described by a wave function. According to the standard probability interpretation, the wave function of an electron is probability amplitude, and its modulus square gives the probability density of finding the electron in a certain position in space. In this article, we show that this central assumption of quantum mechanics may have an ontological extension. It is argued that microscopic particles such as electrons are indeed particles, but their motion is not continuous but discontinuous and random. On this view, the modulus square of the wave function not only gives the probability density of the particles being found in certain locations, but also gives the probability density of the particles being there. In other words, the wave function in quantum mechanics can be regarded as a representation of the state of random discontinuous motion of particles, and at a deeper level, it may represent the dispositional property of the particles that determines their random discontinuous motion.