Harmonic oscillator Hermite equation

It is customary to express the eigenfunctions for the stationary states of the harmonic oscillator in terms of the Hermite polynomials. The infinite set of Hermite polynomials // ( ) is defined in Appendix D, which also derives many of the properties of those polynomials. In particular, equation (D.3) relates the Hermite polynomial of order n to the th-order derivative which appears in equation (4.39)... 

In the last equation Hi(x) is the th Hermite polynomial. The reader may readily recognize that the functions look familiar. Indeed, these functions are identical to the wave functions for the different excitation levels of the quantum harmonic oscillator. Using the expansion (2.56), it is possible to express AA as a series, as has been done before for the cumulant expansion. To do so, one takes advantage of the linearization theorem for Hermite polynomials [42] and the fact that exp(-t2 + 2tx) is the generating function for these polynomials. In practice, however, it is easier to carry out the integration in (2.12) numerically, using the representation of Po(AU) given by expressions (2.56) and (2.57). 

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

The polynomials defined here are different from the Hermite polynomials which occur in the solutions of the Schrodinger equation for the harmonic oscillator. 

They are related to the Hermite polynomials HN(x), which are the solutions to the Schrodingen equation for a harmonic oscilator by... 

A comparison of equation (G.17) with (D.10) shows that the solutions a(f) are the Hermite polynomials, whose properties are discussed in Appendix D. Thus, the functions [Pg.323]

To show that the functions we have defined above are the same as those used in the solution of the harmonic oscillator problem, we look for the differential equation satisfied by // ( ). It is first convenient to derive certain relations between successive Hermite polynomials and their derivatives. We note that since S = its partial derivative with respect... 

On application of this equation it is found that the momentum wave functions for the harmonic oscillator have the same form (Hermite orthogonal functions) as the coordinate wave functions (Prob. 64-1), whereas those for the hydrogen atom afe quite different.1... 

Such a possible feature can be found, as an example, within a typical set of solutions of the Schrodinger equation. The harmonic oscillator provides an obvious particular case of such an EH space. It is well known that harmonic oscillator solutions constitute the set of Hermite polynomials [73], weighted by a gaussian function [65]. These polynomials can be considered related to the GTO basis functions most widely used in contemporary Quantum Chemistry. First derivatives of Hermite polynomials are always well defined, producing another polynomial of the same kind. 

In the q-coordinate system, the vibrational normal coordinates, the SA atom-dimensional Schrodinger equation can be separated into SA atom one-dimensional Schrodinger equations, which are just in the form of a standard harmonic oscillator, with the solutions being Hermite polynomials in the q-coordinates. The eigenvectors of the F G matrix are the (mass-weighted) vibrational normal coordinates, and the eigenvalues ( are related to the vibrational frequencies as shown in eq. (16.42) (analogous to eq. (13.31)). 

The solutions of the harmonic oscillator equation, Eq. (6), Sec. 3-1, have been described in many places. They are called the Hermite orthogonal functions and are of the form... 

The following equations give several of the Hermite polynomials, H z), z = y Q, in terms of which the harmonic oscillator wave functions are expressed in Eq. (1), Sec. 3-3. Additional explicit forms may be readily evaluated by use of the recursion formula, Eq. (4), Sec. 3-3. 

It turns out that the set of harmonic oscillator wavefunctions were already known. This is because differential equations like those of equation 11.6, the rewritten Schrodinger equation, had been studied and solved mathematically before quantum mechanics was developed. The polynomial parts of the harmonic oscillator wavefunctions are called Hermite polynomials after Charles Hermite, the nineteenth-century French mathematician who studied their properties. For convenience, if we define (where is the Greek letter xi, pronounced... 

We have found the ground state for the oscillator, but there are also other functions that can satisfy the Schrodinger equation. These will represent the excited states of the oscillator and become important when the molecule absorbs energy from the light used in IR spectroscopy. The complete set of solutions for the wavefunctions of the harmonic oscillator are actually a product of the Gaussian function discussed above and a Hermite polynomial which ensures that the cancellation of the function we forced by one choice of fi in Equations (A6.21)-(A6.23) also occurs for the excited states. The general solution for state n of the harmonic oscillator is then... 

The Time-Independent Schrodinger Equation for the Harmonic Oscillator (the Hermite Equation)... 

Using equations (28) and (29) we obtain a set of equations for the new expansion coefficients B . The new expansion has better convergence properties than the power series expansion. The functions (x) are harmonic oscillator wave-functions. Thus the hermite expansion involves an expansion around the classical path in a harmonic oscillator basis set. The hermite-corrected GWP can be used in a basis set expansion so as to approach the exact quantum theory fi"om the classical path limit in a systematic fashion. Hence in this sense the method is complementary to the multitrajectory approach discussed below. The advantage of the hermit basis is however that it is an orthorgonal basis and therefore somewhat simpler to work with in practical computations. ... 

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