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Journal of Physics Conference Series

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

Nila Prasetya Aryani, Herry Fridolin Lalus, Siti Aisyah Wahyuni, Fifin Dewi Ratnasari, I Akhlis

Published Jun 1, 2021• 0 citations
Publisher

Abstract

Abstract 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.

DOI
10.1088/1742-6596/1918/2/022024

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Abstract We analyze transition probabilities of harmonic oscillator system with spatial LQC (Linear-Quadratic-Cubic) perturbation in time-dependent. This system initially was in the ground state with no perturbation at t < 0, then at t ≥ 0, the system is perturbed by spacial LQC perturbation in time-dependent until t → ∞. We use the time-dependent perturbation theory to analyze this problem. In the initial state, before there is no perturbation, we define the ground state with the base ket of harmonic oscillator without perturbation. Next, when the perturbation is applied to the system, we compute the transition amplitude base on the system state presented above and then we get total wave function that depends on time. By getting this wave function, we can compute transition probability for the system. As a result, there are three transition probabilities, namely the transitions from the ground state to the first, second, and third excited state. There is no transition to others.
Here a special case of perturbation in quantum harmonic oscillator is studied. Here we assume the perturbed potential to be a Harmonic Oscillator that has been shifted in the position space.We construct the new creation and annihilation operators for the new Hamiltonian to find out its energy eigenstates. What is interesting about the solution of this system is that we find out that the ground state of this system is a coherent state of the ground state of the original harmonic oscillator. We know that the Hamiltonian of electromagnetic field is similar to that of the harmonic oscillator where the electric field and magnetic field are expressed in terms of time evolution of annihilation and creation operators. According to the quantum electromagnetic field theory, photon states are expressed as the coherent states of the ground states of the unperturbed harmonic oscillator.In the previous research works,it has been established how coherent states provide a better picture of Electric field compared to the energy eigenstates. This paper mainly deals with why the coherent states provide a more complete picture of electric and magnetic field theory compared to just energy eigenstates. Here we find how our perturbation act as a coherent state generator and adds a new term in the Hamiltonian. The Perturbation is thought to arise from the only available interaction i.e. observation. The perturbation also explains how observation leads to coherence of photon states. We will find out how the new perturbation affects the time evolution of electromagnetic field. This also explains how the classical electromagnetic field theory takes into account the factor of observation, due to which the classical laws are in perfect agreement with the quantum laws.
Multidimensional harmonic oscillator matrix elements of arbitrary products of exponentials for operators containing only linear and quadratic terms in the creation and annihilation operators are obtained. A simple and straightforward method is presented that leads to recurrence relations and generating functions for such matrix elements. Results are shown to be general enough to cover the cases of Franck–Condon overlaps, transition probabilities for time-dependent harmonic oscillators, and correlation functions for electronic spectra.