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

International Journal of Quantum Chemistry

Laplace transform wave functions

Christine M. Farmer

Published Nov 1, 1969• 6 citations
Publisher

Abstract

Abstract The wave function defining a quantum‐mechanical system is considered as the Laplace transform of some distribution and the consequent form of the Variational Principle derived; an integral equation defines the eigenfunctions of a certain subclass. The model of the hydrogen‐like atom is used to test the theory; the eigenfunctions and associated energy levels of the ground and excited states are obtained for arbitrary values of the orbital quantum number.

DOI
10.1002/qua.560030621

Related papers

Solution of Schrödinger equation by Laplace transform

M. J. Englefield

Laplace transform techniques for solving differential equations do not seem to have been directly applied to the Schrödinger equation in quantum mechanics. This may be because the Laplace transform of a wave function, in contrast to the Fourier transform, has no direct physical significance. However, this paper will show that scattering phase shifts and bound state energies can be determined from the singularities of the Laplace transform of the wave function. The Laplace transform method can thereby simplify calculations if the potential allows a straightforward solution of the transformed Schrödinger equation. Suitable cases are the Coulomb, oscillator and exponential potentials and the Yamaguchi separable non-local potential.

Journal of the Australian Mathematical Society•Aug 1, 1968•14 citations

The Hydrogen-Like Atom

Uri Peskin

The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.

Cambridge University Press eBooks•May 11, 2023•0 citations

Wave mechanics in quantum phase space: hydrogen atom

Jun Lu

The rigorous solutions of the stationary Schrodinger equation for hydrogen atom are solved with the wave-mechanics method within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The Fourier-like projection transformations of wave function from the phase space to position and momentum spaces are extended to three-dimensional systems. The eigenfunctions in general position and momentum spaces could be obtained through the transformations from eigenfunction in the phase space.

Progress in Natural Science Materials International•Feb 1, 2007•0 citations

Schrödinger’s original quantum–mechanical solution for hydrogen

Anna Galler, Jeremy Canfield, James K. Freericks

Abstract In 1926, Erwin Schrödinger wrote a series of papers that invented wave mechanics and set the foundation for much of the single-particle quantum mechanics that we teach today. In his first paper, he solved the Schrödinger equation using the Laplace method, which is a technique that is quite powerful, but rarely taught. This is unfortunate, because it opens the door to examining quantum mechanics from a complex-analysis perspective. Gaining this experience with complex analysis is a useful notion to consider when teaching quantum mechanics, as these techniques can be widely used outside of quantum mechanics, unlike the standard Frobenius summation method, which is normally taught, but rarely used elsewhere. The Laplace method strategy is subtle and no one has carefully gone through the arguments that Schrödinger did in this first paper, instead it is often just stated that the solution was adopted from Schlesinger’s famous differential equation textbook. In this work, we show how the Laplace method can be used to solve for the quantum–mechanical energy eigenfunctions of the hydrogen atom, following Schrödinger’s original solution, with all the necessary details, and illustrate how it can be taught in advanced instruction; it does require familiarity with intermediate-level complex analysis, which we also briefly review.

arXiv:2007.14798•European Journal of Physics•Sep 18, 2020•4 citations

Notions of Quantum Mechanics and Wave Function

Caio Lima Firme

Quantum mechanics began in very early twentieth century when Max Planck found the expression for black body thermal radiation in which the emitted light was not a continuum as postulated by classical physics. Erwin Schrodinger applied his equation to solve the quantum harmonic oscillator, the quantum rigid rotor, and a hydrogen-like atom. Pertaining to the solution of the Schrodinger equation to hydrogen atoms are the quantum numbers. C. K. Ingold’s overview of Schrodinger wave mechanics: “What can be known about an electron, considered as a particle, is of a statistical nature, and is summarized in the behavior of a wave. When considering singlet substances, the normal Zeeman Effect occurs, providing the discrete values of the orbital angular momentum. In an attempt to rationalize the anomalous Zeeman effect, W. Pauli proposed the fourth degree of freedom of the electron which he called “two-valuedness not describable classically”.

