Amlan K. Roy

The generalized pseudospectral approach to the bound states of the Hulthén and the Yukawa potentials

The generalized pseudospectral (GPS) method is employed to calculate the bound states of the Hulthén and the Yukawa potentials in quantum mechanics, with special emphasis onhigher excited states andstronger couplings. Accurate energy eigenvalues, expectation values and radial probability densities are obtained through a non-uniform and optimal spatial discretization of the radial Schrödinger equation. Results accurate up to thirteen to fourteen significant figures are reported for all the 55 eigenstates of both these potentials withn <- 10 for arbitrary values of the screening parameters covering a wide range of interaction. Furthermore, excited states as high asn = 17 have been computed with good accuracy for both these potentials. Excellent agreement with the available literature data has been observed in all cases. Then > 6 states of the Yukawa potential has been considerably improved over all other existing results currently available, while the same for Hulthén potential are reported here for the first time. Excepting the 1s and 2s states of the Yukawa potential, the present method surpasses the accuracy of all other existing results in the stronger coupling region for all other states of both these systems. This offers a simple and efficient scheme for the accurate calculation of these and other screened Coulomb potentials.

Calculation of the spiked harmonic oscillators through a generalized pseudospectral method

The generalized pseudospectral method is employed for the accurate calculation of eigenvalues, densities and expectation values for the spiked harmonic oscillators. This allows nonuniform and optimal spatial discretization of the corresponding single-particle radial Schrodinger equation satisfying the Dirichlet boundary conditions leading to the standard diagonalization of the symmetric matrices. The present results for a large range of potential parameters are in excellent agreement with those from the other accurate methods available in the literature. The ground and excited states (both low as well as high angular momentum states) are obtained with equal ease and accuracy. Some new states including the higher excited states are reported here for the first time. This offers a simple, accurate and efficient method for the treatment of these and a wide variety of other singular potentials of physical and chemical interest in quantum mechanics.

Studies on the 3D confined potentials using generalized pseudospectral approach

The generalized pseudospectral Legendre method is used to carry out accurate calculations of eigenvalues of the spherically confined isotropic harmonic oscillator with impenetrable boundaries. The energy of the confined state is found to be equal to that of the unconfined state when the radius of confinement is suitably chosen as the location of the radial nodes in the unconfined state. This incidental degeneracy condition is numerically shown to be valid in general. Further, the full set of pairs of confined states defined by the quantum numbers [(n+1, \ell) ; (n, \ell+2)], n = 1,2,.., and with the radius of confinement {(2 \ell +3)/2}^{1/2} a.u., which represents the single node in the unconfined (1, \ell) state, is found to display a constant energy level separation exactly given by twice the oscillator frequency. The results of similar numerical studies on the confined Davidson oscillator with impenetrable boundary as well as the confined isotropic harmonic oscillator with finite potential barrier are also reported .The significance of the numerical results are discussed.

Studies on some singular potentials in quantum mechanics

A simple methodology is suggested for the efficient calculation of certain central potentials having singularities. The generalized pseudospectral method used in this work facilitates nonuniform and optimal spatial discretization. Applications have been made to calculate the energies, densities, and expectation values for two singular potentials of physical interest, viz., (i) the harmonic potential plus inverse quartic and sextic perturbation and (ii) the Coulomb potential with a linear and quadratic term for a broad range of parameters. The first 10 states belonging to a maximum of = 8 and 5 for (i) and (ii) have been computed with good accuracy and compared with the most accurate available literature data. The calculated results are in excellent agreement, especially in light of the difficulties encountered in these potentials. Some new states are reported here for the first time. This offers a general and efficient scheme for calculating these and other similar potentials of physical and mathematical interest in quantum mechanics accurately.

