Harmonic oscillator orthogonality

Bigeleisen, J. and Ishida, T. Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. I. Reduced partition function of isotopic molecules, J. Chem. Phys. 48, 1311 (1968). Ishida, T., Spindel, W. and Bigeleisen, J. Theoretical analysis of chemical isotope fractionation by orthogonal polynomial methods, in Spindel, W., ed. Isotope Effects on Chemical Processes. Adv. Chem. Ser. 89, 192 (1969). 

A Schrodinger cat-like state is a superposition of two macroscopically distinguishable classical states, [Schrodinger 1935 (a)], which for the harmonic oscillator are represented by strongly excited and sufficiently well separated (thus orthogonal) coherent states. To evolve a coherent state into a superposition, we may apply a unitary operator... 

We shall discuss various states constructed in FD Hilbert space of harmonic oscillator. Let us denote by // " the (.v + l)-dimensional Hilbert space spanned by number states ( 0), 1 s) fulfilling the completeness and orthogonality... 

The matrix equation (18) is block-diagonal (as shown) only if the coupling modes, a = 2,5e, and 5a, are mutually orthogonal. The two-dimensional harmonic oscillator Hamiltonians for the a = 2 and 5e modes are given by... 

As an alternative that solves the kinetic coupling problem. Miller and co-work-ers suggested an all-Cartesian reaction surface Hamiltonian [27, 28]. Originally this approach partitioned the DOF into atomic coordinates of the reactive particle, such as the H-atom, and orthogonal anharmonic modes of what was called the substrate. If there are N atoms and we have selected reactive coordinates there will he Nyi = 3N - G - N-g harmonic oscillator coordinates and the reaction surface Hamiltonian reads... 

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... 

We introduce now an important characteristic of the harmonic oscillator. In addition to the orthogonality and existence of a normalization factor in the wavefunction, ip (x), the harmonic oscillator has the following interesting characteristic. 

The form of the approximate wavefunctions suggests another choice of basis for this problem, namely one comprising some harmonic oscillator functions centred about one minimum and additional harmonic oscillator functions centred about the other minimum. The only minor difficulty in this calculation is that the basis set is not orthogonal (which should be clear simply by inspecting the overlap of the ground-state harmonic oscillator wavefunctions centred at the two points) and an orthonormalization based on equation... 

Since m n, the first integral vanishes because J/ and ij/ are orthogonal. The normalized harmonic oscillator functions are given by Eq. (21.42),... 

To each value of q, a countable infinite set of characteristic values of a is associated for which x t) is an odd or even function that is njr-periodic in time, n being an integer. Series approximations for the characteristic values are obtained by expressing the integral-order Mathieu function as a series of harmonic oscillations, plugging the resultant expression into Eq. (20.8), and equating coefficients of each (orthogonal) frequency component to zero. These laborious calculations yield infinite series in q where each coefficient of q can be expressed as a continued Iraction [4]. 

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... 

Extending the inequalities to excited states, or to the sum of the first energies does not appear as an easy task. The minimax principle [33] is not of immediate use here for fixed r, the wave function of a ground-state baryon, o(r, r, Tj) and that of an excited state, < i(r, r, r ) are not orthogonal with respect to integration over r and For very special cases (harmonic oscillator, for instance) and for particular values of r, < o nd are not even linearly independent. 

We now calculate the first few terms in the transition moment (3.38) explicitly. The leading term fi Rg) v v y vanishes by orthogonality of the harmonic oscillator states. For the next (first-order) term, the substitution for q using Eq. 3.42 leads to... 

Fm basis sets corresponding to classical orthogonal pol3momials (harmonic oscillator functions, Qiebyshev polynomi, Legendre and associated L endre lynomials, etc.) the approximation 2.5 is equivalent to the approximatimi of V by Gaussian quadrature. For... 

The idea behind the DVR method [8-11] is to use a representation in terms of localized functions obtained by transformation from a global basis [12], Usually, bases constructed from orthogonal polynomials, noted F x), which are solution of one dimensional problems such as the particle in a box (Chebyshev polynomials) or the harmonic oscillator (Hermite polynomials), are used. These polynomial bases verify the general relationship... 

The first term is zero because of the orthogonality of the harmonic oscillator wave functions, and the integral in the second term is, because 9z = Q ... 

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