Teimuraz Nadareishvili - Academia.edu (original) (raw)
Papers by Teimuraz Nadareishvili
arXiv (Cornell University), Dec 8, 2019
It is well-known that owing to the restricted character of the area, in which the system is enclo... more It is well-known that owing to the restricted character of the area, in which the system is enclosed, additional "surface terms" emerge in the traditional form of hypervirial and/or Ehrenfest theorems. Especially, when one considers spherically symmetric potentials and operators the radial distance in spherical coordinates is restricted by a half-plane. Therefore the extra terms arise in this case as well as in view of boundary conditions at the origin of coordinates. We analyze the role of these 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 essential in obtaining correct physical results. Among the well-known results some new ones are also derived.
After a short survey of the Klein Paradox in 3-dimensional relativistic equations, there is a det... more After a short survey of the Klein Paradox in 3-dimensional relativistic equations, there is a detailed consideration of Dirac modified equation, which follows by one particle overweight. It is shown, that the spin-orbital coupling is excluded in this equation as a result of pseudospin symmetry. That is why the separation of variables and reduction to radial equation is possible with standard methods in momentum space. The kernel of obtained radial equation differs from the kernel of spinless Salpeter equation in bounded, regular factor. That is why the equation has confinement type solutions.
It is shown that the squared operation of the Dirac equation which is widely applied may create n... more It is shown that the squared operation of the Dirac equation which is widely applied may create new solutions and moreover may change the inner nature of original equation. Some illustrating examples are considered as well.
After the short survey of the Klein Paradox in 3-dimensional relativistic equations, we present a... more After the short survey of the Klein Paradox in 3-dimensional relativistic equations, we present a detailed consideration of Dirac modified equation, which follows by one particle infinite overweighting in Salpeter Equation. It is shown, that the separation of angular variables and reduction to radial equation is possible by using standard methods in momentum space. The kernel of the obtained radial equation differs from that of spinless Salpeter equation in bounded regular factor. That is why the equation has solutions of confined type for infinitely increasing potential.
The problem of boundary behavior at the origin of coordinates is discussed for D-dimensional Schr... more The problem of boundary behavior at the origin of coordinates is discussed for D-dimensional Schrodinger equation in the framework of hyperspherical formalism, which have been often considered last time. We show that the naive (Dirichlet) condition, which seems as natural, is not mathematically well justified, on the contrary to the 3-dimensional case. The stronger argument in favor of Dirichlet boundary condition is the requirement of time independence of wave function's norm. The problem remains open for singular potentials.
Physics of Particles and Nuclei
It is well-known that owing to the restricted character of the area, in which the system is enclo... more It is well-known that owing to the restricted character of the area, in which the system is enclosed, additional "surface terms" emerge in the traditional form of hypervirial and/or Ehrenfest theorems. Especially, when one considers spherically symmetric potentials and operators the radial distance in spherical coordinates is restricted by a half-plane. Therefore the extra terms arise in this case as well as in view of boundary conditions at the origin of coordinates. We analyze the role of these 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 essential in obtaining correct physical results. Among the well-known results some new ones are also derived.
International Journal of Modern Physics B
Following to the Weil method, we generalize the Heisenberg–Robertson uncertainty relation for arb... more Following to the Weil method, we generalize the Heisenberg–Robertson uncertainty relation for arbitrary two operators. Consideration is made in spherical coordinates, where the distant variable [Formula: see text] is restricted from one side, [Formula: see text]. By this reason, accounting of suitable boundary condition at the origin for radial wavefunctions and operators is necessary. Therefore, there arise extra surface terms in comparison with traditional approaches. These extra terms are calculated for various solvable potentials and their influence is investigated. At last, the time–energy uncertainty relations are also analyzed. Some differences between our approach and that, in which a direct product for separate variances were considered, are discussed.
