Abstract
In this work, we present a rigorous mathematical scheme for the derivation of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by the potential Vper(x) = λxα, where α is a positive integer, using the non-degenerate time-independent perturbation theory. To do so, we derive a generalized formula for the integral I=∫−∞+∞xαexp(−x2)Hn(x)Hm(x)dx, where Hn(x) denotes the Hermite polynomial of degree n, using the generating function of orthogonal polynomials. Finally, the analytical results with α = 3 and α = 4 are discussed in detail and compared with the numerical calculations obtained by the Lagrange-mesh method.
Highlights
We present a rigorous mathematical scheme for the derivation of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by the potential Vper(x) = λxα, where α is a positive integer, using the non-degenerate time-independent perturbation theory
Approximation methods play a crucial role in quantum mechanics since the number of problems that are exactly solvable is small in comparison to those that must be solved approximately
The explicit results of two particular cases in which α = 3 and α = 4 are discussed in detail
Summary
Approximation methods play a crucial role in quantum mechanics since the number of problems that are exactly solvable is small in comparison to those that must be solved approximately. The hydrogen atom, harmonic oscillators, and quantum particles in some specific potential wells have exact solutions, and two cold atoms interacting through a point-like force in a three-dimensional harmonic oscillator potential can be solved analytically. Approximation methods have been developed early since the dawn of quantum mechanics. One of the essential approximation methods is the perturbation theory (PT) established by Schrödinger in 1926.7 Later, it was immediately used to interpret the LoSurdo–Stark effect of the hydrogen atom by Epstein.. The PT is not well convergent at higherorder corrections, many more efficient approximation methods have been developed to treat quantum-mechanically complex problems, it is still a paramount and elementary approximation method in quantum physics. The PT contributes significantly to quantum optics and quantum field theory, as discussed in
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