Harmonic oscillator integrals involving

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

Let r and be the irreducible representations to which the vibrational wave functions in (9.189) belong. According to the italicized statement following (9.187), the integral (9.189) vanishes unless T+<8>riratls contains T, . This is the basic IR selection rule. (Note that its deduction does not involve the harmonic-oscillator approximation or the approximation of... 

The matrix element F0>1 is derived below for the harmonic oscillator, but evaluation of T involves a lengthy integration of equation (12) to obtain F (x), and we accept the result of Jackson and Mott. 

We have demonstrated here that for the one-dimensional harmonic oscillator the integral required in the Euler equation, involving the functional derivative 8t/8p, can be exactly expressed in terms of the total kinetic energy. Indeed, the relation, involving a factor of 3, is exactly that given by the TF statistical theory. This latter theory gives for the density in d dimensions... 

The reaction path Hamiltonian is particularly useful for evaluating the path integral representation of the trace, Eq. (3.1 ), because the flux operator F does not involve the bath" degrees of freedom (i.e., the transverse vibrational modes Q ), and since they are harmonic oscillators the path integrals over them can be carried out analytically. All that remains to be done numerically is the path integral over only the reaction coordinate degrees of freedom itself. 

HERMITE POLYNOMIALS AND SOME INTEGRALS INVOLVING THE HARMONIC OSCILLATOR WAVE FUNCTIONS... 

The normalization constants for the harmonic oscillator wavefunctions follow a certain pattern (because the formulas for the integrals involve Hermite polynomials) and so can be expressed as a formula. The general formula for the harmonic oscillator wavefunctions given below includes an expression for the normalization constant in terms of the quantum number n ... 

Simply using arguments based on odd or even functions, determine whether the following integrals involving harmonic oscillator wavefunctions are identically zero, are not identically zero, or are indeterminate. If indeterminate, state why. 

The quantities lnd>(r (r)) are defined as the quotients (-Sp,lh), where is the so-called action for the problem under consideration and involves an integration of kinetic and potential contributions over the period 0dimensionless quantity - In (r" (t)), its relation to the product of the density matrix elements in Eqs. (14) and (16) being clear [28]. A few simple examples (e.g., free particle and harmonic oscillator) admit the exact application of the PI formahsm in the P t form [12, 13], but for general many-body quantum systems this is not possible. However, some analytic developments related to Eq. (15) have given rise to the so-called Feynman s semiclassical approaches, which will be considered in Section 111. To exploit the power of the PI formahsm computational schemes utilize finite-P discretizations. In this regard, given that approximations to calculate density matrix... 

If, e.g., V(r) is a harmonic or Morse oscillator, or the Coulomb potential, as for the hydrogen atom, eq. (7) can be solved analytically, otherwise numerical solution is necessary, but involves only a one-dimensional integration and nowadays can be carried out routinely when needed, as for the analysis of elastic scattering experiments in atomic and nuclear physics. 

---