Classical and Quantum Mechanics of Noncentral Potentials, 1998
Departure for his Heavenly abode. Na sato vidyate bha vo nabhavo vidyate satah, Ubhayorapi drasto... more Departure for his Heavenly abode. Na sato vidyate bha vo nabhavo vidyate satah, Ubhayorapi drastoantastvanayostatva darsibhih. -Srimad Bhagvad-Gita (2/16) The unreal has no existence, and the real never ceases to be; the reality of both (real and unreal) has thus been perceived by the seers of Truth.
With a view to having further insight into the mathematical content of the non-Hermitian Hamilton... more With a view to having further insight into the mathematical content of the non-Hermitian Hamiltonian associaterd with the diffusion-reaction (D-R) equation in one dimension, we investigate (a) the solitary wave solutions of certain types of its nonlinear versions, and (b) the problem of real eigenvalue spectrum associated with its linear version or with this class of non-Hermitian Hamiltonians. For the case (a) we use the standard techniques to handle the quadratic and cubic nonlinearities in the D-R equation whereas for the case (b) a newly proposed method, based on an extended complex phase space, is employed. For a particular class of solutions, an Ermakov system of equations is also found for the linear case. Further, corresponding to the 'classical' version of the above one-dimensional complex Hamiltonian, an equivalent integrable system of two, two-dimensional real Hamiltonians is suggested.
The construction of the second constant of motion of second order for two-dimensional classical s... more The construction of the second constant of motion of second order for two-dimensional classical systems is carried out in terms ofz=q 1 +iq 2 andq=q 1 −iq 2. As a result a class of Toda-type potentials admitting second order invariants is explored.
For a time-dependent system characterized by the potential V(x,t)ϭa 2 (t)x 2 ϩa 1 /x 2 , we first... more For a time-dependent system characterized by the potential V(x,t)ϭa 2 (t)x 2 ϩa 1 /x 2 , we first demonstrate the construction of classical and quantum invariants and then obtain an exact solution of the Schrödinger equation for this potential. The role of the invariant so constructed is further discussed in the context of studies of the coherent states. ͓S1050-2947͑97͒02704-2͔
Summary Ground-state masses and the mass spectra of lowest-lying beauty mesons B, B*, Bs, B s * ... more Summary Ground-state masses and the mass spectra of lowest-lying beauty mesons B, B*, Bs, B s * and Bc are predicted in a simple analytically solvable (forl = 0) quark-confining two-step potential model used earlier to study charmonium and bottonium and even the light mesons like π, ρ, K, …, etc. Results are found to be in good agreement with the values predicted by others.
A general method is used to construct explicitly the quadratic invariants for two dimensional cla... more A general method is used to construct explicitly the quadratic invariants for two dimensional classical dynamical systems. The solutions of the “potential” equations are considered for both time dependent (TD) and time independent (TID) systems dealing mainly with the noncentral potentials. While several interesting integrable TID systems are found, which may have applications in solid state physics and molecular chemistry, an explicit construction of the invariants for a large class of TD systems is carried out which may again be useful in quantum optics and astronomy. In particular, the problem of noncentral TD harmonic oscillator in its varied form is dealt with in some details.
For the purposes 01 making a viable cosmological scenario, it is known that the universe must evo... more For the purposes 01 making a viable cosmological scenario, it is known that the universe must evolve into the radiation dominated Friedmann stage after enough inflation and sUbcessful reheating.
Previous two-step potential model calculations for the pion form factor in the space-like region ... more Previous two-step potential model calculations for the pion form factor in the space-like region are now extended to include the form factor of the K meson, in an anticipation that the experimentalists will shortly be able to measure it for q2 up to, say, 2.0 GeV 2 also, as Dally et al. have already measured it for q2 up to 0.2 GeV 2. Although the curve does not fit well the present experimental data (q2 up to 0.2 GeV2), it is, however, anticipated that the model will probably fit the experimental data fairly well for slightly higher values ofq 2 , say, up to 2.0 GeV 2 , when measured, as it does very successfully in the case of the pion over a wide range of momentum transfer q2.