Following to the Weil method we generalize the Heisenberg-Robertson uncertainty relation for arbi... more Following to the Weil method we generalize the Heisenberg-Robertson uncertainty relation for arbitrary two operators. Consideration is made in spherical coordinates, where the distant variable is restricted from one side, . By this reason accounting of suitable boundary condition at the origin for radial wave functions and operators is necessary. Therefore, there arise extra surface terms in comparison with traditional approaches. These extra terms are calculated for various solvable potentials and their influence is investigated. At last, the time-energy uncertainty relations are also analyzed. Some differences between our approach and that, in which a direct product for separate variances were considered, is discussed.
arXiv: Mathematical Physics, Jan 19, 2010
Exploring the idea that equation for radial wave function must be compatible with the full Schrod... more Exploring the idea that equation for radial wave function must be compatible with the full Schrodinger equation, a boundary condition () 0 u 0 = is derived.
arXiv: Quantum Physics, Jun 2, 2018
Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking... more 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. потенциалов и получено поправки к правилам сумм Крамерса. Кроме того, выведена исправленная теорема Еренфеста и продемонстрирована ее соответствие с реальной физической картиной
It is shown that the well-known potentials, which give a good description of hadron spectra, can ... more It is shown that the well-known potentials, which give a good description of hadron spectra, can be constructed from infrared asymptotics of the gluon propagator, obtained according to the Dyson-Schwinger equation in gluodynamics. © 2010 Bull. Georg. Natl. Acad. Sci.
Virial theorem has a wide application in the classical as well as in the quantum mechanics. This ... more Virial theorem has a wide application in the classical as well as in the quantum mechanics. This theorem connects average values of kinetic and potential energies for the systems confined in limited areas. Moreover it allows making definite conclusions about some interesting problems without solving equations of motion. There are many generalizations of virial theorem, especially in relativistic quantum mechanics, for investigating bound states [1]. Recently much attention was devoted to singular potentials, namely, to potentials, behaving like , at r in the Schrodinger equation, and as for in the Klein-Gordon and Dirac equations. ( ) 0 2 V r V r − → ) 0 ( 0 > V 0 → 0 V rV − = 0 → r
In case of spinless particles there appear additional (singular) solutions in the framework of re... more In case of spinless particles there appear additional (singular) solutions in the framework of relativistic Klein-Gordon equation for Coulomb potential. These solutions obey to all requirements of quantum mechanical general principles. Observation of such states (hydrino, small hydrogen) should be important for manifestation of various physical phenomena. In this article the same problem is considered for spin-1/2 particle (electron) in the Dirac equation. It is shown that such kind of solutions really occurs, but the rate of singularity is more higher than in spinless case. By this reason we have no time- independence of total probability (norm). Moreover the orthogonality property is also failed, while the total probability is finite in the certain area of the model-parameters. Therefore, we are inclined to conclude that this additional solution in the Dirac equation must be ignored and restrict ourselves only by normal (standard) solutions.
