arXiv:quant-ph/9907005

Randomly Walking 1D Quantum Harmonic Oscillator. Averaged Transition Probabilities

Ashot S. Gevorkyan, A. A. Udalov

Published Jul 1, 1999• 0 citations

Abstract

One-dimensional problem for quantum harmonic oscillator with "regular+random" frequency subjected to the external "regular+random" force is considered. Averaged transition probabilities are found.

DOI: 10.48550/arxiv.quant-ph/9907005

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Exactly solvable models of quantum mechanics including fluctuations in the framework of representation of the wave function by random process

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Random walk as a model of motion in a quantum harmonic oscillator

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arXiv:quant-ph/9907005

Randomly Walking 1D Quantum Harmonic Oscillator. Averaged Transition Probabilities

Ashot S. Gevorkyan, A. A. Udalov

Published Jul 1, 1999• 0 citations

arXiv HTML PDF

Abstract

One-dimensional problem for quantum harmonic oscillator with "regular+random" frequency subjected to the external "regular+random" force is considered. Averaged transition probabilities are found.

DOI: 10.48550/arxiv.quant-ph/9907005

Related papers

Exactly solvable models of quantum mechanics including fluctuations in the framework of representation of the wave function by random process

Ashot S. Gevorkyan, Udalov, A. A.

arXiv:quant-ph/0007108Jul 28, 20000 citations

The driven oscillator

Smith, T. B.

arXiv:1508.02379Aug 10, 20150 citations

Random walk as a model of motion in a quantum harmonic oscillator

Zbigniew S. Kowalski

The one-dimensional quantum harmonic oscillator is considered. The wave solution of the Schrödinger equation determines the model of motion of particles in the $n$ -quantum state as orbits of a stationary random walk at the moment of jump from the $n-1$ -quThe one-dimensional quantum harmonic oscillator is considered. The wave solution of the Schrödinger equation determines the model of motion of particles in the $n$-quantum state as orbits of a stationary random walk at the moment of jump from the $n-1$-qu

Applicationes MathematicaeJan 1, 20221 citation

Quantum Theory of the Forced Harmonic Oscillator

D.M. Gilbey, F. O. Goodman

A unified treatment by the methods of the operator mechanics is given of the quantum theory of the harmonic oscillator subjected to an applied force which is a given, but arbitrary, function of time. Few of the results are new, and most have appeared before scattered among papers devoted to a variety of topics and using a variety of mathematical techniques. Emphasis is placed on numerical methods for the evaluation of the relevant transition probabilities, and on the equality, under certain conditions, of the classical and quantum values of the average energy transfer to an oscillator by an impressed force. Generating functions for the transition probabilities are derived. Some numerical results are presented for illustration.A unified treatment by the methods of the operator mechanics is given of the quantum theory of the harmonic oscillator subjected to an applied force which is a given, but arbitrary, function of time. Few of the results are new, and most have appeared before scattered among papers devoted to a variety of topics and using a variety of mathematical techniques. Emphasis is placed on numerical methods for the evaluation of the relevant transition probabilities, and on the equality, under certain conditions, of the classical and quantum values of the average energy transfer to an oscillator by an impressed force. Generating functions for the transition probabilities are derived. Some numerical results are presented for illustration.

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Transitions of a Quantum Harmonic Oscillator as a Result of Interaction with an Ultrashort Pulse Field

Dmitry N. Makarov, Mikhail Y. Arkhipov, Polina A. Stremilova

The quantum harmonic oscillator is a model system that is actively used in many applications of quantum and nonlinear physics, molecular chemistry and biophysics. In this paper, a formula is obtained for the probability amplitude of the transition of a quantum harmonic oscillator from an oscillatory state with the number $\nu$ to a state with the number $\nu+n$ , which is affected by an external force depending on time using a method based on the Magnus and Baker-Campbell-Hausdorff decomposition. A formula is obtained for the special case of interaction of a quantum harmonic oscillator with an ultrashort pulse field. As an example, graphs of the probabilities of the oscillator transition from the first oscillatory state to the second and third, the average energy received and the rms fluctuation of the energy received for different values of the ratio of the carrier frequency of the pulse and the frequency of the oscillator are constructed. Conclusions are drawn about the correctness of using the concept of average energy in relation to this problem.The quantum harmonic oscillator is a model system that is actively used in many applications of quantum and nonlinear physics, molecular chemistry and biophysics. In this paper, a formula is obtained for the probability amplitude of the transition of a quantum harmonic oscillator from an oscillatory state with the number$\nu$to a state with the number$\nu+n$, which is affected by an external force depending on time using a method based on the Magnus and Baker-Campbell-Hausdorff decomposition. A formula is obtained for the special case of interaction of a quantum harmonic oscillator with an ultrashort pulse field. As an example, graphs of the probabilities of the oscillator transition from the first oscillatory state to the second and third, the average energy received and the rms fluctuation of the energy received for different values of the ratio of the carrier frequency of the pulse and the frequency of the oscillator are constructed. Conclusions are drawn about the correctness of using the concept of average energy in relation to this problem.

