Trial introduction of a complex potential function for perturbation that causes quantum mechanical transition in a one‐dimensional harmonic oscillator

Mitsuru Yamada

International Journal of Quantum ChemistryJan 1, 20040 citations

arXiv:quant-ph/0702007 · Quantum Information Processing

Quantum Mechanical Search and Harmonic Perturbation

Jie-Hong Roland Jiang, Dah-Wei Chiou, Cheng-En R. Wu

Published Sep 19, 2007• 1 citation

arXiv HTML PDF

DOI: 10.1007/s11128-007-0062-5

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Trial introduction of a complex potential function for perturbation that causes quantum mechanical transition in a one‐dimensional harmonic oscillator

Mitsuru Yamada

Abstract Some results of computer simulation of the behavior of a one‐dimensional quantum mechanical oscillator are reported in this article. This harmonic oscillator comprises a particle trapped within a hyperbolic potential V (x) = x2. Further, a perturbation potential function V′(x, t) was superposed upon the hyperbolic potential in order to induce a quantum mechanical transition. This perturbation function V′(x, t) is a function of both of space and time variables, and is set to represent a wave packet that is enveloped by a Gaussian bell‐shaped curve. A wave that probably has an appropriate wave number and angular frequency was inputted into the expression for the wave packet. In the initial phase, while the harmonic oscillator was allowed to oscillate almost freely, the wave packet was allowed to approach the harmonic oscillator. In the middle phase, the wave packet passes through the harmonic oscillator, affecting the shape of the quantum mechanical wave that represents the physical state of the system. In the last phase, when the wave packet left the system of the harmonic oscillator, the system settled onto an energetically stable state. The main objective of the simulation was to simulate the instance of a quantum mechanical transition from one eigenstate to another. After several trials, it was found that the perturbation function consisting of a complex function was, at least superficially, able to cause one desired transition, that is, a transition from one eigenstate to another eigenstate. By using such a complex perturbation function, a transition from the first excited state to the ground state was observed to occur. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004Abstract Some results of computer simulation of the behavior of a one‐dimensional quantum mechanical oscillator are reported in this article. This harmonic oscillator comprises a particle trapped within a hyperbolic potential V (x) = x2. Further, a perturbation potential function V′(x, t) was superposed upon the hyperbolic potential in order to induce a quantum mechanical transition. This perturbation function V′(x, t) is a function of both of space and time variables, and is set to represent a wave packet that is enveloped by a Gaussian bell‐shaped curve. A wave that probably has an appropriate wave number and angular frequency was inputted into the expression for the wave packet. In the initial phase, while the harmonic oscillator was allowed to oscillate almost freely, the wave packet was allowed to approach the harmonic oscillator. In the middle phase, the wave packet passes through the harmonic oscillator, affecting the shape of the quantum mechanical wave that represents the physical state of the system. In the last phase, when the wave packet left the system of the harmonic oscillator, the system settled onto an energetically stable state. The main objective of the simulation was to simulate the instance of a quantum mechanical transition from one eigenstate to another. After several trials, it was found that the perturbation function consisting of a complex function was, at least superficially, able to cause one desired transition, that is, a transition from one eigenstate to another eigenstate. By using such a complex perturbation function, a transition from the first excited state to the ground state was observed to occur. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

International Journal of Quantum ChemistryJan 1, 20040 citations

{x}^{4}

perturbed harmonic oscillator. This is a consequence of five facts: (i) The quantum perturbation series for the energy can be solved order by order in closed form. (ii) The quantum series rearranges directly into the classical canonical perturbation series plus quantum corrections proportional to successively higher powers of ${\ensuremath{\Elzxh}}^{2}$. (iii) The classical series and the subseries for the quantum corrections converge (subseries by subseries) to the terms of the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) expansion for the energy (to all orders of \ensuremath{\Elzxh}). (iv) The (all-order) JWKB expansion, conversely, yields the quantum perturbation series and consequently uniquely determines the exact eigenvalue. (v) The ratio of the quantum energy to the classical energy at action (n+1/2)\ensuremath{\Elzxh} differs from 1 asymptotically by 1/[9\ensuremath{\pi}(n

{+1/2)}^{2}

], independent of both \ensuremath{\Elzxh} and the perturbation coupling constant, as the harmonic-oscillator quantum number n approaches \ensuremath{\infty}. An interesting additional feature is the mechanism by which the convergence of the classical perturbation series and of the series for the quantum corrections gets turned into the divergence of the quantum perturbation series.