A Laplace transform approach to the quantum harmonic oscillator

Abstract

The one-dimensional quantum harmonic oscillator problem is examined via the Laplace transform method. The stationary states are determined by requiring definite parity and good behaviour of the eigenfunction at the origin and at infinity.

Lie algebraic approach to the time-dependent quantum general harmonic oscillator and the bi-dimensional charged particle in time-dependent electromagnetic fields

“…Once again, we observe that Eqs. (44)- (45) are the classical Euler equations corresponding to the Lagrangian…”

“…The transformation parameters must be calculated from the restrictions (44), (45), (46), (86), (90), (97), (102) and (107). By successively applying the six transformations above tox andp we can workout the Heisenberg picture position and momentum operators newly obtaining the symplectic form…”

“…It has been studied by diverse mathematical methods such as the group-theoretical approach [27], the path integral approach [28], unitary transformations [7,29], and the Lewis and Riesenfeld [30] invariant theory [31,32,33,34,35,18]. The linear potential, a particular case of the GHO, has been treated through powerful methods as the Lewis and Riesenfeld [30] invariant theory [36,37,38], Feynman's path integrals [39,40,41,42,43], the generalization of the well known ladder operators [44], Laplace transform techniques [45], time-space transformation methods [46] and others [47,48]. The charge particle in electromagnetic fields has been studied through different methods that include the Lewis and Riesenfeld [30] invariant theory [49,50], path integral method [51], unitary transformation approach [52,19], and through quadratic invariants [53].…”

A Laplace transform approach to find the exact solution of the $$N$$ N -dimensional Schrödinger equation with Mie-type potentials and construction of Ladder operators

“…Here the inverse transformation is done through the convolution of Faltungs theorem which is also a new aspect of this paper. More recent works on LT can be found in the ref [18][19][20][21][22][23][24][25].…”

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

“…For example, it has been shown that the one-dimensional quantum harmonic oscillator problem is examined via the Laplace transform method. 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.…”