We consider a Hamiltonian $\hat H$ with a (partially) continuous spectrum and examine the zero-overlap condition which involves the projection onto exact continuum eigenstates of a set of pseudostates obtained from the diagonalization of $\hat H$ in a finite basis of square-integrable functions. For each projected pseudostate the condition implies the occurrence of zeros at all energies that correspond to the pseudo-continuum matrix eigenvalues, except for the eigenenergy associated with that pseudostate. This feature was observed for the Coulomb continuum represented in a Laguerre basis [M. McGovern et al., Phys. Rev. A 79, 042707 (2009)] and later explained using special properties of the Laguerre functions [I. B. Abdurakhmanov et al., J. Phys. B 44, 075204 (2011)]. We establish that a sufficient condition for the zero-overlap condition to occur is that the image space of the operator $\hat Q \hat H \hat P$ , where $\hat P$ is the projection operator onto the subspace spanned by the basis and $\hat Q = \hat 1 - \hat P$ its complement, has dimension one. We show that the condition is met for the one-dimensional free-particle problem by a basis of harmonic oscillator eigenstates and by a minimal set of simple momentum-space wave functions, and for the Coulomb problem by a Laguerre basis, thus offering an alternative proof for the latter case. The zero-overlap condition ensures that in, e.g., an ionizing collision or laser-atom interaction process, transition probabilities obtained from the projection of a time-propagated pseudostate-expanded system wave function onto eigenstates of $ \hat H $ are asymptotically stable.We consider a Hamiltonian $\hat H$ with a (partially) continuous spectrum and examine the zero-overlap condition which involves the projection onto exact continuum eigenstates of a set of pseudostates obtained from the diagonalization of $\hat H$ in a finite basis of square-integrable functions. For each projected pseudostate the condition implies the occurrence of zeros at all energies that correspond to the pseudo-continuum matrix eigenvalues, except for the eigenenergy associated with that pseudostate. This feature was observed for the Coulomb continuum represented in a Laguerre basis [M. McGovern et al., Phys. Rev. A 79, 042707 (2009)] and later explained using special properties of the Laguerre functions [I. B. Abdurakhmanov et al., J. Phys. B 44, 075204 (2011)]. We establish that a sufficient condition for the zero-overlap condition to occur is that the image space of the operator $\hat Q \hat H \hat P$, where $\hat P$ is the projection operator onto the subspace spanned by the basis and $\hat Q = \hat 1 - \hat P$ its complement, has dimension one. We show that the condition is met for the one-dimensional free-particle problem by a basis of harmonic oscillator eigenstates and by a minimal set of simple momentum-space wave functions, and for the Coulomb problem by a Laguerre basis, thus offering an alternative proof for the latter case. The zero-overlap condition ensures that in, e.g., an ionizing collision or laser-atom interaction process, transition probabilities obtained from the projection of a time-propagated pseudostate-expanded system wave function onto eigenstates of $ \hat H $ are asymptotically stable.

physics.atom-phCross-listedarXiv:2603.29750Apr 1, 2026

The overlap integrals of two harmonic-oscillator wavefunctions: some remarks on originals and reproductions

S. Waldenstrøm, Kalbe Razi Naqvi

Chemical Physics LettersJan 1, 198253 citations

Adiabatic continuity, wave-function overlap, and topological phase transitions

Jiahua Gu, Kai Sun

In this paper, we study the relation between wave-function overlap and adiabatic continuity in gapped quantum systems. We show that for two band insulators, a scalar function can be defined in the momentum space, which characterizes the wave-function overlap between Bloch states in the two insulators. If this overlap is nonzero for all momentum points in the Brillouin zone, these two insulators are adiabatically connected, i.e., we can deform one insulator into the other smoothly without closing the band gap. In addition, we further prove that this adiabatic path preserves all the symmetries of the insulators. The existence of such an adiabatic path implies that two insulators with nonzero wave-function overlap belong to the same topological phase. This relation, between adiabatic continuity and wave-function overlap, can be further generalized to correlated systems. The generalized relation cannot be applied to study generic many-body systems in the thermodynamic limit, because of the orthogonality catastrophe. However, for certain interacting systems (e.g., quantum Hall systems), the quantum wave-function overlap can be utilized to distinguish different quantum states. Experimental implications are also discussed.In this paper, we study the relation between wave-function overlap and adiabatic continuity in gapped quantum systems. We show that for two band insulators, a scalar function can be defined in the momentum space, which characterizes the wave-function overlap between Bloch states in the two insulators. If this overlap is nonzero for all momentum points in the Brillouin zone, these two insulators are adiabatically connected, i.e., we can deform one insulator into the other smoothly without closing the band gap. In addition, we further prove that this adiabatic path preserves all the symmetries of the insulators. The existence of such an adiabatic path implies that two insulators with nonzero wave-function overlap belong to the same topological phase. This relation, between adiabatic continuity and wave-function overlap, can be further generalized to correlated systems. The generalized relation cannot be applied to study generic many-body systems in the thermodynamic limit, because of the orthogonality catastrophe. However, for certain interacting systems (e.g., quantum Hall systems), the quantum wave-function overlap can be utilized to distinguish different quantum states. Experimental implications are also discussed.

