Composite Hermite and Anti-Hermite Polynomials
Abstract/ Overview
The Weber-Hermite differential equation, obtained as the dimensionless form of the stationary
Schroedinger equation for a linear harmonic oscillator in quantum mechanics, has been expressed
in a generalized form through introduction of a constant conjugation parameter according to
the transformation
x x
d d
d d
→ , where the conjugation parameter is set to unity ( = 1 ) at the end
of the evaluations. Factorization in normal order form yields -dependent composite eigenfunctions, Hermite polynomials and corresponding positive eigenvalues, while factorization in the anti-normal order form yields the partner composite anti-eigenfunctions, anti-Hermite polynomials
and negative eigenvalues. The two sets of solutions are related by an -sign reversal conjugation
rule → − . Setting = 1 provides the standard Hermite polynomials and their partner antiHermite polynomials. The anti-Hermite polynomials satisfy a new differential equation, which is
interpreted as the conjugate of the standard Hermite differential equation