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Modern Physics Letters A

ENERGY-DEPENDENT POTENTIAL AND NORMALIZATION OF WAVE FUNCTION

Amar Benchikha, L. Chétouani

Published Jun 6, 2013• 23 citations
Publisher

Abstract

The problem of normalization related to energy-dependent potentials is examined in the context of the path integral approach, and a justification is given. As examples, the harmonic oscillator and the hydrogen atom (radial) where, respectively the frequency and the Coulomb's constant depend on energy, are considered and their propagators determined. From their spectral decomposition, we have found that the wave functions extracted are correctly normalized.

DOI
10.1142/s021773231350079x

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The problem of normalization related to energy-dependent potentials is examined in the context of the path integral approach, and a justification is given. As examples, the harmonic oscillator and the hydrogen atom (radial) where, respectively the frequency and the Coulomb's constant depend on energy, are considered and their propagators determined. From their spectral decomposition, we have found that the wave functions extracted are correctly normalized.
de Broglie’s concept of particles as sources of phase is extended to the analysis of systems in which the potential is not constant. The treatment assumes that the phase information propagates as waves according to classical electrodynamics including reflection with phase changes from potential energy barriers. The energies and weighting functions in such cases are obtained by comparison of the phase of the accelerated particle relative to the phases of reflections from the change in potential. Application to the chemically interesting systems of the harmonic oscillator and hydrogen atom are considered using the time evolution of the relative phases of the reflected de Broglie phase waves, whereby the weighting functions are obtained from the distance dependence of the phase differences between the reflections and the source. This is accomplished conveniently by taking time derivatives of the relative de Broglie phases and considering the periodicity of the motion. Optimum trajectories are determined by minimization of the action 2i(mv2-r∂V/∂r)/ħγ which depends on the potential V and the velocity v. The treatment leads to the Newtonian force law for circular orbits and affords the same energies as derived using the Schrödinger equation.
In the model of a compressed atom (or ion) considered in the present paper the boundary condition associated with the corresponding uncompressed atom, i.e., the condition that the radial wave function must vanish at r=∞, is replaced by the boundary condition that the radial wave function must have a node at the finite distance r=a. The treatment of the problem of obtaining the energy shift due to the compression is based on the phase-integral method developed by Fröman and Fröman, an essential feature of which is that one can use exact formulas in the calculations and make all approximations in the final stage. The treatment of the problem of obtaining the relative change of the wave function due to the compression is based on the rigorous evaluation of the normalization integral developed by Furry [Phys. Rev. 71, 360 (1947)] and Yngve [J. Math. Phys. 13, 324 (1972)], in which one also uses exact formulas in the calculations and makes all approximations in the final stage. Since compression of an atom gives rise to very subtle effects, rigorous methods are indispensible for obtaining accurate and reliable analytical final formulas. As an application, the resulting general formulas are particularized to the case of a hydrogenic atom, and a numerical illustration of the accuracy of the formulas is given.