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.
The Weber-Hermite differential equation arises as the dimensionless form of the one-dimensional stationary Schroedinger equation for a linear harmonic oscillator of mass m, angular frequency
Introducing parameters s and
we easily transform Equation (1a) into the dimensionless form
which we call the Weber-Hermite differential equation, since its general solutions are the Weber-Hermite func- tions composed of the Hermite polynomials [
It is convenient to replace
to express Equation (1c) in the familiar mathematical form
We provide conjugate pairs of solutions of this equation through factorization.
We define a conjugation parameter and develop the factorization procedure in Section 2. Normal-order solutions in terms of composite Hermite polynomials, their recurrence relations, positive eigenvalues and differential equation are presented in Section 3.1, while the composite anti-Hermite polynomials, their recurrence relations, negative eigenvalues and differential equation arising from the anti-normal order solutions are contained in Section 3.2.
Factorization and the Conjugation ParameterFactorization is a powerful technique for solving second-order ordinary differential equations. An important feature of factorization is factor ordering in the resulting product of factors, especially if the factors are operators [
to express the Weber-Hermite Equation (1e) in the general form
which is the same as Equation (1e) for
Even though the main motivation for introducing the parameter
Noting that the operator
The operators are related by
giving
The operators are said to be
where we have adopted the usual Hermitian conjugation notation using the symbol
We note that in a case where
the
mitian conjugation notation adopted here. We observe that the mathematical operation in Equation (3f) applies to the factorization of a second order operator of the form
According to the conjugation rule in Equation (3c), the factorized forms (3a) and (3b) are
For reasons which may become clear below, we recognize
Since Equations (3a) and (3b) are related by the
We start by considering that the normal order form (3a) is an eigenvalue equation with eigenvalue
where
Applying Hermitian conjugation of the operators
which on multiplying from the left by the ε-sign reversal conjugate
The basic equation for the lowest order eigenfunction
with a simple solution
noting that the integration constant evaluated at
Eigenfunctions
which on substituting
To evaluate higher order eigenfunctions
and then apply the general relation
which follows easily from Equation (5c) by setting
For
which on substituting
Proceeding in the same manner for
easily gives the forms
We arrive at the important general result that higher order eigenfunctions are obtained in the form of a re- currence relation
Setting
where
Using Equation (5b) in Equation (5c) and substituting the result on the l.h.s. of Equation (7a) provides the general relation for generating the composite Hermite polynomials in the form
Using Equation (5b) together with its
in Equation (7b) defines the composite Hermite polynomials in terms of the lowest order eigenfunction accord- ing to
Explicit forms of
Setting
which is easily evaluated to obtain the first recurrence relation for the polynomials
Setting
Setting
taking the general expansion
The symbol
Substituting
into Equation (6e) gives the second recurrence relation for the composite Hermite polynomials in the form
Comparing the first recurrence relation (8b) and the second recurrence relation (8f) easily provides the third recurrence relation for the composite Hermite polynomials in the form
Applying
Using Equation (8e) together with the result of setting
which we substitute into Equation (9a) to obtain the differential equation for the composite Hermite polynomials in the form
which differs from the familiar Hermite differential equation [
Substituting
from Equation (7a) into Equation (9c) and reorganizing gives the final result
which confirms that the eigenfunctions
Comparing Equations (1e) and (10b), noting
which correspond to the eigenfunctions
We now set
satisfying
The eigenfunctions
Setting
The first five Hermite polynomials are the same as Equation (8d) with
Finally, we set
The anti-normal order form (3b) is an eigenvalue equation with eigenvalue
where
Applying Hermitian conjugation according to Equation (3e), we express Equation (12b) in the form
which on multiplying from the left by the (ε-sign reversal) Hermitian conjugate
The basic equation for the highest order anti-eigenfunction
with a simple solution
noting that the integration constant evaluated at
Anti-eigenfunctions
operator
which substituting
To evaluate lower order anti-eigenfunctions
and apply the general relation
which follows easily from Equation (13c) by setting
For
which on substituting
Proceeding in the same manner for
easily gives the important general result that lower order anti-eigenfunctions are obtained in the form of a re- currence relation
Setting
where
Using Equation (13b) in Equation (13c) and substituting the result on the l.h.s. of Equation (15a) provides the general relation for generating the composite anti-Hermite polynomials in the form
Using Equation (13b) together with its (
in Equation (15b) defines the composite anti-Hermite polynomials in terms of the highest order anti-eigenfunction according to
Explicit forms of
Setting
which is easily evaluated to obtain the first recurrence relation for the polynomials
Setting
Setting
taking the general expansion
Substituting
into Equation (14d) gives the second recurrence relation for the composite anti-Hermite polynomials in the form
Comparing the first recurrence relation (16b) and the second recurrence relation (16f) easily provides the third recurrence relation for the composite anti-Hermite polynomials in the form
Applying
Using Equation (16f) together with the result of setting
which we substitute into Equation (17a) to obtain the differential equation for the composite Hermite poly- nomials in the form
which is a new differential equation. It is the conjugate of the composite Hermite differential Equation (9c). Applying the conjugation rule
Substituting
from Equation (15a) into Equation (17c) and reorganizing gives the final result
which confirms that the eigenfunctions
Comparing Equations (1e) and (18b), noting
which correspond to the anti-eigenfunctions
We now set
satisfying
The anti-eigenfunctions
Setting
The first five anti-Hermite polynomials (
Finally, we set
We observe that the anti-eigenfunctions
We have established that the Weber-Hermite differential equation, which is the dimensionless form of the stationary Schroedinger equation for a linear harmonic oscillator, has two sets of solutions characterized by positive and negative eigenvalues. Factorization in the normal order form yields the standard eigenfunctions, Hermite polynomials and the corresponding positive eigenvalues, while factorization in the anti-normal order form yields the partner anti-eigenfunctions, anti-Hermite polynomials and the corresponding negative eigenvalues. The two sets of solutions are related by a fundamental conjugation rule.
I thank Maseno University and Technical University of Kenya for providing facilities and conducive work environment during the preparation of the manuscript.
Joseph Akeyo Omolo, (2015) Composite Hermite and Anti-Hermite Polynomials. Advances in Pure Mathematics,05,817-827. doi: 10.4236/apm.2015.514076