Revisiting the quantum harmonic oscillator via unilateral Fourier transforms

Abstract

The literature on the exponential Fourier approach to the one-dimensional quantum harmonic oscillator problem is revised and criticized. It is shown that the solution of this problem has been built on faulty premises. The problem is revisited via the Fourier sine and cosine transform method and the stationary states are properly determined by requiring definite parity and square-integrable eigenfunctions. *

More on the quantum harmonic oscillator via unilateral Fourier transform

“…is able to accomplish the purpose. The factorization prescribed by (12) dictates that φ(ξ) obeys the equation…”

“…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]. In Ref.…”

“…In Ref. [12], the quantum harmonic oscillator was approached by the unilateral Fourier transform method and the eigenfunctions were obtained by recurring to a few properties of the sometimes clumsy confluent hypergeometric function (Kummer's function).…”

Time-independent Green’s function of a quantum simple harmonic oscillator system and solutions with additional generic delta-function potentials

“…The stationary states are determined by requiring definite parity and good behaviour of the eigenfunction at the origin and at infinity [11]. Recently, in [12], the exponential Fourier approach in the literature to the one-dimensional quantum harmonic oscillator problem is revised and criticized. The problem is revisited via the Fourier sine and cosine transform method and the stationary states are properly determined by requiring definite parity and square-integrable eigenfunctions [12].…”

A Full-Fledged Analytical Solution to the Quantum Harmonic Oscillator for Undergraduate Students of Science and Engineering

“…We only use standard mathematical tools available in natural science, technology, engineering and mathematics disciplines generally from the second college year. We believe this document can help the beginning reader to understand, enjoy, and take advantage of other smart and elegant ways to approach the QHO problem [21][22][23][24][25][26]. This methodology could also be useful for the first-year physics student taking introductory subjects in quantum physics or modern physics.…”