Supersymmetric Quantum Mechanics, Excited State Energies and Wave Functions, and the Rayleigh−Ritz Variational Principle: A Proof of Principle Study

Abstract

In addition to ground state wave functions and energies, excited states and their energies are also obtained in a standard Rayleigh-Ritz variational calculation. However, their accuracy is generally much lower. Using the super-symmetric (SUSY) form of quantum mechanics, we show that better accuracy and more rapid convergence can be obtained by taking advantage of calculations of the ground states of higher sector SUSY Hamiltonians, followed by application of the SUSY "charge operators". Our proof of principle study uses a general family of one-dimensional anharmonic oscillator models. We first obtain the exact, analytic ground states for a general family of anharmonic systems. We give the general, factorized form of the Hamiltonian for the hierarchy that arises in SUSY theory. The "charge" operators can then be used to convert states among the sectors. We illustrate the approach with two specific anharmonic oscillator models. Using the ground state of the second sector Hamiltonian, we show that the corresponding excited state energies and wave functions of the first sector are accurately obtained by applying the charge operators, using significantly smaller basis sets than are required in a standard variational approach applied to the original Schrodinger equation. This is a consequence of the higher accuracy of the Rayleigh-Ritz variational method when applied for ground states.

Near‐exact supersymmetric partner potentials: Construction and exploitation

“…On this point, we differ significantly from other works (e.g., Ref. [22]). Consider now the role of parity [9].…”

“…The observations are specifically noteworthy because the latter entails a first order error. Our error analysis is particularly relevant to studies [22] involving a hierarchy of SUSY Hamiltonians that are constructed with a view to allowing one to employ a smaller basis set for calculations of excited-state energies. We have also indicated a way of simplifying the forms of the partner potentials.…”

New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations

“…Developments and applications of one-dimensional SUSY-QM can be found in relevant reviews and books ( [7,9,11,15,26,32,33]). Recently, SUSY-QM has been developed as a computational tool to provide much more accurate excitation energies using the standard Rayleigh-Ritz variational method ( [5,19,20]). …”

Generalized Non-Relativistic Supersymmetric Quantum Mechanics

“…We 5 www.intechopen.com therefore asked whether this could be the case for SUSY. It turns out that SUSY does, in fact, lead to significant computational advantages Kouri et al (2010a); Kouri, Markovich, Maxwell & Bittner (2009) ;Kouri, Markovich, Maxwell & Bodman (2009). In particular, the structure of the degeneracies between sector Hamiltonians makes it possible to achieve significant progress in more accurate calculations of excited state energies and wave functions.…”