18 More Special Functions 1 Hermite Functions 2

18. More Special Functions 1. Hermite Functions 2. Applications of Hermite Functions 3. Laguerre Functions 4. Chebyshev Polynomials 5. Hypergeometric Functions 6. Confluent Hypergeometric Functions 7. Dilogarithm 8. Elliptic Integrals

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

Recurrence Relations All Hn can be generated by recursion.

Table & Fig. 18. 1. Hermite Polynomials Mathematica

Special Values

Hermite ODE

Rodrigues Formula

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

Schlaefli Integral

Orthogonality & Normalization Orthogonal 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

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 Excitation = quantum / quasiparticle : a+ a = number operator a+ = creation operator a = annihilation operator with ground state

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 Properties, e. g. , transition probabilities require Hvib = sum of SHOs. m = 3, 4

Example 18. 2. 1. Threefold Hermite Formula for i, j, k = cyclic permuation of 1, 2, 3 Triangle condition

Consider

Hermite Product Formula Set Range of set by q! q 0

Mathematica

Example 18. 2. 2. Fourfold Hermite Formula Mathematica

Product Formula with Weight exp( a 2 x 2) Ref: Gradshteyn & Ryzhik, p. 803