16
Hermite Polynomials
Abstract
Topics of this chapter are Hermite polynomials in the complex domain and its
generalization to non-integer, complex degrees. The evaluation is either based on
recurrence relations or the confluent hypergeometric function. The quantum harmonic oscillator is a particular important quantum system. Therefore in addition
code for evaluating the harmonic oscillator wave function is derived, applications
like the anharmonic oscillator, Wigner functions in phase space, and coherent
states are discussed. Hermite polynomials are orthogonal polynomials. Thus
all methods related to orthogonal polynomials can be additionally applied. The
corresponding MATLAB code for evaluating Hermite polynomials, respectively,
functions, computing their nodes and for all applications mentioned above can
be downloaded.
In Chap. 15, we started to discuss orthogonal polynomials. This chapter is intended
to introduce Hermite polynomials and some applications.
The harmonic oscillator is a particularly important quantum system and its
wave function in coordinate or momentum representation can be evaluated with the
help of Hermite polynomials. Thus, we discuss in addition the harmonic oscillator
eigenfunctions related to Wigner functions, the anharmonic oscillator, and coherent
states. I suppose there is no quantum dynamic textbook without discussing the
harmonic oscillator.
16.1 Function Overview
Hermite polynomial The SPECFUNPHYS class hermitepoly supports the
evaluation of the polynomial coefficients of Hermite polynomials. The
SPECFUNPHYS class hermitex allows the computation of the corresponding
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_16
199
Hermite Polynomials
Abstract
Topics of this chapter are Hermite polynomials in the complex domain and its
generalization to non-integer, complex degrees. The evaluation is either based on
recurrence relations or the confluent hypergeometric function. The quantum harmonic oscillator is a particular important quantum system. Therefore in addition
code for evaluating the harmonic oscillator wave function is derived, applications
like the anharmonic oscillator, Wigner functions in phase space, and coherent
states are discussed. Hermite polynomials are orthogonal polynomials. Thus
all methods related to orthogonal polynomials can be additionally applied. The
corresponding MATLAB code for evaluating Hermite polynomials, respectively,
functions, computing their nodes and for all applications mentioned above can
be downloaded.
In Chap. 15, we started to discuss orthogonal polynomials. This chapter is intended
to introduce Hermite polynomials and some applications.
The harmonic oscillator is a particularly important quantum system and its
wave function in coordinate or momentum representation can be evaluated with the
help of Hermite polynomials. Thus, we discuss in addition the harmonic oscillator
eigenfunctions related to Wigner functions, the anharmonic oscillator, and coherent
states. I suppose there is no quantum dynamic textbook without discussing the
harmonic oscillator.
16.1 Function Overview
Hermite polynomial The SPECFUNPHYS class hermitepoly supports the
evaluation of the polynomial coefficients of Hermite polynomials. The
SPECFUNPHYS class hermitex allows the computation of the corresponding
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_16
199