Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics Conference Series

Transition probabilities of harmonic oscillator system with spatial Linear-Quadratic-Cubic (LQC) perturbation in time-dependent

Herry Fridolin Lalus, Nila Prasetya Aryani

Published Jun 1, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/1918/2/022025

Related papers

Exact formulated approach to transition probabilities of linearly driven parametric oscillators

Wensen Liu

The transition probabilities of linearly driven parametric oscillators that model collinear collisions of an atom and a diatomic molecule are studied within the formulation for treating the time evolution of time-dependent quadratic Hamiltonian systems. It is shown that the transition probabilities of three oscillator models: (i.e. the oscillator with varying effective mass and frequency plus linear potential, the harmonic oscillator with linear potential and the harmonic oscillator with linear plus quadratic potentials) obey the Poisson distribution in common, and evaluations of these probabilities are reduced to performing a simple operation and numerical integrations on time in terms of the parameters of the considered systems. The reasons for the difference between the previous results and ours are analyzed.

Physica Scripta•Mar 1, 2009•0 citations

Time-Dependent Perturbation Theory in Quantum Phase Space

Qian‐Shu Li

Time-dependent perturbation theory is discussed within the quantum phase space representation as formulated by Torres-Vega and Frederick(T-F) and the quantum transition probability is studied.Time-dependent perturbation formula in T-F quantum phase space is established,in accordance with the calculation theory in T-F phase space.As an example,the one dimensional harmonic oscillator system,affected by an external electric field,is used to calculate in T-F phase space,the result indicate that the principle of the method is correct.Quantum transition probability of the system's first excited state in T-F phase space is discussed.

Transactions of Beijing Institute of Technology•Jan 1, 2009•0 citations

Time-dependent perturbation and exact results for the periodically driven quantum harmonic oscillator

Russell Akridge

Transition amplitudes and probabilities for the harmonic oscillator with a forcing function proportional to cos(ωt) beginning at time zero are calculated to lowest nonvanishing order using time-dependent perturbation theory. The results are compared with the exact amplitudes and probabilities. When the exact amplitude is expanded in a Taylor series in powers of the coupling constant, the individual terms turn out to be the perturbation amplitudes, showing that the complete series of perturbation amplitudes converges to the exact amplitude.

American Journal of Physics•Feb 1, 1995•5 citations

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

Nila Prasetya Aryani, Herry Fridolin Lalus, Siti Aisyah Wahyuni +2 more

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.

Journal of Physics Conference Series•Jun 1, 2021•0 citations

Explicit Computation of the Quantum Phase of the Time Perturbed Harmonic Oscillator

Henryk Gzyl

We consider a simple harmonic oscillator subject to a time-dependent, spatially homogeneous perturbation and show that it is canonically equivalent to a non-perturbed oscillator. The canonical transformation that relates both systems can be implemented as a unitary transformation mapping the perturbed system and the non-perturbed systems onto each other. This unitary transformation allows us to explicitly compute the time dependence of the states of the perturbed system, and in particular to compute the phase that affects all states. The phase turns out to be a classical action along the classical trajectory of the origin (in phase space), along its motion under the action of perturbation. In this simple system, the transition probabilities due to the perturbation can also be computed explicitly. Our approach is independent of the magnitude of the perturbation, and does not require an adiabaticity assumption.

Acta Physica Polonica A•Jan 1, 2023•0 citations

Joint excitation probability for two harmonic oscillators in dimension one and the Mott problem

Gianfausto Dell’Antonio, Rodolfo Figari, Alessandro Teta

We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions $a_1$, $a_2$, with $a_1 >0$, $|a_2|>a_1$, in the two possible situations: $a_2>0$ and $a_2 <0$. At time zero the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. %$\pm M v_0$. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability $\mathcal{P}^{-}_{n_1 n_2} (t)$ (resp. $\mathcal{P}^{+}_{n_1 n_2} (t)$) that both oscillators are in the excited states labelled by $n_1$, $n_2 >0$ at time $t > |a_2| v_0^{-1}$ when $a_2 <0$ (resp. $a_2 >0$). We prove that $\mathcal{P}_{n_1 n_2}^- (t)$ is negligible with respect to $\mathcal{P}_{n_1 n_2}^+ (t)$, up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott in \cite{m}, where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schrödinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schrödinger equation for the three-particle system, with no reference to the wave packet collapse postulate.

