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

Zeitschrift für Naturforschung A

Orthogonalitätsrelationen der mehrzeitigen erzeugenden Funktionale des Harmonischen Oszillators

R Weber

Published Feb 1, 1971• 2 citations
Publisher

Abstract

Abstract We treat the one-dimensional harmonic oscillator completely in the field theoretic calculus of many time generating functionals. Without the results of common quantum mechanics we compute eigen values and functionals of the energy preparing all information of the harmonic oscillator. As an example of functional integration and for applications in scattering theory we prove orthonormality relations of these functionals.

DOI
10.1515/zna-1971-0206

Related papers

Harmonic oscillator integrals by operator methods, a simple unified approach

Wolfgang Witschel

One- and two-centre matrix elements for a wide class of operators in an harmonic oscillator basis are derived by operator methods. The advantage of the method is that all calculations are reduced to the well known Hausdorff transform. The elementary operator steps in the harmonic oscillator occupation number representation make this technique suitable for introductory quantum mechanics classes. Zusammenfassung. Fur eine Vielzahl von Operatoren wurden Ein- und Zweizentren-Matrixelemente des harmonischen Oszillators mit Operatormethoden abgeleitet. Die elementaren Operatorschritte in der Besetzungszahldarstellung wurden auf die bekannte Hausdorff-Formel zuruckgefuhrt.

European Journal of Physics•Nov 1, 1996•5 citations

Berechnung des anharmonischen Oszillators mittels funktionaler Integration

GERD RÖPKE

Energy eigenvalues and transition-probabilities for the anharmonic oscillator are calculated by evaluating functional integrals in a lattice-space and reordering infinite series in summable series. The results are compared with the known values.

Zeitschrift für Naturforschung A•Jun 1, 1967•2 citations

The quantum-mechanical harmonic oscillator

Francisco Marcelo Fernández, Eduardo Alberto Castro

The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.

CRC Press eBooks•Jan 16, 2020•0 citations

The Harmonic Oscillator

Owen L. de Lange, R. E. Raab

Abstract In both classical and quantum physics the harmonic oscillator is one of the most important physical systems [1–4]. Furthermore, because of the simplicity and symmetry of the Hamiltonian operator, the quantum mechanical oscillator is particularly suited to analysis by abstract operator methods. For this reason the study of the one dimensional oscillator is a useful introduction to the application of operator methods in quantum mechanics. In Sections 2.1–2.4 we use these methods to determine energy eigenvalues and eigenkets, and matrix elements; the analysis is extended to isotropic multidimensional oscillators in Section 2.10. Quasiclassical states of the oscillator are discussed in Section 2.5. In Section 2.6 shift operators are used to calculate wave functions. In Sections 2.7–2.11 we discuss some applications of the theory of the oscillator.

•Nov 28, 1991•0 citations

One dimensional (1-D) harmonic oscillators revisited

Nilesh P. Barde, Pranav P. Bardapurkar

The concept of harmonic oscillator particularly one dimensional (1-D) is mentioned in literature repeatedly and is explained in more complex manner by using various methods. This creates difficulties in understanding the description of the concept for new learners. The purpose of this article was to enlighten different methods to formulate harmonic oscillator in improving knowledge about detailed steps to derive eigen energy values in more comprehensible manner for the beginners. The energy values are derived by using classical method, quantum mechanically, Schrodinger time independent equation, perturbation technique, variation method, WKB approximation etc. A coherent way of derivation of eigen values using various approaches makes this article as unique.

•May 1, 2018•0 citations

Unified Representation of Quantum Mechanics on One-dimensional Harmonic Oscillator

Yongqin Wang

A quantum state corresponds to a specific wave function. We adopt a new mathematical method [1] to improve Dirac's ladder operator method. A set of orthonormal wave functions will be used to associate the operator with the square matrix corresponding to it. These allow us to determine the matrix elements by using the operator relations without having to know the specific wave functions. As a result, we can get the direct results of matrix mechanics and wave mechanics on one-dimensional Harmonic oscillator and their descriptions will be also unified.

•Mar 7, 2013•1 citation
Abstract We treat the one-dimensional harmonic oscillator completely in the field theoretic calculus of many time generating functionals. Without the results of common quantum mechanics we compute eigen values and functionals of the energy preparing all information of the harmonic oscillator. As an example of functional integration and for applications in scattering theory we prove orthonormality relations of these functionals.
One- and two-centre matrix elements for a wide class of operators in an harmonic oscillator basis are derived by operator methods. The advantage of the method is that all calculations are reduced to the well known Hausdorff transform. The elementary operator steps in the harmonic oscillator occupation number representation make this technique suitable for introductory quantum mechanics classes. Zusammenfassung. Fur eine Vielzahl von Operatoren wurden Ein- und Zweizentren-Matrixelemente des harmonischen Oszillators mit Operatormethoden abgeleitet. Die elementaren Operatorschritte in der Besetzungszahldarstellung wurden auf die bekannte Hausdorff-Formel zuruckgefuhrt.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.
Abstract In both classical and quantum physics the harmonic oscillator is one of the most important physical systems [1–4]. Furthermore, because of the simplicity and symmetry of the Hamiltonian operator, the quantum mechanical oscillator is particularly suited to analysis by abstract operator methods. For this reason the study of the one dimensional oscillator is a useful introduction to the application of operator methods in quantum mechanics. In Sections 2.1–2.4 we use these methods to determine energy eigenvalues and eigenkets, and matrix elements; the analysis is extended to isotropic multidimensional oscillators in Section 2.10. Quasiclassical states of the oscillator are discussed in Section 2.5. In Section 2.6 shift operators are used to calculate wave functions. In Sections 2.7–2.11 we discuss some applications of the theory of the oscillator.
The concept of harmonic oscillator particularly one dimensional (1-D) is mentioned in literature repeatedly and is explained in more complex manner by using various methods. This creates difficulties in understanding the description of the concept for new learners. The purpose of this article was to enlighten different methods to formulate harmonic oscillator in improving knowledge about detailed steps to derive eigen energy values in more comprehensible manner for the beginners. The energy values are derived by using classical method, quantum mechanically, Schrodinger time independent equation, perturbation technique, variation method, WKB approximation etc. A coherent way of derivation of eigen values using various approaches makes this article as unique.
A quantum state corresponds to a specific wave function. We adopt a new mathematical method [1] to improve Dirac's ladder operator method. A set of orthonormal wave functions will be used to associate the operator with the square matrix corresponding to it. These allow us to determine the matrix elements by using the operator relations without having to know the specific wave functions. As a result, we can get the direct results of matrix mechanics and wave mechanics on one-dimensional Harmonic oscillator and their descriptions will be also unified.