IEEE International Conference on Problems of Informatics, Electronics and Radio Engineering (PIERE)Nov 15, 20240 citations

Equivalence of quantum harmonic oscillators and classical oscillators subject to random forces

Can Gokler

We show that the Schrödinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals $O( m / \hbar)$ , when $\hbar / m \ll 1$ . Conversely, every solution to the Schrödinger equation, including all the superposition states, arises this way. In other words, the quantum harmonic oscillator is approximately nothing but the classical harmonic oscillator, with the same mass and frequency as the quantum harmonic oscillator, subject to a random force. We generalize the result to multiple non-interacting oscillators. We show that the Schrödinger equation for $n$ non-interacting quantum harmonic oscillators with masses $m_1, ..., m_n$ and frequencies $ω_1, ..., ω_n$ can be derived as an approximation to the Newtonian mechanics of $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This is valid for time intervals $O( \tilde{m} / \hbar)$ , where $\tilde{m}$ is the mass of the minimum mass oscillator, when $\hbar/\tilde{m} \ll 1$ . Conversely, every solution, including all the entangled states, to the Schrödinger equation arises this way. In other words, $n$ non-interacting quantum harmonic oscillators are approximately nothing but $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This provides a local Newtonian model of entanglement of non-interacting quantum oscillators. The correlations required by entangled states are embedded in the phase space probability density of the classical oscillators.We show that the Schrödinger equation for the quantum harmonic oscillator can be derived as an approximation to the Newtonian mechanics of a classical harmonic oscillator subject to a random force for time intervals $O( m / \hbar)$, when $\hbar / m \ll 1$. Conversely, every solution to the Schrödinger equation, including all the superposition states, arises this way. In other words, the quantum harmonic oscillator is approximately nothing but the classical harmonic oscillator, with the same mass and frequency as the quantum harmonic oscillator, subject to a random force. We generalize the result to multiple non-interacting oscillators. We show that the Schrödinger equation for $n$ non-interacting quantum harmonic oscillators with masses $m_1, ..., m_n$ and frequencies $ω_1, ..., ω_n$ can be derived as an approximation to the Newtonian mechanics of $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This is valid for time intervals $O( \tilde{m} / \hbar)$, where $\tilde{m}$ is the mass of the minimum mass oscillator, when $\hbar/\tilde{m} \ll 1$. Conversely, every solution, including all the entangled states, to the Schrödinger equation arises this way. In other words, $n$ non-interacting quantum harmonic oscillators are approximately nothing but $n$ non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This provides a local Newtonian model of entanglement of non-interacting quantum oscillators. The correlations required by entangled states are embedded in the phase space probability density of the classical oscillators.

arXiv:2010.14930Oct 26, 20202 citations

The one-dimensional quantum harmonic oscillator is considered. The wave solution of the Schrödinger equation determines the model of motion of particles in the

n

-quantum state as orbits of a stationary random walk at the moment of jump from the

n-1

-qu

\nu

to a state with the number

\nu+n

, which is affected by an external force depending on time using a method based on the Magnus and Baker-Campbell-Hausdorff decomposition. A formula is obtained for the special case of interaction of a quantum harmonic oscillator with an ultrashort pulse field. As an example, graphs of the probabilities of the oscillator transition from the first oscillatory state to the second and third, the average energy received and the rms fluctuation of the energy received for different values of the ratio of the carrier frequency of the pulse and the frequency of the oscillator are constructed. Conclusions are drawn about the correctness of using the concept of average energy in relation to this problem.

O( m / \hbar)

, when

\hbar / m \ll 1

. Conversely, every solution to the Schrödinger equation, including all the superposition states, arises this way. In other words, the quantum harmonic oscillator is approximately nothing but the classical harmonic oscillator, with the same mass and frequency as the quantum harmonic oscillator, subject to a random force. We generalize the result to multiple non-interacting oscillators. We show that the Schrödinger equation for

n

non-interacting quantum harmonic oscillators with masses

m_1, ..., m_n

and frequencies

ω_1, ..., ω_n

can be derived as an approximation to the Newtonian mechanics of

n

non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This is valid for time intervals

O( \tilde{m} / \hbar)

, where

\tilde{m}

is the mass of the minimum mass oscillator, when

\hbar/\tilde{m} \ll 1

. Conversely, every solution, including all the entangled states, to the Schrödinger equation arises this way. In other words,

n

non-interacting quantum harmonic oscillators are approximately nothing but

n

non-interacting classical harmonic oscillators, with the same set of masses and frequencies as the quantum oscillators, subject to random forces. This provides a local Newtonian model of entanglement of non-interacting quantum oscillators. The correlations required by entangled states are embedded in the phase space probability density of the classical oscillators.