Physics of Particles and Nuclei, 2020
Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking... more 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. потенциалов и получено поправки к правилам сумм Крамерса. Кроме того, выведена исправленная теорема Еренфеста и продемонстрирована ее соответствие с реальной физической картиной
physica status solidi (c), 2014
Semiconductor nanowires are believed to act as key elements in future nanoscaled optoelectronic d... more Semiconductor nanowires are believed to act as key elements in future nanoscaled optoelectronic devices, as they offer intriguing electrical and optoelectronic properties. However, the future of any semiconductor nanowire technology will essentially rely on their doping capability. The availability of both n- and p-type semiconductors is important for the realization of nanowire-based electronics. Wide band gap semiconductors, such as ZnO, suffer from doping polarity. They can be easily doped n- (or p-type) to the expense of difficulties for doping of opposite type. Space confinement changes donor and acceptor ionization energies. The main factor that makes difficult to obtain n- or p-conductivity is the formation of compensating defects. Compensating processes are strongly affected by electronic structure of the system: band gap, ionization energies of donors, acceptors and their compensation centers. In the presented work we calculated energy levels of an electron bound to Coulomb impurity that is incorporated in semiconductor nanowire. Effect of dielectric confinement on ionization energies are considered as well. For analyzing perspectives of suppressing processes of compensation and achieving low ohmic p-conductivity Kroger method of quasi-chemical equations is used. (© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
MRS Proceedings, 2014
ABSTRACTWe present calculation of electronic structure of impurity in nanowire. Ionization energy... more ABSTRACTWe present calculation of electronic structure of impurity in nanowire. Ionization energy of impurities are calculated in dependence on nanowire radius. Direct Hamiltonian matrix diagonalization method with the physically reasonable approximate potential is employed for finding the exact solution of Schrödinger equation in the effective-mass approximation. It is shown that shallow donors are strongly influences by space confinement, which is expressed in sharp increase of ionization energy. Calculations show that effect of space confinement on deep impurities is less pronounced. The obtained results give hope that by selecting optimal value of nanowire radius compensation processes can be suppressed.
Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Disc... more Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Discussion is carried on for most general second order differential equation, which involves all physically interesting cases, such as Schr\"odinger and two-body Klein-Gordon equations with singular potentials. Some physical consequences are discussed. The connection with Feynman-Hellmann like theorems are also considered and some relevant differences are underlined.
We obtain the extra delta-like singularity while reduction of the Laplace operator in spherical c... more We obtain the extra delta-like singularity while reduction of the Laplace operator in spherical coordinates, elimination of which restricts the radial wave functions at the origin. This restriction has the form of boundary condition for the radial wave function. Comment: 7 pages
We show that equation for radial wave function in its traditional form is compatible with the ful... more We show that equation for radial wave function in its traditional form is compatible with the full Schrodinger equation if and only if a definite additional constraint required. This constraint has a boundary condition form at the origin. Some of consequences are also discussed. Comment: 6 pages
Physics of Particles and Nuclei Letters, 2015
arXiv (Cornell University), Dec 8, 2019
It is well-known that owing to the restricted character of the area, in which the system is enclo... more It is well-known that owing to the restricted character of the area, in which the system is enclosed, additional "surface terms" emerge in the traditional form of hypervirial and/or Ehrenfest theorems. Especially, when one considers spherically symmetric potentials and operators the radial distance in spherical coordinates is restricted by a half-plane. Therefore the extra terms arise in this case as well as in view of boundary conditions at the origin of coordinates. We analyze the role of these 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 essential in obtaining correct physical results. Among the well-known results some new ones are also derived.
After a short survey of the Klein Paradox in 3-dimensional relativistic equations, there is a det... more After a short survey of the Klein Paradox in 3-dimensional relativistic equations, there is a detailed consideration of Dirac modified equation, which follows by one particle overweight. It is shown, that the spin-orbital coupling is excluded in this equation as a result of pseudospin symmetry. That is why the separation of variables and reduction to radial equation is possible with standard methods in momentum space. The kernel of obtained radial equation differs from the kernel of spinless Salpeter equation in bounded, regular factor. That is why the equation has confinement type solutions.
It is shown that the squared operation of the Dirac equation which is widely applied may create n... more It is shown that the squared operation of the Dirac equation which is widely applied may create new solutions and moreover may change the inner nature of original equation. Some illustrating examples are considered as well.
After the short survey of the Klein Paradox in 3-dimensional relativistic equations, we present a... more After the short survey of the Klein Paradox in 3-dimensional relativistic equations, we present a detailed consideration of Dirac modified equation, which follows by one particle infinite overweighting in Salpeter Equation. It is shown, that the separation of angular variables and reduction to radial equation is possible by using standard methods in momentum space. The kernel of the obtained radial equation differs from that of spinless Salpeter equation in bounded regular factor. That is why the equation has solutions of confined type for infinitely increasing potential.
