Stochastic diagonalization of Hamiltonian: A genetic algorithm‐based approach

Abstract

ABSTRACT:We propose and demonstrate the workability of a genetic-algorithmbased technique of finding the eigenvalues and eigenfunctions of a real symmetric Hamiltonian matrix. The algorithm can be tailored to extract all the eigenvalues at a time or to extract only the lowest or the highest eigenvalue first and then sequentially determine the next few higher (lower) eigenvalues. The algorithm is generalized to handle diagonalization problems in which the basis functions themselves are optimized simultaneously. Several model applications are reported.

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

“…(9). Let us define at this point a time-dependent Hamiltonian H(t) which continuously, adiabatically interpolates between H (0) and its SUSY partner H (1) through an appropriate switching function S(t):…”

“…x ð Þ we generate the first excited state of H (1) and second excited state of H (0) , respectively. By repeating the whole sequence of steps, we can construct the ground states of H (0) , H (1) , and H (2) from which the ground, the first excited, the second excited states, etc. of H can be recovered.…”

“…One can repeat the procedure to generate H 2 , H 3 , …, H n and their ground states, thereby obtaining the whole set of eigenstates of H 0 . We have recently experimented with a numerical recipe of this kind for obtaining the eigenvalues and vectors of a Hamiltonian matrix [1][2][3] using Genetic Algorithm. [4] Could a similar approach be useful within the framework of supersymmetric (SUSY) quantum mechanics (QMs)?…”

“…from which the entire set of n-eigenstates of the Hamiltonian H (0) (and therefore of H) can be recovered by operation with appropriate sequence of charge operators.Alternatively, after adiabatically switching from the ground state W then use the first excited state of H (0) as the initial state for adiabatically switching into the first excited state of H (1) , i.e., W can lead to the spectra of both the SUSY partner Hamiltonians H (0) and H(1) , simultaneously.Let us consider a number of examples in detail to illustrate how the proposed method works.…”

Direct search for wave operator by a Genetic Algorithm (GA): Route to few eigenvalues of a Hamiltonian

“…The population of initial solution strings are allowed to undergo a Roulette-wheel selection process ( fitness proportional selection) to form the mating pool and randomly chosen strings from the mating pool are allowed to undergo an arithmetic crossover process with a fixed crossover probability p c . If the i th and the j th strings are selected for crossover, a pair of random integers (m, n) are selected and the arithmetic crossover between the strings are carried out as follows [12]:…”