Calculation of energy eigenvalues via supersymmetric quantum mechanics

1989 ◽  
Vol 67 (10) ◽  
pp. 931-934 ◽  
Author(s):  
Francisco M. Fernández ◽  
Q. Ma ◽  
D. J. DeSmet ◽  
R. H. Tipping
Keyword(s):  
Quantum Mechanics ◽  
Ground State ◽  
Potential Energy Function ◽  
Supersymmetric Quantum Mechanics ◽  
Energy Eigenvalues ◽  
Arbitrary Power ◽  
Rational Function Approximation ◽  
Ricatti Equation ◽  
Original Potential ◽  
Ground State Energies

A systematic procedure using supersymmetric quantum mechanics is presented for calculating the energy eigenvalues of the Schrödinger equation. Starting from the Hamiltonian for a given potential-energy function, a sequence of supersymmetric partners is derived such that the ground-state energy of the kth one corresponds to the kth eigen energy of the original potential. Various theoretical procedures for obtaining ground-state energies, including a method involving a rational-function approximation for the solution of the Ricatti equation that is outlined in the present paper, can then be applied. Illustrative numerical results for two one-dimensional parity-invariant model potentials are given, and the results of the present procedure are compared with those obtainable via other methods. Generalizations of the method for arbitrary power-law potentials and for radial problems are discussed briefly.

Quantum Mechanics

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Quantum Mechanics ◽  
Ground State ◽  
Variational Principles ◽  
Free Particle ◽  
Uncertainty Relations ◽  
Probabilistic Interpretation ◽  
Historical Account ◽  
Potential Wells ◽  
Harmonic Oscillator Potential ◽  
Ground State Energies

In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.

SUPERSYMMETRIC QUANTUM MECHANICS APPLIED TO NONRELATIVISTIC QUARK MODELS

1993 ◽  
Vol 08 (08) ◽  
pp. 1437-1455 ◽  
Author(s):  
E.J.O. GAVIN ◽  
H. FIEDELDEY ◽  
H. LEEB ◽  
S.A. SOFIANOS
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Approximation Scheme ◽  
Energy Levels ◽  
Supersymmetric Quantum Mechanics ◽  
S Wave ◽  
Nucleon Spectrum ◽  
Quark Potential ◽  
Quark Models ◽  
Original Potential

We examine the effect of changing the energy levels and normalization constants of bound states corresponding to baryons and mesons in nonrelativistic quark models. We do this by applying the transformations of supersymmetric quantum mechanics (SUSYQM) to the potentials used in these models. In particular, we fit the spectra and leptonic decay widths of [Formula: see text] and [Formula: see text] mesons by modifying several existing [Formula: see text] potentials by means of supersymmetric transformations. It is found that the potentials are unchanged beyond 2 fm, and that fitting the widths induces greater oscillations in the potentials than those generated by adjusting the energy levels only. Transformations of SUSYQM are applied to the hypercentral potential in order to accommodate the Roper resonance in the s-wave nucleon spectrum. The quark-quark potential found by inverting the transformed hypercentral potential via a new exact Abel transform differs significantly from the original potential up to 5 fm from the origin and violates the concavity requirement. The [Formula: see text] potential related to this potential by Lipkin’s rule does not reproduce the meson spectrum. As the Hall-Post lower bound is also accurate for baryons, the results of the application of supersymmetric transformations in this approximation scheme are also considered and compared to the upper bound of the hypercentral approximation.

scholarly journals PARABOSE-PARAFERMI SUPERSYMMETRY

1996 ◽  
Vol 11 (16) ◽  
pp. 2957-2975 ◽  
Author(s):  
ALI MOSTAFAZADEH
Keyword(s):  
Quantum Mechanics ◽  
Quantum System ◽  
Algebraic Structure ◽  
Central Charge ◽  
Supersymmetric Quantum Mechanics ◽  
Eigenvalues And Eigenvectors ◽  
Energy Eigenvalues ◽  
General Terms ◽  
Symmetry Transformations ◽  
Definition Of

The (p=2) parabose–parafermi supersymmetry is studied in general terms. It is shown that the algebraic structure of the (p=2) parastatistical dynamical variables allows for (symmetry) transformations which mix the parabose and parafermi coordinate variables. The example of a simple parabose-parafermi oscillator is discussed and its symmetries investigated. It turns out that this oscillator possesses two parabose-parafermi supersymmetries. The combined set of generators of the symmetries forms the algebra of supersymmetric quantum mechanics supplemented with an additional central charge. In this sense there is no relation between the parabose–parafermi supersymmetry and the parasupersymmetric quantum mechanics. A precise definition of a quantum system involving this type of parabose-parafermi supersymmetry is offered, thus introducing (p=2) supersymmetric paraquantum mechanics. The spectrum degeneracy structure of general (p=2) supersymmetric paraquantum mechanics is analyzed in detail. The energy eigenvalues and eigenvectors for the parabose–parafermi oscillator are then obtained explicitly. The latter confirms the validity of the results obtained for general supersymmetric paraquantum mechanics.

CONFINED LENNARD–JONES POTENTIAL: A VARIATIONAL TREATMENT

2010 ◽  
Vol 25 (08) ◽  
pp. 641-648 ◽  
Author(s):  
F. R. SILVA ◽  
E. DRIGO FILHO
Keyword(s):  
Quantum Mechanics ◽  
Variational Method ◽  
Energy Levels ◽  
Numerical Results ◽  
Supersymmetric Quantum Mechanics ◽  
Lennard Jones Potential ◽  
Jones Potential ◽  
Energy Eigenvalues ◽  
Lennard Jones ◽  
Variational Treatment

In this work, the energy eigenvalues for the confined Lennard–Jones potential are calculated through the Variational Method allied to the Supersymmetric Quantum Mechanics. Numerical results are obtained for different energy levels, parameters of the potential and values of confinement radius. In the limit, where this radius assumes great values, the results for the non-confined case are recovered.

scholarly journals INDUCED VARIATIONAL METHOD FROM SUPERSYMMETRIC QUANTUM MECHANICS AND THE SCREENED COULOMB POTENTIAL

2000 ◽  
Vol 15 (19) ◽  
pp. 1253-1259 ◽  
Author(s):  
ELSO DRIGO FILHO ◽  
REGINA MARIA RICOTTA
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Variational Method ◽  
Coulomb Potential ◽  
Trial Wave Function ◽  
Supersymmetric Quantum Mechanics ◽  
Exact Results ◽  
Energy Eigenvalues ◽  
Screened Coulomb Potential

The formalism of supersymmetric quantum mechanics supplies a trial wave function to be used in the variational method. The screened Coulomb potential is analyzed within this approach. Numerical and exact results for energy eigenvalues are compared.

Sign in / Sign up

Export Citation Format

Share Document