In the spirit of quark–diquark model, used earlier for the nucleon [Ann. Phys. (N.Y.) 108 (1977) ... more In the spirit of quark–diquark model, used earlier for the nucleon [Ann. Phys. (N.Y.) 108 (1977) 198] we investigate here some properties of the newly discovered exotic pentaquark systems, viz., θ+(1540) and (say) ζ0(3099). For this purpose, in the first stage, we consider these multiquark states as three-body systems—two diquarks of (ud)-type and an antiquark with a Coulomb-type color force
An ansatz for the eigenfunction is used to obtain an exact closed form solution to the Schrödinge... more An ansatz for the eigenfunction is used to obtain an exact closed form solution to the Schrödinger wave equation, for the potential V(r) = ar2+br-4+cr-6. The results arrived at in this framework are discussed in the light of those obtained by Znojil using the continued fraction method for a similar potential.
Dynamical invariants for time-dependent (TD) oscillator systems in two dimensions are derived. Th... more Dynamical invariants for time-dependent (TD) oscillator systems in two dimensions are derived. The applications of the dynamical algebraic approach known already for one-dimensional TD systems [R. S. Kaushal and H. J. Korsch, J. Math. Phys. 22, 1904 (1981)] are now carried out to two-dimensional TD systems. A possible generalization of Ermakov systems to higher dimensions is discussed.
The quark contining two-step potential used earlier for mesons is now applied to nucleons. By usi... more The quark contining two-step potential used earlier for mesons is now applied to nucleons. By using the Coulomb part of this potential for the quark-quark pair we investigate the predictions of the quark-diquark (QDQ) model for the nucleons. It is found that this kind of composite model, consistent for both the proton and the neutron, not only accounts for the recent data on the nucleon form factors but also provides a deeper understanding of the already available results in such a model. Further, it is noticed that the present QDQ model, accounting satisfactorily for various properties of the nucleon, offers an alternative way to investigate the ground state of deuteron and the nature of nucleon-nucleon force in terms of the pre-Yukawa picture of chemical-type forces. The results obtained in this alternative approach are found to be in agreement with those obtained by other conventional calculations.
With a view to exploring a new kind of phase transition in the process of hadronization of quark-... more With a view to exploring a new kind of phase transition in the process of hadronization of quark-gluon plasma (QGP) we investigate the occurrence of pentaquark baryons and tetraquark mesons in the system. For this purpose, the frame work of an analoguous Saha's ionization formula for the colored ions in the system is used. The study of color-ionic-fraction (CIF) of multiply (color) ionized to unionized quark clusters (termed as "quarkons") as a function of temperature is carried out. It is pointed out that not only the temperature of the fire-ball in the relativistic heavy ion collisions evolves with respect to space and time but also the CIF associated with a particular stage of ionization. Further, for the case of single color-ionization a correspondence of the present results with those available for the bubble nucleation mechanism in QGP is demonstrated.
Using the auxiliary equation method, we obtain exact solutions of certain nonlinear chemotaxis di... more Using the auxiliary equation method, we obtain exact solutions of certain nonlinear chemotaxis diffusion reaction equations in the presence of a stimulant. In particular, we account for the nonlinearities arising not only from the density-dependent source terms contributed by the particles and the stimulant but also from the coupling term of the stimulant. In addition to this, the diffusion of the stimulant and the effect of long-range interactions are also accounted for in the constructed coupled differential equations. The results obtained here could be useful in the studies of several biological systems and processes, e.g., in bacterial infection, chemotherapy, etc.