The problem of boundary behavior at the origin of coordinates is discussed for D-dimensional Schr... more The problem of boundary behavior at the origin of coordinates is discussed for D-dimensional Schrodinger equation in the framework of hyperspherical formalism, which have been often considered last time. We show that the naive (Dirichlet) condition, which seems as natural, is not mathematically well justified, on the contrary to the 3-dimensional case. The stronger argument in favor of Dirichlet boundary condition is the requirement of time independence of wave function's norm. The problem remains open for singular potentials.
Physics of Particles and Nuclei
It is well-known that owing to the restricted character of the area, in which the system is enclo... more It is well-known that owing to the restricted character of the area, in which the system is enclosed, additional "surface terms" emerge in the traditional form of hypervirial and/or Ehrenfest theorems. Especially, when one considers spherically symmetric potentials and operators the radial distance in spherical coordinates is restricted by a half-plane. Therefore the extra terms arise in this case as well as in view of boundary conditions at the origin of coordinates. We analyze the role of these 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 essential in obtaining correct physical results. Among the well-known results some new ones are also derived.
International Journal of Modern Physics B
Following to the Weil method, we generalize the Heisenberg–Robertson uncertainty relation for arb... more Following to the Weil method, we generalize the Heisenberg–Robertson uncertainty relation for arbitrary two operators. Consideration is made in spherical coordinates, where the distant variable [Formula: see text] is restricted from one side, [Formula: see text]. By this reason, accounting of suitable boundary condition at the origin for radial wavefunctions and operators is necessary. Therefore, there arise extra surface terms in comparison with traditional approaches. These extra terms are calculated for various solvable potentials and their influence is investigated. At last, the time–energy uncertainty relations are also analyzed. Some differences between our approach and that, in which a direct product for separate variances were considered, are discussed.
Following to the Weil method we generalize the Heisenberg-Robertson uncertainty relation for arbi... more Following to the Weil method we generalize the Heisenberg-Robertson uncertainty relation for arbitrary two operators. Consideration is made in spherical coordinates, where the distant variable is restricted from one side, . By this reason accounting of suitable boundary condition at the origin for radial wave functions and operators is necessary. Therefore, there arise extra surface terms in comparison with traditional approaches. These extra terms are calculated for various solvable potentials and their influence is investigated. At last, the time-energy uncertainty relations are also analyzed. Some differences between our approach and that, in which a direct product for separate variances were considered, is discussed.
arXiv: Mathematical Physics, Jan 19, 2010
Exploring the idea that equation for radial wave function must be compatible with the full Schrod... more Exploring the idea that equation for radial wave function must be compatible with the full Schrodinger equation, a boundary condition () 0 u 0 = is derived.
arXiv: Quantum Physics, Jun 2, 2018
Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking... more 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. потенциалов и получено поправки к правилам сумм Крамерса. Кроме того, выведена исправленная теорема Еренфеста и продемонстрирована ее соответствие с реальной физической картиной
It is shown that the well-known potentials, which give a good description of hadron spectra, can ... more It is shown that the well-known potentials, which give a good description of hadron spectra, can be constructed from infrared asymptotics of the gluon propagator, obtained according to the Dyson-Schwinger equation in gluodynamics. © 2010 Bull. Georg. Natl. Acad. Sci.
