Hermite orthonormal function

An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, hamonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the bajectoiy, and the idea has been taken up by Billing [151,152]. The basic problem with this approach is the slow convergence of the basis set. 

Now let us see how an approximate form for the memory function for F k,t), i.e., Ki, k,t) in Eq. (5.23), comes about on the basis of the results of the generalized kinetic theory. This can be done by relating the phase-space correlation function to more familiar ones, such as F k,t), Ch k,t) and Ct(A , t). For this purpose it is convenient to switch from a continuous to a discrete matrix representation of the phase-space correlation function by introducing a complete set of orthonormal momentum functions these are generally chosen to be the Hermite... 

Equation of quantum state. The Dirac bra-ket formalism of quantum mechanics. Representation of the wave-momentum and coordinates. The adjunct operators. Hermiticity. Normal and adjunct operators. Scalar multiplication. Hilbert space. Dirac function. Orthogonality and orthonormality. Commutators. The completely set of commuting operators. 

Use again the functions x" to construct (i) polynomials orthonormal in the interval (0, ) with weight factor e" and (ii) polynomials orthonormal in (-00, +oo) with weight factor These are the polynomials of Laguerre and Hermite respectively. Now define the Laguerre and Hermite functions, which are orthonormal in the usual sense but for the infinite intervals. 

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