arXiv:1605.07627Physical review. B./Physical review. BSep 7, 201613 citations

The overlap interal of two harmonic-oscillator wave functions

W. L. Smith

Journal of Physics B Atomic and Molecular PhysicsAug 1, 19690 citations

Particle Existence Probability Spectrum (PEPS)

Jincheng Zhang

This paper introduces the Particle Existence Probability Spectrum (PEPS), a novel theoretical framework that challenges the conventional understanding of wave function collapse in quantum mechanics. The core concept posits that the probability distribution of a particle's existence itself possesses a structured spectrum, denoted as *Sp(ω)*. This spectrum represents a dynamic and inherently probabilistic organization of potential locations rather than a sudden, deterministic collapse of a pre-existing wave function. The theoretical underpinnings are developed through a rigorous mathematical treatment of quantum mechanics, emphasizing the continuous nature of probability. The experimental validation of PEPS relies on high-precision interference experiments, designed to reveal the underlying structure of the probability distribution. The implications of PEPS extend to a re-evaluation of wave function interpretation and could potentially unlock new insights into the fundamental nature of reality. The central equation for the PEPS is: *Sp(ω) = ∫ |ψ(x, ω)|^2 dx* Where: * *Sp(ω)* is the Particle Existence Probability Spectrum. * *|ψ(x, ω)|^2* is the probability density of finding the particle at position *x* with frequency *ω*. * *∫* denotes the integral over all possible positions *x*.This paper introduces the Particle Existence Probability Spectrum (PEPS), a novel theoretical framework that challenges the conventional understanding of wave function collapse in quantum mechanics. The core concept posits that the probability distribution of a particle's existence itself possesses a structured spectrum, denoted as *Sp(ω)*. This spectrum represents a dynamic and inherently probabilistic organization of potential locations rather than a sudden, deterministic collapse of a pre-existing wave function. The theoretical underpinnings are developed through a rigorous mathematical treatment of quantum mechanics, emphasizing the continuous nature of probability. The experimental validation of PEPS relies on high-precision interference experiments, designed to reveal the underlying structure of the probability distribution. The implications of PEPS extend to a re-evaluation of wave function interpretation and could potentially unlock new insights into the fundamental nature of reality. The central equation for the PEPS is: *Sp(ω) = ∫ |ψ(x, ω)|^2 dx* Where: * *Sp(ω)* is the Particle Existence Probability Spectrum. * *|ψ(x, ω)|^2* is the probability density of finding the particle at position *x* with frequency *ω*. * *∫* denotes the integral over all possible positions *x*.

Aug 17, 20260 citations

We consider a Hamiltonian

\hat H

with a (partially) continuous spectrum and examine the zero-overlap condition which involves the projection onto exact continuum eigenstates of a set of pseudostates obtained from the diagonalization of

\hat H

in a finite basis of square-integrable functions. For each projected pseudostate the condition implies the occurrence of zeros at all energies that correspond to the pseudo-continuum matrix eigenvalues, except for the eigenenergy associated with that pseudostate. This feature was observed for the Coulomb continuum represented in a Laguerre basis [M. McGovern et al., Phys. Rev. A 79, 042707 (2009)] and later explained using special properties of the Laguerre functions [I. B. Abdurakhmanov et al., J. Phys. B 44, 075204 (2011)]. We establish that a sufficient condition for the zero-overlap condition to occur is that the image space of the operator

\hat Q \hat H \hat P

, where

\hat P

is the projection operator onto the subspace spanned by the basis and

\hat Q = \hat 1 - \hat P

its complement, has dimension one. We show that the condition is met for the one-dimensional free-particle problem by a basis of harmonic oscillator eigenstates and by a minimal set of simple momentum-space wave functions, and for the Coulomb problem by a Laguerre basis, thus offering an alternative proof for the latter case. The zero-overlap condition ensures that in, e.g., an ionizing collision or laser-atom interaction process, transition probabilities obtained from the projection of a time-propagated pseudostate-expanded system wave function onto eigenstates of $ \hat H $ are asymptotically stable.