More on the quantum harmonic oscillator via unilateral Fourier transform

Vieira, Douglas Willian; Castro, Antonio S. de

doi:10.1590/1806-9126-rbef-2022-0038

Cited by 1 publication

(1 citation statement)

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“…Notwithstanding these issues, the unilateral Fourier transform has been shown to be a useful tool for solving bound-state solution problems in non-relativistic quantum mechanics (see, e.g. [16][17][18][19]), and even for the homogeneous differential equation of the classical harmonic oscillator (in the sense of the Dirac delta distributions) [20]. To be accurate, all the aforementioned problems were solved under the convenient "special conditions" alluded by Butkov. In this work, we venture to utilize the unilateral Fourier transform to solve a first-order ordinary differential equation that entails a combination of derivatives of both odd and even orders, and we obtain the Green function in the process.…”

Section: Introductionmentioning

confidence: 99%

Motion under a time-dependent driving force plus a linear velocity-dependent friction: From the unilateral Fourier transform to the Green function

Correa,

Castro

2024

Rev. Bras. Ensino Fís.

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The problem of a particle subject to a time-dependent driving force plus a linear velocity-dependent friction can be addressed by utilizing the unilateral Fourier transform, despite the presence of derivatives of odd and even order in the differential equation. This technique yields a system of algebraic equations that combine the Fourier sine and cosine transforms. While this method is useless for solving the homogeneous equation, it can be effectively used to obtain an integral representation of the particular solution. Remarkably, this integral representation of the particular solution is expressed in terms of the Green function. This type of exactly solvable problem is relevant for students who are studying mathematical methods applied in the fields of physics and engineering at the undergraduate level, as it can serve as a useful illustration of how unilateral Fourier transforms can be employed to solve problems and to develop an understanding of Green functions, even in introductory calculus courses.

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