Calculation of the bound states of power-law and logarithmic potentials through a generalized pseudospectral method

Bound states of the power-law and logarithmic potentials are calculated using a generalized pseudospectral method. The solution of the single-particle Schrodinger equation in a nonuniform and optimal spatial discretization offers accurate eigenvalues, densities and expectation values. The calculations are carried out for states with arbitrary n and l quantum numbers. Comparisons are made with the available literature data and excellent agreement is observed. In all the cases, the present method yields considerably improved results over the other existing calculations. Some new states are reported.

Direct calculation of ground-state electronic densities and properties of noble gas atoms through a single time-dependent hydrodynamical equation

Ground-state electronic densities and properties of noble gas atoms (He, Ne, Ar, Kr and Xe) have been calculated through a single time-dependent quantum fluid dynamical equation of motion. The equation has been transformed through imaginary time into a diffusion equation which is then numerically solved in order to reach a global minimum. The present results compare favourably with other available values.

Clusters of glycolic acid with three to six water molecules

Semiempirical, ab initio, and density functional theory calculations are used to locate many low-energy minima on the potential energy surfaces of the CH2OHCOOH-(H2O)n complexes with n = 3,4,5,6. In the clusters with three, four, and five water molecules, the lowest-energy structure consists of a (H2O)n complex, not necessarily of lowest energy, hydrogen bonded to the carboxylic group of the glycolic acid. The lowest-energy structure for n = 6 is similar except that the water hexamer is hydrogen bonded to both the carboxylic and alpha-hydroxyl groups of the acid. In all the lowest-energy clusters, the intramolecular hydrogen bond remains intact in the glycolic acid.

Bound state spectra of the 3D rational potential

We present bound state spectra of the 3D rational potential, V(r) = r2 + λr2/(1 + gr2), g > 0, by means of the generalized pseudospectral method. All the 30 states corresponding to n = 0–9 are considered for the first time for a broad range of coupling parameters. These results surpass the accuracy of all other existing calculations published so far except the finite-difference method, which yields similar accuracy as ours. Variation of energies and radial distribution functions is followed with respect to the interaction parameters. Special emphasis has been laid on higher excitations and negative values of the interaction, where relatively less work has been reported. The energy sequence is found to be different for positive and negative interaction; numerically following a mirror-image relationship usually, if not always. Additionally, 20 energy splittings arising from certain levels belonging to n = 0–9 are systematically studied as functions of the potential parameters. Several new states (including the higher ones) are presented.

Density-functional calculations for doubly excited states of He, and

More than 200 low-lying and moderately high-lying doubly excited states of He-isoelectronic systems (Z = 2 - 5) have been studied in the nonrelativistic Hohenberg - Kohn - Sham density-functional framework by incorporating the nonvariational work-function exchange potential suggested by Harbola and Sahni and a simple parametrized local Wigner-type correlation functional. In essence, a Kohn - Sham-type equation has been solved numerically. Our results on excited-state energies, excitation energies and differences between excitation energies are in good agreement with available experimental and theoretical results. In 50% of cases, the percentage deviation in the calculated excitation energy is less than unity. The estimates of the relatively small distances (0.02 - 0.1 au) of the He states below the corresponding N = 2, 3, 4 ionization thresholds are in error by 0.5 - 34.9%, indicating that the Wigner correlation functional needs to be improved for greater accuracy. The excited radial densities display expected structures. Some new states are reported.

Information entropy as a measure of tunneling and quantum confinement in a symmetric double-well potential

Information entropic measures such as Fisher information, Shannon entropy, Onicescu energy and Onicescu Shannon entropy of a symmetric double-well potential are calculated in both position and momentum space. Eigenvalues and eigenvectors of this system are obtained through a variation-induced exact diagonalization procedure. The information entropy-based uncertainty relation is shown to be a better measure than conventional uncertainty product in interpreting purely quantum mechanical phenomena, such as, tunneling and quantum confinement in this case. Additionally, the phase-space description provides a semiclassical explanation for this feature. Total information entropy and phase-space area show similar behavior with increasing barrier height.