Hamiltonian matrix representation of harmonic oscillator system with Linear-Cubic-Quartic (LCQ) perturbation

In this paper, we analyze the matrix representation of the energy operator (Hamiltonian) of the harmonic oscillator system when this system is perturbed by an LCQ perturbation. The LCQ perturbation is simultaneously acted on this system, so the total Hamiltonian is a summation of the pure oscillator harmonic operator term (in the annihilation and creation operator statement) and the three perturbation terms. In this work, we use three different small parameters for each term of the perturbation for keeping the generality. We use the simple algebraic method by using the Dirac notation and the well-known base vector of harmonic oscillator to analyze this problem. Next, we present every term of this operator in matrix representation and then adding them for finding matrix representation for the total Hamiltonian. (paper)