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18. More Special Functions

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

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18. More Special Functions

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  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 Hncan 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 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 Orthogonal Let   

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

  13. Eq.18.19 is erronous  

  14. Fig.18.2. n Mathematica

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

  16. Set  or 

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

  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 : Hvibquadratic 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  for Triangle condition i,j,k= cyclic permuation of 1,2,3  for

  22. Consider

  23.  

  24. Hermite Product Formula  Set Range of  set by q!  q  0

  25. Mathematica

  26. Example 18.2.2.Fourfold Hermite Formula  Mathematica 

  27. Product Formula with Weight exp(a2 x2) Ref: Gradshteyn & Ryzhik, p.803

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