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Some Quantum-Mechanical Relations in case of singular operators and potentials

Anzor Khelashvili, Teimuraz Nadareishvili

Published Jun 2, 2018• 0 citations
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Abstract

Elaboration of some fundamental relations in three dimensional quantum mechanics is considered in connection to taking into account the restriction on domains in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. In particular, we derive the relation between the average value of operator time derivative and the time derivative of mean value of this operator, which usually is considered as the same by definition. The deviation from known result is deduced manifested by extra term, which depends on boundary behavior mentioned above. The general form for this extra term participates to the hypervirial like theorems. As a particular case, the virial theorem for a Coulomb potential is considered and correction to the Kramers sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced also and its consistency with real physical picture is demonstrated.

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Hypervirial and Ehrenfest Theorems in Spherical Coordinates: Systematic Approach

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Abstract Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case, the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover, the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.

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Some Quantum Mechanical Relations in case of Singular Operators

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Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case,the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.

European Physical Society Conference on High Energy Physics (EPS-HEP)•Jul 1, 2019•0 citations

Some Quantum Mechanical Relations in case of Singular Operators

Teimuraz Nadareishvili, Anzor Khelashvili

Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case,the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.

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Anzor Khelashvili, Teimuraz Nadareishvili

Abstract It is well-known that if the physical system is located in the restricted area, additional “surface terms” emerge in the traditional form of hypervirial and/or Erenfest theorems. In the current literature mainly one-dimensional Schrodinger equation was considered in this respect. Our observation consists in that this situation emerges automatically in spherical coordinates, as well as one of the coordinates, namely radial distance, is restricted by a half-line. In particular, these considerations are clearly manifested, when one consider spherically symmetric potentials and operators, depended only on distance. Evidently, these additional terms give rise owing the boundary conditions for wave functions and the behavior of the considered operators at the origin of coordinates. We have analyzed the role of these additional terms for various model- potentials in the Schrodinger equation. We consider regular, as well as soft-singular potentials and show that the inclusion of these terms is very important for obtaining correct physical results. Some complicated integrals for hypergeometric functions are also derived. The modified virial relations, derived below, is converted into the usual relations, when the additional terms are absent and when present, they give reasonable corrections in correct direction. This fact also provides its legitimacy.

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Operators and bilinear densities in the Dirac formal 1D Ehrenfest theorem

Salvatore De Vincenzo

We calculate formal time derivatives of the mean values of the standard position, velocity, and mechanical momentum operators, i. e., the Ehrenfest theorem for a one-dimensional Dirac particle in the coordinate representation.We show that these derivatives contain boundary terms that essentially depend on the values taken there by characteristic bilinear densities.We do not automatically take the boundary terms to vanish (as is usually done); nevertheless, we relate the boundary terms to similar terms that must be zero if one requires the hermiticity of certain specific unbounded operators.Throughout the article, we thoroughly discuss and illustrate all these aspects, which include the relations to certain boundary conditions.To clarify, we call our approach formal because all operations involving operators (for example, some operators products) are performed without respecting the restrictions imposed by the sets of functions on which the self-adjoint operators can act.Moreover, the Dirac Hamiltonian that we consider in our calculations contains a potential that is the time component of a Lorentz two-vector; nevertheless, we also obtain and concisely discuss the Ehrenfest theorem for a Hamiltonian with the most general Lorentz potential in (1+1) dimensions.

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What is the boundary condition for the radial wave function of the Schrödinger equation?

Anzor Khelashvili, Teimuraz Nadareishvili

There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.

arXiv:1009.2694•American Journal of Physics•May 13, 2011•31 citations
Elaboration of some fundamental relations in three dimensional quantum
mechanics is considered in connection to taking into account the restriction on
domains in radial distance. In such cases the boundary behavior of the radial
wave function and singularity of operators at the origin of coordinates
contribute to these relations. In particular, we derive the relation between
the average value of operator time derivative and the time derivative of mean
value of this operator, which usually is considered as the same by definition.
The deviation from known result is deduced manifested by extra term, which
depends on boundary behavior mentioned above. The general form for this extra
term participates to the hypervirial like theorems. As a particular case, the
virial theorem for a Coulomb potential is considered and correction to the
Kramers sum rule is derived. Moreover the corrected Ehrenfest theorem is
deduced also and its consistency with real physical picture is demonstrated.
Abstract Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case, the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover, the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.
Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case,the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.
Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case,the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.
Abstract It is well-known that if the physical system is located in the restricted area, additional “surface terms” emerge in the traditional form of hypervirial and/or Erenfest theorems. In the current literature mainly one-dimensional Schrodinger equation was considered in this respect. Our observation consists in that this situation emerges automatically in spherical coordinates, as well as one of the coordinates, namely radial distance, is restricted by a half-line. In particular, these considerations are clearly manifested, when one consider spherically symmetric potentials and operators, depended only on distance. Evidently, these additional terms give rise owing the boundary conditions for wave functions and the behavior of the considered operators at the origin of coordinates. We have analyzed the role of these additional terms for various model- potentials in the Schrodinger equation. We consider regular, as well as soft-singular potentials and show that the inclusion of these terms is very important for obtaining correct physical results. Some complicated integrals for hypergeometric functions are also derived. The modified virial relations, derived below, is converted into the usual relations, when the additional terms are absent and when present, they give reasonable corrections in correct direction. This fact also provides its legitimacy.
We calculate formal time derivatives of the mean values of the standard position, velocity, and mechanical momentum operators, i. e., the Ehrenfest theorem for a one-dimensional Dirac particle in the coordinate representation.We show that these derivatives contain boundary terms that essentially depend on the values taken there by characteristic bilinear densities.We do not automatically take the boundary terms to vanish (as is usually done); nevertheless, we relate the boundary terms to similar terms that must be zero if one requires the hermiticity of certain specific unbounded operators.Throughout the article, we thoroughly discuss and illustrate all these aspects, which include the relations to certain boundary conditions.To clarify, we call our approach formal because all operations involving operators (for example, some operators products) are performed without respecting the restrictions imposed by the sets of functions on which the self-adjoint operators can act.Moreover, the Dirac Hamiltonian that we consider in our calculations contains a potential that is the time component of a Lorentz two-vector; nevertheless, we also obtain and concisely discuss the Ehrenfest theorem for a Hamiltonian with the most general Lorentz potential in (1+1) dimensions.
There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.