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

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

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

4. Table & Fig. 18.1. Hermite Polynomials
Mathematica

5. Special Values  

6. Hermite ODE  
Hermite ODE

7. Rodrigues Formula  
Rodrigues Formula

8. 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 &  

9. Schlaefli Integral  

10. Orthogonality & Normalization
Let   

11.  

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

13. 
Eq is erronous  

14. Fig n
Mathematica

15. Operator Appoach 
see § 5.3
Factorize H :
Let 

16. Set  or 

17. 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

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

19. ODE for 0 

20. 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

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

22. Consider 

23.   

24. Hermite Product Formula

25. Example 18.2.2. Fourfold Hermite Formula

26. 3. Laguerre Functions

27. 4. Chebyshev Polynomials

28. 5. Hypergeometric Functions

29. 6. Confluent Hypergeometric Functions

30. 7. Dilogarithm

31. 8. Elliptic Integrals