Matrix elements harmonic oscillator

To evaluate these integrals, we start with the harmonic-oscillator matrix element (3.61), which reads in the notation of the present chapter... 

The harmonic oscillator matrix elements for a normal coordinate displacement, Q, and the conjugate momentum, P, operators are converted into dimensionless quantities, Q and P,... 

Here v is the frequency of the exciting light and Ar the Raman shift or the frequency of the vibration. The factor containing Av in the denominator is the harmonic oscillator matrix element squared,... 

The corresponding level broadening equals half. In fact is the diagonal kinetic coefficient characterizing the rate of phonon-assisted escape from the ground state [Ambegaokar 1987]. In harmonic approximation for the well the only nonzero matrix element is that with /= 1,K0 Q /> = <5o, where is the zero-point spread of the harmonic oscillator. For an anharmonic potential, other matrix elements contribute to (2.52). 

Next we discuss the effect of deuteratlon on low frequency modes Involving the protons> Because of the anharmonlc variation of the energy as a function of tilt angle a (Fig. 4b), the hindered rotations of H2O and D2O turn out to be qualitatively different. The first vibrational excited state of H2O Is less localized than that of D2O, because of Its larger effective mass. The oscillation frequency of the mode decreases by a factor 1.19 and the matrix elements by a factor 1.51 upon deuteratlon. Therefore, the harmonic approximation, which yields an Isotopic factor 1.4 for both the frequency and the Intensity, Is quite Inappropriate for this mode. 

In the application to an oscillator of some quantum-mechanical procedures, the matrix elements of x" and p" for a harmonic oscillator are needed. In this section we derive the matrix elements n x n), n x n), n p n), and n p n), and show how other matrix elements may be determined. 

From equations (4.34) and the orthonormality of the harmonic oscillator eigenfunctions n), we find that the matrix elements of a and are... 

The matrix elements n x n) for the unperturbed harmonic oscillator are given by equations (4.50). The first-order correction term is obtained by substituting equations (9.50) and (4.50e) into (9.24), giving the result... 

Some of the more useful matrix elements for the harmonic oscillator are presented in the following table. They are given as functions of the dimensionless quantities = Inx vmfh and a = 2sf /tv, as defined in Section 6.2. 

The eigenvalues of this Hamitonian can calculated by numerical diagonalization of the truncated matrix of the quantum system in the basis of the harmonic oscillator wave functions. The matrix elements of Hq and V are... 

In Eq. (17) the matrix element (q exp -i(Hq + f)t q) is formally equivalent to that of a harmonic oscillator in an electric field, a problem whose analytic solution is well known. This fact enables the reduction of the multi-dimensional integrations over q in Eq. (17) to a product of 1-dimensional integrations. 

The derivation above may be generalized to wave functions other than electronic ones. By evaluation of transition dipole matrix elements for rigid-rotor and harmonic-oscillator rotational and vibrational wave functions, respectively, one arrives at the well-known selection rules in those systems that absorptions and emissions can only occur to adjacent levels, as previously noted in Chapter 9. Of course, simplifications in the derivations lead to many forbidden transitions being observable in the laboratory as weakly allowed, both in the electronic case and in the rotational and vibrational cases. 

Fermi resonance occurs in C02. We note from (6.100) that 2vl e (= 1346 cm-1) is very close to e (= 1354 cm-1). Hence harmonic-oscillator levels of C02 for which 2v + v 2 = 2vx +1>2 (where the primed and unprimed quantum numbers refer to different vibrational levels) are quite close together and we have Fermi resonance. For example, (6.99) and (6.100) predict the levels (10°0), (02°0), and (0220) to lie 1335, 1339, and 1335 cm-1, respectively, above the ground vibrational level the observed spectrum (Table 6.2) shows these levels actually lie 1388, 1285, and 1335 cm-1, respectively, above the ground level (Problems 6.20 and 6.21). Clearly, the (10°0) and (02°0) levels have interacted with each other, thus shifting their energies. The (0220) level is unaffected because the matrix element H j is zero if states i and j have different values of / (Problem 6.22) Fermi resonance occurs only between states of the same symmetry. Fermi resonance between two levels increases the energy of the upper level and decreases the energy of the lower level the levels repel each other. 

Figure 4. Time dependence of selected density matrix elements for a harmonic oscillator obtained using the full Redfleld tensor. The oscillator is described by < > = 100 cm-1, 7 1(1 - 0) = 2.0 ps, and 7 2(A = 1) = oo, where n denotes vibrational levels. The system is initially prepared in a superposition of levels 6 and 7. (a) p T, (b) P34 (c) poi (d) dashed line, P66 and the solid line. P77. (From Ref. 24.)...

Since the selection rule for nonzero Qi and Pi matrix elements in the harmonic oscillator basis is Av = 1, and since the definition of a polyad is such that all pairs of states differing by only one vibrational quantum number... 

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. 

The evaluation of the vibrational matrix elements is quite straightforward however, the problem has been formulated in terms of the amplitude of vibration with respect to the centre of gravity of the oscillator, so that a transformation of the usual oscillator wave functions is required. For a diatomic harmonic-oscillator... 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

In this last equation, the right-hand side matrix elements are those of the IP time evolution operator of the driven damped quantum harmonic oscillator describing the H-bond bridge when the fast mode is in its first excited state ... 

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