CRC Press eBooks•Aug 28, 2019•0 citations

Quantization problem and variational principle in the phase-space formulation of quantum mechanics

Leon W. Cohen

The problem of quantization in the phase-space formulation of quantum mechanics is considered. An integral equation for the phase-space eigenfunctions is derived which is equivalent to the standard eigenvalue equation for a quantum mechanical operator. A differential form is also given. A variational principle is derived for quasiprobability distributions. It is shown that the expected value of the classical Hamiltonian calculated with a trial quasiprobability distribution will be greater than the ground state energy only if the distribution is chosen from a certain class of functions. The notion of ψ-representability is introduced to classify these functions. They represent distributions which correspond to possible quantum mechanical states. Also, a general relation is given between different distribution functions.

Journal of Mathematical Physics•Oct 1, 1976•71 citations
Abstract The wave function defining a quantum‐mechanical system is considered as the Laplace transform of some distribution and the consequent form of the Variational Principle derived; an integral equation defines the eigenfunctions of a certain subclass. The model of the hydrogen‐like atom is used to test the theory; the eigenfunctions and associated energy levels of the ground and excited states are obtained for arbitrary values of the orbital quantum number.
Laplace transform techniques for solving differential equations do not seem to have been directly applied to the Schrödinger equation in quantum mechanics. This may be because the Laplace transform of a wave function, in contrast to the Fourier transform, has no direct physical significance. However, this paper will show that scattering phase shifts and bound state energies can be determined from the singularities of the Laplace transform of the wave function. The Laplace transform method can thereby simplify calculations if the potential allows a straightforward solution of the transformed Schrödinger equation. Suitable cases are the Coulomb, oscillator and exponential potentials and the Yamaguchi separable non-local potential.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
The rigorous solutions of the stationary Schrodinger equation for hydrogen atom are solved with the wave-mechanics method within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The Fourier-like projection transformations of wave function from the phase space to position and momentum spaces are extended to three-dimensional systems. The eigenfunctions in general position and momentum spaces could be obtained through the transformations from eigenfunction in the phase space.
Abstract In 1926, Erwin Schrödinger wrote a series of papers that invented wave mechanics and set the foundation for much of the single-particle quantum mechanics that we teach today. In his first paper, he solved the Schrödinger equation using the Laplace method, which is a technique that is quite powerful, but rarely taught. This is unfortunate, because it opens the door to examining quantum mechanics from a complex-analysis perspective. Gaining this experience with complex analysis is a useful notion to consider when teaching quantum mechanics, as these techniques can be widely used outside of quantum mechanics, unlike the standard Frobenius summation method, which is normally taught, but rarely used elsewhere. The Laplace method strategy is subtle and no one has carefully gone through the arguments that Schrödinger did in this first paper, instead it is often just stated that the solution was adopted from Schlesinger’s famous differential equation textbook. In this work, we show how the Laplace method can be used to solve for the quantum–mechanical energy eigenfunctions of the hydrogen atom, following Schrödinger’s original solution, with all the necessary details, and illustrate how it can be taught in advanced instruction; it does require familiarity with intermediate-level complex analysis, which we also briefly review.
Quantum mechanics began in very early twentieth century when Max Planck found the expression for black body thermal radiation in which the emitted light was not a continuum as postulated by classical physics. Erwin Schrodinger applied his equation to solve the quantum harmonic oscillator, the quantum rigid rotor, and a hydrogen-like atom. Pertaining to the solution of the Schrodinger equation to hydrogen atoms are the quantum numbers. C. K. Ingold’s overview of Schrodinger wave mechanics: “What can be known about an electron, considered as a particle, is of a statistical nature, and is summarized in the behavior of a wave. When considering singlet substances, the normal Zeeman Effect occurs, providing the discrete values of the orbital angular momentum. In an attempt to rationalize the anomalous Zeeman effect, W. Pauli proposed the fourth degree of freedom of the electron which he called “two-valuedness not describable classically”.
The problem of quantization in the phase-space formulation of quantum mechanics is considered. An integral equation for the phase-space eigenfunctions is derived which is equivalent to the standard eigenvalue equation for a quantum mechanical operator. A differential form is also given. A variational principle is derived for quasiprobability distributions. It is shown that the expected value of the classical Hamiltonian calculated with a trial quasiprobability distribution will be greater than the ground state energy only if the distribution is chosen from a certain class of functions. The notion of ψ-representability is introduced to classify these functions. They represent distributions which correspond to possible quantum mechanical states. Also, a general relation is given between different distribution functions.