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Zeitschrift für Naturforschung A

On a Unified Theory of Harmonic Oscillator Two-Centre Integrals

W . Wltschel

Published Feb 1, 1972• 2 citations
Publisher

Abstract

Abstract Occupation number representation and operator-technique are used in the calculation of harmonic oscillator matrixelements for one and two centres and for equal and different frequencies. The potentials treated are generalized Gauss-potentials of the form p̑k x̑l exp{a x̑2 }, x̑k p̑l exp{a p̑2 }, and p̑k x̑l exp{a x̑ p̑} which by application of an operator identity could be reduced to the same form. Applications in nuclear and molecular physics, in molecular spectroscopy and in quantum chemistry are discussed briefly.

DOI
10.1515/zna-1972-0202

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Abstract Occupation number representation and operator-technique are used in the calculation of harmonic oscillator matrixelements for one and two centres and for equal and different frequencies. The potentials treated are generalized Gauss-potentials of the form p̑k x̑l exp{a x̑2 }, x̑k p̑l exp{a p̑2 }, and p̑k x̑l exp{a x̑ p̑} which by application of an operator identity could be reduced to the same form. Applications in nuclear and molecular physics, in molecular spectroscopy and in quantum chemistry are discussed briefly.
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.
A field theoretical formulation of the rather easy problem of N identical, one-dimensional bosons, interacting pairwise via harmonic oscillator potentials of equal force constants is given, in a way similar to the way the nonlinear Schrodinger model provides a field theoretical formulation of a one-dimensional, delta -function Bose gas. The nonlinear integro-differential equation obtained for the complex scalar field of the model is treated classically, and solved in the semiclassical limit of a large number of particles. Working in the centre of mass frame of the field, a sequence of operators that would raise or lower the energy of a state by definite amounts without changing its particle number are constructed from certain bilinear expressions in the field. These operators have, in the limit of a large number of particles, commutation relations similar to those of the Fourier modes of a compact free field, except that the first mode is missing as a result of the fact that the field in the centre of mass frame is subject to the obvious constraint that its centre of mass always coincides with the coordinate origin. The quantum states can be labelled by the total centre of mass momentum, and the quasi-particle numbers in each of the independent oscillator modes. The energy spectrum of the system is calculated, giving the same results as the exact, quantum-mechanical N-body analysis.
Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
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.
Abstract We discuss the application of operator methods to one-dimensional systems other than the harmonic oscillator. In Section 3.1 the method of ‘oneway factorization’ is used to obtain generalized quantum harmonic potentials. In Section 3.2 factorization and shift operators for an arbitrary one-dimensional potential are considered, and a classification is presented of the problems that can be treated by this method [1]. The use of the hypervirial theorem in the calculation of expectation values for one-dimensional systems is discussed in Section 3.3. In Sections 3.4–3.7 four examples are used to illustrate the applications of the preceding material. In these examples we consider in some detail how shift operators for energy can be derived and we also use these shift operators to calculate wavefunctions. Supersymmetric quantum mechanics and shape invariance are discussed in Sections 3.8 and 3.9.