Virial theorem has a wide application in the classical as well as in the quantum mechanics. This ... more Virial theorem has a wide application in the classical as well as in the quantum mechanics. This theorem connects average values of kinetic and potential energies for the systems confined in limited areas. Moreover it allows making definite conclusions about some interesting problems without solving equations of motion. There are many generalizations of virial theorem, especially in relativistic quantum mechanics, for investigating bound states [1]. Recently much attention was devoted to singular potentials, namely, to potentials, behaving like , at r in the Schrodinger equation, and as for in the Klein-Gordon and Dirac equations. ( ) 0 2 V r V r − → ) 0 ( 0 > V 0 → 0 V rV − = 0 → r
In case of spinless particles there appear additional (singular) solutions in the framework of re... more In case of spinless particles there appear additional (singular) solutions in the framework of relativistic Klein-Gordon equation for Coulomb potential. These solutions obey to all requirements of quantum mechanical general principles. Observation of such states (hydrino, small hydrogen) should be important for manifestation of various physical phenomena. In this article the same problem is considered for spin-1/2 particle (electron) in the Dirac equation. It is shown that such kind of solutions really occurs, but the rate of singularity is more higher than in spinless case. By this reason we have no time- independence of total probability (norm). Moreover the orthogonality property is also failed, while the total probability is finite in the certain area of the model-parameters. Therefore, we are inclined to conclude that this additional solution in the Dirac equation must be ignored and restrict ourselves only by normal (standard) solutions.
Physics of Particles and Nuclei, 2020
Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking... more 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. потенциалов и получено поправки к правилам сумм Крамерса. Кроме того, выведена исправленная теорема Еренфеста и продемонстрирована ее соответствие с реальной физической картиной
physica status solidi (c), 2014
Semiconductor nanowires are believed to act as key elements in future nanoscaled optoelectronic d... more Semiconductor nanowires are believed to act as key elements in future nanoscaled optoelectronic devices, as they offer intriguing electrical and optoelectronic properties. However, the future of any semiconductor nanowire technology will essentially rely on their doping capability. The availability of both n- and p-type semiconductors is important for the realization of nanowire-based electronics. Wide band gap semiconductors, such as ZnO, suffer from doping polarity. They can be easily doped n- (or p-type) to the expense of difficulties for doping of opposite type. Space confinement changes donor and acceptor ionization energies. The main factor that makes difficult to obtain n- or p-conductivity is the formation of compensating defects. Compensating processes are strongly affected by electronic structure of the system: band gap, ionization energies of donors, acceptors and their compensation centers. In the presented work we calculated energy levels of an electron bound to Coulomb impurity that is incorporated in semiconductor nanowire. Effect of dielectric confinement on ionization energies are considered as well. For analyzing perspectives of suppressing processes of compensation and achieving low ohmic p-conductivity Kroger method of quasi-chemical equations is used. (© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
MRS Proceedings, 2014
ABSTRACTWe present calculation of electronic structure of impurity in nanowire. Ionization energy... more ABSTRACTWe present calculation of electronic structure of impurity in nanowire. Ionization energy of impurities are calculated in dependence on nanowire radius. Direct Hamiltonian matrix diagonalization method with the physically reasonable approximate potential is employed for finding the exact solution of Schrödinger equation in the effective-mass approximation. It is shown that shallow donors are strongly influences by space confinement, which is expressed in sharp increase of ionization energy. Calculations show that effect of space confinement on deep impurities is less pronounced. The obtained results give hope that by selecting optimal value of nanowire radius compensation processes can be suppressed.
Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Disc... more Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Discussion is carried on for most general second order differential equation, which involves all physically interesting cases, such as Schr\"odinger and two-body Klein-Gordon equations with singular potentials. Some physical consequences are discussed. The connection with Feynman-Hellmann like theorems are also considered and some relevant differences are underlined.
We obtain the extra delta-like singularity while reduction of the Laplace operator in spherical c... more We obtain the extra delta-like singularity while reduction of the Laplace operator in spherical coordinates, elimination of which restricts the radial wave functions at the origin. This restriction has the form of boundary condition for the radial wave function. Comment: 7 pages
We show that equation for radial wave function in its traditional form is compatible with the ful... more We show that equation for radial wave function in its traditional form is compatible with the full Schrodinger equation if and only if a definite additional constraint required. This constraint has a boundary condition form at the origin. Some of consequences are also discussed. Comment: 6 pages
Physics of Particles and Nuclei Letters, 2015