arXiv:0711.2230•Nov 14, 2007•1 citation
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.
The transition probabilities of linearly driven parametric oscillators that model collinear collisions of an atom and a diatomic molecule are studied within the formulation for treating the time evolution of time-dependent quadratic Hamiltonian systems. It is shown that the transition probabilities of three oscillator models: (i.e. the oscillator with varying effective mass and frequency plus linear potential, the harmonic oscillator with linear potential and the harmonic oscillator with linear plus quadratic potentials) obey the Poisson distribution in common, and evaluations of these probabilities are reduced to performing a simple operation and numerical integrations on time in terms of the parameters of the considered systems. The reasons for the difference between the previous results and ours are analyzed.
Time-dependent perturbation theory is discussed within the quantum phase space representation as formulated by Torres-Vega and Frederick(T-F) and the quantum transition probability is studied.Time-dependent perturbation formula in T-F quantum phase space is established,in accordance with the calculation theory in T-F phase space.As an example,the one dimensional harmonic oscillator system,affected by an external electric field,is used to calculate in T-F phase space,the result indicate that the principle of the method is correct.Quantum transition probability of the system's first excited state in T-F phase space is discussed.
Transition amplitudes and probabilities for the harmonic oscillator with a forcing function proportional to cos(ωt) beginning at time zero are calculated to lowest nonvanishing order using time-dependent perturbation theory. The results are compared with the exact amplitudes and probabilities. When the exact amplitude is expanded in a Taylor series in powers of the coupling constant, the individual terms turn out to be the perturbation amplitudes, showing that the complete series of perturbation amplitudes converges to the exact amplitude.
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.
We consider a simple harmonic oscillator subject to a time-dependent, spatially homogeneous perturbation and show that it is canonically equivalent to a non-perturbed oscillator. The canonical transformation that relates both systems can be implemented as a unitary transformation mapping the perturbed system and the non-perturbed systems onto each other. This unitary transformation allows us to explicitly compute the time dependence of the states of the perturbed system, and in particular to compute the phase that affects all states. The phase turns out to be a classical action along the classical trajectory of the origin (in phase space), along its motion under the action of perturbation. In this simple system, the transition probabilities due to the perturbation can also be computed explicitly. Our approach is independent of the magnitude of the perturbation, and does not require an adiabaticity assumption.
We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions a1a_1a1​, a2a_2a2​, with a1>0a_1 >0a1​>0, ∣a2∣>a1|a_2|>a_1∣a2​∣>a1​, in the two possible situations: a2>0a_2>0a2​>0 and a2<0a_2 <0a2​<0. At time zero the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. %±Mv0\pm M v_0±Mv0​. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability Pn1n2−(t)\mathcal{P}^{-}_{n_1 n_2} (t)Pn1​n2​−​(t) (resp. Pn1n2+(t)\mathcal{P}^{+}_{n_1 n_2} (t)Pn1​n2​+​(t)) that both oscillators are in the excited states labelled by n1n_1n1​, n2>0n_2 >0n2​>0 at time t>∣a2∣v0−1t > |a_2| v_0^{-1}t>∣a2​∣v0−1​ when a2<0a_2 <0a2​<0 (resp. a2>0a_2 >0a2​>0). We prove that Pn1n2−(t)\mathcal{P}_{n_1 n_2}^- (t)Pn1​n2​−​(t) is negligible with respect to Pn1n2+(t)\mathcal{P}_{n_1 n_2}^+ (t)Pn1​n2​+​(t), up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott in \cite{m}, where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schrödinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schrödinger equation for the three-particle system, with no reference to the wave packet collapse postulate.