18. More Special Functions Hermite Functions Applications of Hermite Functions Laguerre Functions Chebyshev Polynomials Hypergeometric Functions Confluent Hypergeometric Functions Dilogarithm Elliptic Integrals

1. Hermite Functions Hermite ODE : Hermite functions Hermite polynomials ( n = integer ) Hermitian form  Rodrigues formula  Assumed starting point here. Generating function :

Recurrence Relations     All Hn can be generated by recursion.

Table & Fig. 18.1. Hermite Polynomials Mathematica

Special Values  

Hermite ODE   Hermite ODE

Rodrigues Formula   Rodrigues Formula

Series Expansion   For n odd, j & k can run only up to m 1, hence & consistent only if n is even   For n odd, j & k can run only up to m 1, hence &  

Schlaefli Integral  

Orthogonality & Normalization Let   

 

2. Applications of Hermite Functions Simple Harmonic Oscillator (SHO) : Let  Set  

 Eq.18.19 is erronous  

Fig.18.2. n Mathematica

Operator Appoach  see § 5.3 Factorize H : Let 

Set  or 

i.e., a is a lowering operator c = const   with i.e., a is a lowering operator   with i.e., a+ is a raising operator

Since  we have ground state  Set m = 0  with ground state   Excitation = quantum / quasiparticle : a+ a = number operator a+ = creation operator a = annihilation operator

ODE for 0 

Molecular Vibrations For molecules or solids : For molecules : For solids : R = positions of nuclei r = positions of electron Born-Oppenheimer approximation :  R treated as parameters Harmonic approximation : Hvib quadratic in R. Transformation to normal coordinates  Hvib = sum of SHOs. Properties, e.g., transition probabilities require m = 3, 4

Example 18.2.1. Threefold Hermite Formula i,j,k = cyclic permuation of 1,2,3 Triangle condition  for

Consider 

  

Hermite Product Formula 

Example 18.2.2. Fourfold Hermite Formula

3. Laguerre Functions

4. Chebyshev Polynomials

5. Hypergeometric Functions

6. Confluent Hypergeometric Functions

7. Dilogarithm

8. Elliptic Integrals