The role played by the pendulum and its variants in the development of mathematical sciences is r... more The role played by the pendulum and its variants in the development of mathematical sciences is reviewed. In particular, the case of a pendulum viz. harmonic oscillator in physics is highlighted in the context of mathematical, conceptual, conventional and engineering disciplines besides the one in mathematical physics. It is pointed out that in all these applications the concept of pendulum is used as a vehicle of knowledge mainly on the basis of structural analogy. Finally, its role is an example of the consciousness-manifesting phenomenon, will be discussed.
We investigate the problem el finiteness and negative sign of the mean squxre charge radiu~ of tk... more We investigate the problem el finiteness and negative sign of the mean squxre charge radiu~ of tke ~eucron, (r~). In general, ~x~.ous q~ark^Oa~vtan va~d~l~ ~e ~tch ~rov~<~ z 9oai~(v~ Si~ f~v (r~). It iS Ium~d that th~ u~e of a spatial dist~ibutio~ ia the longitudinal direction ovt o~,ly reproduces the right sign ,t (v~, but also provides a somewhat csmparable magnitude for .(r~) m tl~e framework of dif~ ferent quark-patton models. We also ealaulatc the neutron charge form factor at thv values of the square momentam transfer different from zero, in these models. I.-Introductiou.
Recent data on few-body reactions have focussed attention on the desirability of treating the sin... more Recent data on few-body reactions have focussed attention on the desirability of treating the singlet deuteron (d*) on par with the triplet deuteron (d). This question is examined in the context of a serious ambiguity existing in the literature on the basic status of d*, viz. the conventional virtual-level point of view (equivalent to a (~ ghost in field theory language) vs. a less-known resonant interpretation (due to Nomoto). Curiously enough, the lattcr interpretation seems to prodnce better overlaps with experiment on the d-to-d* ratio of threebody vertices, as deduced from appropriate break-up reaction data. Moreover, the distinction appears to be qualitative enough to survive the full-fledged mathematical machinery of the few-body theory, which is avoided in the present analysis in favour of eiIcctiv(; three-body wave functions which fit the corresponding charge form factor as well as the Coulomb energy. 1.-Introduction and summary.
Classical and Quantum Mechanics of Noncentral Potentials, 1998
Departure for his Heavenly abode. Na sato vidyate bha vo nabhavo vidyate satah, Ubhayorapi drasto... more Departure for his Heavenly abode. Na sato vidyate bha vo nabhavo vidyate satah, Ubhayorapi drastoantastvanayostatva darsibhih. -Srimad Bhagvad-Gita (2/16) The unreal has no existence, and the real never ceases to be; the reality of both (real and unreal) has thus been perceived by the seers of Truth.
With a view to having further insight into the mathematical content of the non-Hermitian Hamilton... more With a view to having further insight into the mathematical content of the non-Hermitian Hamiltonian associaterd with the diffusion-reaction (D-R) equation in one dimension, we investigate (a) the solitary wave solutions of certain types of its nonlinear versions, and (b) the problem of real eigenvalue spectrum associated with its linear version or with this class of non-Hermitian Hamiltonians. For the case (a) we use the standard techniques to handle the quadratic and cubic nonlinearities in the D-R equation whereas for the case (b) a newly proposed method, based on an extended complex phase space, is employed. For a particular class of solutions, an Ermakov system of equations is also found for the linear case. Further, corresponding to the 'classical' version of the above one-dimensional complex Hamiltonian, an equivalent integrable system of two, two-dimensional real Hamiltonians is suggested.
The construction of the second constant of motion of second order for two-dimensional classical s... more The construction of the second constant of motion of second order for two-dimensional classical systems is carried out in terms ofz=q 1 +iq 2 andq=q 1 −iq 2. As a result a class of Toda-type potentials admitting second order invariants is explored.
