Behavior of the Dirichlet boundary for wave functions in a class of singular potentials

Quantum mechanical variation principle in the form of energy minimization is applicable only to ground states of systems, or, at best, states of lowest energies of given symmetries, provided the symmetry information is embedded in chosen trial functions. Thus, for bound quantum states with specified choices of trial functions involving nonlinear parameters, scope of the principle is severely restricted. A pedagogic way out is to enforce exact orthogonality of the chosen function with all exact lower energy states. In actual practice, this limits one to opt for linear variations where upper bound to each state is obtained in a single run. In this work, the motivation is to explore if there exists at all a way to determine optimized wave functions and energies for excited states via nonlinear variations but without any constraints, even for simple systems. Realizing that the major problem in excited-state nonlinear variations is concerned with the variations of nodal positions, at least for problems reducible to one dimension, we seek a route via which nodes could be fixed beforehand, so that the information gained may be subsequently utilized to construct a suitable nonlinear trial function and carry out a straightforward optimization. To achieve this end, the idea of supersymmetric quantum mechanics has been used quite profitably, yielding the nodal structure of the excited states. Workability of the strategy for several excited-state wave functions and their properties is demonstrated by choosing the problems of spherical Stark effect on hydrogen atom and anharmonic oscillator. V C 2011 Wiley Periodicals, Inc. Int J Quantum Chem 112: [960][961][962][963][964][965][966][967][968][969][970][971] 2012

Section: Discussionmentioning

confidence: 99%

Rayleigh–ritz method for excited quantum states via nonlinear variations without constraints: Role of supersymmetry

Mukherjee

Bhattacharyya

2011

“…The singular or spiked oscillator is defined by the family of quantum Hamiltonians where α and λ take positive values only and the domain of x is [0,∞). This problem has received enough attention since the middle 70s 1–42 because of its applications in physical chemistry and nuclear physics 1, 3, 38–42 and mainly for its interesting mathematical behavior. As Aguilera‐Navarro and Guardiola pointed out 9, none of the two terms on the potential dominate for the extreme values of λ.…”

Section: Introductionmentioning

confidence: 99%

“…Detwiller and Klauder 3 pointed out that the conventional perturbation theory could not be applied for α ≥ 5/2, but they were able to advance the form that the ground state energy should have for small λ. Now, it is known that for α > 2, the perturbative series for the ground state has not a usual Taylor expansion but some terms of the form λ n ln λ (with n an integer) appear in its development 36, 37. Furthermore, the exponents in the power series become fractional for α > 3 4, 31.…”

Section: Introductionmentioning

confidence: 99%

“…The number of works on this topic has been steadily increasing 1–42. A variety of methods have been used to solve this problem, and few of them are as follows: perturbation theory, the linear variational method, Rayleigh‐Ritz variational method, and numerical approaches.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Bandypadhyay and Bhattacharyya 37 recognized the advantage of solving the confined problem, i.e., the problem subject to Dirichlet boundary conditions. In this work, we propose a numerical method to find the eigenvalues and the eigenfunctions of the spiked oscillator with very high precision, considering the system inside a box of infinite walls.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Accurate solutions for the spiked oscillators

Campoy¹,

Aquino²

2008

ABSTRACT:The one-dimensional harmonic oscillator potential plus a term of the form /x ␣ is known as the spiked oscillator potential; it constitutes a very interesting system because of its difficulties to accept perturbative and variational solutions for certain regimes of the ␣ parameter. By the use of a numerical method, we obtain accurate energy eigenvalues and eigenfunctions for a wide range of values and a few ␣ values. The accuracy of the present results is by much higher than previously published results.

Surprises in nonlinear perturbations: Case of a multiple well potential problem

Dhatt

Bhattacharyya

2011

ABSTRACT:We study an atypical nonlinear perturbation problem on the lowest three states of a harmonic oscillator. The perturbation is such that the total potential is a single-well at low values of coupling strength, turns to a double-well in the intermediate coupling range and finally becomes a triple-well at high values of the coupling strength. Proceeding via logarithmic perturbation theory, we notice that the perturbation furnishes exact results for both the energy and wave function of the second excited state but shows divergence for the first excited one. However, the ground state wave function is found to be exactly summable to the correct result with a finite energy series. Effects of varying the coupling constant, and hence the number of wells, on energies and wave functions of the concerned states are noted. Our results show further that the perturbed ground state acquires a pre-exponential factor and, strangely, an exponent same as the second excited state. The origin of such an outcome is analyzed. We also highlight the reason behind the Klauder phenomenon that tells of increased sustenance of memory effects in wave functions, when compared with the Hamiltonian, in the limit of zero coupling strength.