Quantum states of a particle in a box via unilateral Fourier transform

Abstract

The quantum problem of stationary states of a particle in a box is revisited by means of the unilateral Fourier transform. Homogeneous Dirichlet boundary conditions demand a finite Fourier sine transform which is actually the Fourier sine series.

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

“…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.…”

More on the quantum harmonic oscillator via unilateral Fourier transform

“…The unilateral Fourier transform has proved to be a straightforward and efficient manner to deal with a few bound-state solution problems in nonrelativistic quantum mechanics [12][13][14]. In recent times, the quantum harmonic oscillator has also been approached by the Laplace transform [15][16][17][18], by the exponential Fourier transform [19][20][21][22], and also by the unilateral Fourier transform [12].…”