For a time-dependent system characterized by the potential V(x,t)ϭa 2 (t)x 2 ϩa 1 /x 2 , we first... more For a time-dependent system characterized by the potential V(x,t)ϭa 2 (t)x 2 ϩa 1 /x 2 , we first demonstrate the construction of classical and quantum invariants and then obtain an exact solution of the Schrödinger equation for this potential. The role of the invariant so constructed is further discussed in the context of studies of the coherent states. ͓S1050-2947͑97͒02704-2͔
Summary Ground-state masses and the mass spectra of lowest-lying beauty mesons B, B*, Bs, B s * ... more Summary Ground-state masses and the mass spectra of lowest-lying beauty mesons B, B*, Bs, B s * and Bc are predicted in a simple analytically solvable (forl = 0) quark-confining two-step potential model used earlier to study charmonium and bottonium and even the light mesons like π, ρ, K, …, etc. Results are found to be in good agreement with the values predicted by others.
A general method is used to construct explicitly the quadratic invariants for two dimensional cla... more A general method is used to construct explicitly the quadratic invariants for two dimensional classical dynamical systems. The solutions of the “potential” equations are considered for both time dependent (TD) and time independent (TID) systems dealing mainly with the noncentral potentials. While several interesting integrable TID systems are found, which may have applications in solid state physics and molecular chemistry, an explicit construction of the invariants for a large class of TD systems is carried out which may again be useful in quantum optics and astronomy. In particular, the problem of noncentral TD harmonic oscillator in its varied form is dealt with in some details.
For the purposes 01 making a viable cosmological scenario, it is known that the universe must evo... more For the purposes 01 making a viable cosmological scenario, it is known that the universe must evolve into the radiation dominated Friedmann stage after enough inflation and sUbcessful reheating.
Previous two-step potential model calculations for the pion form factor in the space-like region ... more Previous two-step potential model calculations for the pion form factor in the space-like region are now extended to include the form factor of the K meson, in an anticipation that the experimentalists will shortly be able to measure it for q2 up to, say, 2.0 GeV 2 also, as Dally et al. have already measured it for q2 up to 0.2 GeV 2. Although the curve does not fit well the present experimental data (q2 up to 0.2 GeV2), it is, however, anticipated that the model will probably fit the experimental data fairly well for slightly higher values ofq 2 , say, up to 2.0 GeV 2 , when measured, as it does very successfully in the case of the pion over a wide range of momentum transfer q2.
In the spirit of quark–diquark model, used earlier for the nucleon [Ann. Phys. (N.Y.) 108 (1977) ... more In the spirit of quark–diquark model, used earlier for the nucleon [Ann. Phys. (N.Y.) 108 (1977) 198] we investigate here some properties of the newly discovered exotic pentaquark systems, viz., θ+(1540) and (say) ζ0(3099). For this purpose, in the first stage, we consider these multiquark states as three-body systems—two diquarks of (ud)-type and an antiquark with a Coulomb-type color force
An ansatz for the eigenfunction is used to obtain an exact closed form solution to the Schrödinge... more An ansatz for the eigenfunction is used to obtain an exact closed form solution to the Schrödinger wave equation, for the potential V(r) = ar2+br-4+cr-6. The results arrived at in this framework are discussed in the light of those obtained by Znojil using the continued fraction method for a similar potential.
Dynamical invariants for time-dependent (TD) oscillator systems in two dimensions are derived. Th... more Dynamical invariants for time-dependent (TD) oscillator systems in two dimensions are derived. The applications of the dynamical algebraic approach known already for one-dimensional TD systems [R. S. Kaushal and H. J. Korsch, J. Math. Phys. 22, 1904 (1981)] are now carried out to two-dimensional TD systems. A possible generalization of Ermakov systems to higher dimensions is discussed.
The quark contining two-step potential used earlier for mesons is now applied to nucleons. By usi... more The quark contining two-step potential used earlier for mesons is now applied to nucleons. By using the Coulomb part of this potential for the quark-quark pair we investigate the predictions of the quark-diquark (QDQ) model for the nucleons. It is found that this kind of composite model, consistent for both the proton and the neutron, not only accounts for the recent data on the nucleon form factors but also provides a deeper understanding of the already available results in such a model. Further, it is noticed that the present QDQ model, accounting satisfactorily for various properties of the nucleon, offers an alternative way to investigate the ground state of deuteron and the nature of nucleon-nucleon force in terms of the pre-Yukawa picture of chemical-type forces. The results obtained in this alternative approach are found to be in agreement with those obtained by other conventional calculations.
