If two potentials V 1 (x) and V 2 (x) are ordered V 1 <V 2 , then the comparison theorem tells the author that the corresponding Schrodinger eigenvalues are ordered E 1 <E 2 . He presents some simple conditions which allow him to predict such spectral ordering for the ground state, even when the graphs of the potentials cross over. As illustrations, the truncated quartic oscillator and the Yukawa potential are studied. By allowing Coulomb 'tangents' to cross over the Yukawa in various ways, he is able to improve earlier energy upper bounds which he had obtained by representing the Yukawa potential as a smooth transformation of the Coulomb potential, and also to augment these results with lower bounds.
If two potentials V 1 (x) and V 2 (x) are ordered V 1 <V 2 , then the comparison theorem tells the author that the corresponding Schrodinger eigenvalues are ordered E 1 <E 2 . He presents some simple conditions which allow him to predict such spectral ordering for the ground state, even when the graphs of the potentials cross over. As illustrations, the truncated quartic oscillator and the Yukawa potential are studied. By allowing Coulomb 'tangents' to cross over the Yukawa in various ways, he is able to improve earlier energy upper bounds which he had obtained by representing the Yukawa potential as a smooth transformation of the Coulomb potential, and also to augment these results with lower bounds.
We derive eigenstate comparisons from ordered Schrödinger operators, extending energy bounds to overlaps, spatial distributions and nodes. For three form-ordered operators with discrete comparison ground states, suppose the upper ground-state energy lies in the lower operator's ground-state gap. Three comparison spectral values then bound all three squared ground-state overlaps and guarantee a simple discrete intermediate ground state. Related gap estimates control low-energy subspaces and individual excited-state overlaps under additional gap conditions. In one dimension and radially, a single crossing of the potential difference relative to the energy shift orders ground-state likelihood ratios and monotone observables; a nondecreasing difference suffices. The results cover asymmetric wells and shortened right Dirichlet walls, with coordinatewise comparisons for submodular reference potentials and coordinatewise nondecreasing perturbations. A two-sided Sturm argument orders corresponding internal nodes for nondecreasing potential differences, even between orthogonal one-dimensional excited states. For Yukawa ground states, Hulthén comparisons bound overlaps, radial moments and the density at the nucleus; the latter two-sided bounds hold for screening 0<δ<2/6. Optimizing scaled Hulthén majorants gives a lower bound of approximately 1.13611 for n2δc(n,0), where δc(n,0) is the critical screening for the nth s-wave bound state. This bound is optimal within that two-parameter comparison family.
Abstract: In this study, the Schrödinger equation was solved with a superposition of the Hulthen potential and generalized inverse quadratic Yukawa potential model using the Nikiforov-Uvarov (NU) method. For completeness, we also calculated the wave function. To validate our results, the numerical bound state energy eigenvalues was computed for various principal n and angular momentum l quantum numbers. With the aid of the Hellmann-Feynman theorem, the expressions for the expectation values of the square of inverse of position, r^(-2), inverse of position, r^(-1), kinetic energy, T ̂ and square of momentum, p ̂ are calculated. By adjusting the potential parameters, special cases of the potential were considered, resulting in Generalized Inverse Quadratic Yukawa potential, Hulthen potential, Coulomb potential, Kratzer potential, Inversely Quadratic Yukawa potential and Coulomb plus inverse square potential, respectively. Their energy eigenvalue expressions and numerical computations agreed with the literature. Keywords: Schrödinger equation, Hulthen potential (HP), Generalized inverse quadratic Yukawa potential (GIQYP), Nikiforov-Uvarov method. PACS: 03.65.−w, 03.65.Ca, 03.65.Ge.
An approach related to the teaching of the chapter on bound and scattering states in a short-range potential, which forms a standard part of quantum-mechanics course, is outlined with an example. The proposed interaction closely approximates the realistic Yukawa potential. It also represents the Coulomb potential in a limiting case. Investigation of the Schrödinger equation with the potential may serve as an excellent demonstration to be incorporated in the undergraduate quantum-mechanics or nuclear physics course.
The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.
We study the ground state energy and the critical screening parameter of the Yukawa potential in nonrelativistic quantum mechanics. After a short review of the existing literature on these quantities, we apply fifth-order perturbation theory to the calculation of the ground state energy, using the exact solutions of the Coulomb potential together with a cutoff on the principal number summations. We also perform a variational calculation of the ground state energy using a Coulomb-like radial wave function and the exact solution of the corresponding minimization condition. For not too large values of the screening parameter, close agreement is found between the perturbative and variational results. For the critical screening parameter, we devise a novel method that permits us to determine it to 10 digits. This is the most precise calculation of this quantity to date, and allows us to resolve some discrepancies between previous results.
Splitting methods are among the most classical and fundamental tools for the simulation of quantum dynamics, and their importance has grown further with the rise of quantum computing. In this work, we analyze the Schrödinger equation with Yukawa potential, a physically relevant and widely used model potential. It may be viewed as a Coulomb interaction with exponential decay at spatial infinity, preserving the Coulomb singularity at the origin while removing the long-range Coulomb tail. We prove that the operator splitting for this unbounded Hamiltonian achieves a global 1/4-order convergence rate in the time step for many-body Yukawa interactions, with explicit polynomial dependence on the number of particles. The result holds for all initial wavefunctions in H2(R3N), the natural domain of the Hamiltonian, and our numerical experiments are consistent with the theoretical estimates. To identify the sharp obstruction behind this rate, we prove a short-time lower bound in the one-body setting of order t5/4 for the one-step error, which rules out any uniform global estimate of order better than 1/4 in general. This agreement with the optimal 1/4 rate in the Coulomb case is particularly interesting, as Yukawa potential is short-ranged compared to Coulomb potential. For the many-body upper bound, one of the new technical ingredients is the explicit polynomial-in-system-size Sobolev estimates of many-body Yukawa systems. These estimates are crucial for obtaining fully a priori bounds that depend only on the norms of the initial states, rather than on the solution at time t. For the one-body lower bound, we leverage a new analysis argument based on Fourier analysis and Kato smoothing.