Abstract
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 , where the conjugation parameter is set to unity () at the end of the evaluations. Factorization in normal order form yields 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 reversal conjugation rule . Setting provides the standard Hermite polynomials and their partner anti-Hermite polynomials. The anti-Hermite polynomials satisfy a new differential equation, which is interpreted as the conjugate of the standard Hermite differential equation.
Highlights
The Weber-Hermite differential equation arises as the dimensionless form of the one-dimensional stationary
To take account of operator factor ordering in general form, we introduce a constant parameter, which is set to unity ( = 1 ) at the end of the evaluations, according to a transformation rule d → d dx dx to express the Weber-Hermite Equation (1e) in the general form
2x dH n dx which we substitute into Equation (9a) to obtain the differential equation for the composite Hermite polynomials in the form
Summary
Schroedinger equation for a linear harmonic oscillator of mass m, angular frequency ω , total energy E and displacement x obtained in quantum mechanics in the form [1]-[4],. Even though the main motivation for introducing the parameter is to account for operator ordering, it turns out that plays a fundamental role as a conjugation parameter, which provides a conjugation rule relating the two alternate normal and anti-normal order factorized forms of Equation (2b). The general solutions of the normal or anti-normal order forms are conjugate polynomials related by the -conjugation rule. For operators or eigenfunctions expressible in matrix form, the Hermitian conjugation under the -sign reversal conjugation is effected by applying the conjugation rule (3c) to every element and taking the transpose. We observe that the mathematical operation in Equation (3f) applies to the factorization of a second order operator of the form d2 + f 2 . The factorized form (3a) is said to be in normal order, while the form (3b) is in anti-normal order
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