With a view to exploring a new kind of phase transition in the process of hadronization of quark-... more With a view to exploring a new kind of phase transition in the process of hadronization of quark-gluon plasma (QGP) we investigate the occurrence of pentaquark baryons and tetraquark mesons in the system. For this purpose, the frame work of an analoguous Saha's ionization formula for the colored ions in the system is used. The study of color-ionic-fraction (CIF) of multiply (color) ionized to unionized quark clusters (termed as "quarkons") as a function of temperature is carried out. It is pointed out that not only the temperature of the fire-ball in the relativistic heavy ion collisions evolves with respect to space and time but also the CIF associated with a particular stage of ionization. Further, for the case of single color-ionization a correspondence of the present results with those available for the bubble nucleation mechanism in QGP is demonstrated.
Using the auxiliary equation method, we obtain exact solutions of certain nonlinear chemotaxis di... more Using the auxiliary equation method, we obtain exact solutions of certain nonlinear chemotaxis diffusion reaction equations in the presence of a stimulant. In particular, we account for the nonlinearities arising not only from the density-dependent source terms contributed by the particles and the stimulant but also from the coupling term of the stimulant. In addition to this, the diffusion of the stimulant and the effect of long-range interactions are also accounted for in the constructed coupled differential equations. The results obtained here could be useful in the studies of several biological systems and processes, e.g., in bacterial infection, chemotherapy, etc.
The role played by the pendulum and its variants in the development of mathematical sciences is r... more The role played by the pendulum and its variants in the development of mathematical sciences is reviewed. In particular, the case of a pendulum viz. harmonic oscillator in physics is highlighted in the context of mathematical, conceptual, conventional and engineering disciplines besides the one in mathematical physics. It is pointed out that in all these applications the concept of pendulum is used as a vehicle of knowledge mainly on the basis of structural analogy. Finally, its role is an example of the consciousness-manifesting phenomenon, will be discussed.
We investigate the problem el finiteness and negative sign of the mean squxre charge radiu~ of tk... more We investigate the problem el finiteness and negative sign of the mean squxre charge radiu~ of tke ~eucron, (r~). In general, ~x~.ous q~ark^Oa~vtan va~d~l~ ~e ~tch ~rov~<~ z 9oai~(v~ Si~ f~v (r~). It iS Ium~d that th~ u~e of a spatial dist~ibutio~ ia the longitudinal direction ovt o~,ly reproduces the right sign ,t (v~, but also provides a somewhat csmparable magnitude for .(r~) m tl~e framework of dif~ ferent quark-patton models. We also ealaulatc the neutron charge form factor at thv values of the square momentam transfer different from zero, in these models. I.-Introductiou.
Recent data on few-body reactions have focussed attention on the desirability of treating the sin... more Recent data on few-body reactions have focussed attention on the desirability of treating the singlet deuteron (d*) on par with the triplet deuteron (d). This question is examined in the context of a serious ambiguity existing in the literature on the basic status of d*, viz. the conventional virtual-level point of view (equivalent to a (~ ghost in field theory language) vs. a less-known resonant interpretation (due to Nomoto). Curiously enough, the lattcr interpretation seems to prodnce better overlaps with experiment on the d-to-d* ratio of threebody vertices, as deduced from appropriate break-up reaction data. Moreover, the distinction appears to be qualitative enough to survive the full-fledged mathematical machinery of the few-body theory, which is avoided in the present analysis in favour of eiIcctiv(; three-body wave functions which fit the corresponding charge form factor as well as the Coulomb energy. 1.-Introduction and summary.
Uploads
Papers by R.S. Kaushal