Exploring sequential quantum adiabatic switching across supersymmetric partners for finding the eigenstates of a system

Abstract

We demonstrate that one can exhaustively determine the n‐bound eigenstates of a Hamiltonian H by constructing a sequence of supersymmetric (SUSY) partner Hamiltonians and invoking a time‐dependent quantum adiabatic switching algorithm for passage from the ground state of one to the other. The ground states of the initial pair H(0) and H(1) are constructed by solving the Riccati equation for the superpotential ϕ(0) for H(0) and adiabatically switching from the ground state Ψ 0(0) of H(0) to the ground state Ψ 0(1) of H(1). The charge operator Q 0+ is then used to recover the first excited state Ψ 1(0) of H(0). The procedure is repeated for the ground states of SUSY pairs H(n + 1) and H(n + 2), and appropriate charge operators lead to the excited states Ψ italicn + 2(0) of H(0) with $n = 0,1,2, \cdots.$, thereby exhausting the full eigenspectrum of H(0). The workability of the proposed method is shown with several well‐known examples. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Adiabatic switching approach to multidimensional supersymmetric quantum mechanics for several excited states

“…Thus, the timedependent Hamiltonian is given by in Equation (14) with the switching function S(t) ¼ t 2 / 2 . This is the same switching function that has been used for onedimensional systems in [15].…”

“…As time progresses, the energy evolves from the ground state energy E ¼ 0 of the sector one Hamiltonian to the ground state energy E ¼ 4.58467 of the sector two Hamiltonian. In particular, the total number of time steps is 10,000, and we note that this number of time steps is orders of magnitude smaller than that used by Kar and Bhattacharyya [15]. Table 1 [8], where the basis set is composed of the direct product of the eigenstates of a harmonic oscillator in each direction and N u 1 and N u 2 are the numbers of basis functions for the variables u 1 and u 2 .…”

“…In a recent study [15], Kar and Bhattacharyya showed that bound states of a one-dimensional Hamiltonian can be obtained through quantum adiabatic switching between pairs of SUSY partner Hamiltonians. In this process, we start with the ground state of the sector one Hamiltonian H 1 .…”

“…The purpose of the current study is to extend the quantum adiabatic approach of Kar and Bhattacharyya [15] to multidimensional systems with an aim of obtaining more than a single excited state for sector one. First, we use our vector approach to generate the tensor sector two Hamiltonian, which is isospectral with the original scalar sector one Hamiltonian above the ground state of the sector one Hamiltonian.…”

“…It has been shown that 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. Subsequently, Kar and Bhattacharyya showed that bound states of a one-dimensional Hamiltonian can be obtained through quantum adiabatic switching [14] between the ground states of pairs of SUSY partner Hamiltonians [15]. In this approach, one starts with the ground state of the sector one Hamiltonian.…”