Wave function harmonic oscillator

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

Some of the Hermite polynomials and the corresponding harmonic-oscillator wave functions are presented in Thble 1. The importance of the parity of these functions under the inversion operation, cannot be overemphasized. 

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... 

Other types of radial functions have been applied, including Gaussian-type functions (Stewart 1980), and harmonic oscillator wave functions (Kurki-Suonio 1977b). 

While Eq. (9.49) has a well-defined potential energy function, it is quite difficult to solve in the indicated coordinates. However, by a clever transfonnation into a unique set of mass-dependent spatial coordinates q, it is possible to separate the 3 Ai-dirncnsional Eq. (9.49) into 3N one-dimensional Schrodinger equations. These equations are identical to Eq. (9.46) in form, but have force constants and reduced masses that are defined by the action of the transformation process on the original coordinates. Each component of q corresponding to a molecular vibration is referred to as a normal mode for the system, and with each component there is an associated set of harmonic oscillator wave functions and eigenvalues that can be written entirely in terms of square roots of the force constants found in the Hessian matrix and the atomic masses. 

Now consider a one-dimensional harmonic oscillator of charge q. We must evaluate q(m x n). The harmonic-oscillator wave functions are given by (1.133). Using (1.133) and the Hermite-polynomial identity (1.138) with z = a 2x, we have... 

Consider the integral (m x2 n, where the wave functions are onedimensional harmonic-oscillator wave functions. For what values of m — n is this integral nonzero ... 

For the three-dimensional harmonic oscillator, V = kxx2 + h.kyy2+ kzz2. Separation of variables gives the wave function as the product of three one-dimensional harmonic oscillator wave functions, and gives the energy as... 

Tanczos35 has extended the theory (for V-T and V-V transfer) to polyatomic molecules, and a detailed comparison with experiment was recently given by Stretton33. Considering each surface atom, energy transfer depends on how the intermolecular potential varies with the oscillation of the atom. In deriving the result for the diatomic molecule from the harmonic-oscillator wave functions, we substituted... 

For the reflection symmetric two-level electron-phonon models with linear coupling to one phonon mode (exciton, dimer) Shore et al. [4] introduced variational wave function in a form of linear combination of the harmonic oscillator wave functions related with two levels. Two asymmetric minima of elfective polaron potential turn coupled by a variational parameter (VP) respecting its anharmonism by assuming two-center variational phonon wave function. This approach was shown to yield the lowest ground state energy for the two-level models [4,5]. 

The application ofc Tyir to the analysis of the vibrational-inversion-rotation spectra of ammonia will be discussed in detail in Sections 5.1-5.4. Here we mention only that if the interaction between the inversion, vibration and rotation states is neglected, the overall wave function pvit can be written as a product of the harmonic oscillator wave functions the inversion wave function, p), and the symmetric rotor wave function Sj/cm( > 4>) exp (i/cx) ... 

Here, Xni l) denotes the harmonic oscillator wave-functions, (pa(r) and (pb(r) correspond to the states of electrons localized on a and b ions, the index v numbers the hybrid cluster states in the molecular field. It should be noted that, within the scope of the adopted approach, the quantum properties of the vibronic states in a self-consistent field are taken into account. Therefore, it is reasonable to call the proposed approximation quasidynamical. The vibronic states obtained within the scope of the quasidynamical approach are hybrid, i.e. retaining the quantum properties of both electronic and vibrational states. In the case of strong vibronic coupling, i.e. in the case of adiabatic potentials possessing deep minima both the... 

Let s return to the characteristics of the harmonic oscillator wave functions (see Fig. 4.31). There are a number of interesting features to point out. 

The harmonic oscillator wave functions explain how a particle can tunnel through a potential barrier to arrive at regions forbidden by classical mechanics. 

The vibronic wave functions for the LnX63 system are written as Herz-berg-Teller products of harmonic oscillator wave functions and electronic wave functions. We consider an absorption transition from the zeroth vibrational level of the electronic ground state to the /-th electronic excited state accompanied by the excitation of one quantum of the v-th odd-parity normal mode Qv (i.e. the occupation number k(v)=1). The vibrational wavefunctions are then Xoo and Xfi f°r the ground and/-th excited electronic state, respectively. The EDV transition element of the q-th component of the moment M00>fi in Eq. (37) for the 00—>/l transition can be written in terms of static and dynamic coupling parts ... 

Wc). The initial wave function ) =, ) r) t) where the is the quantum number associated with the harmonic oscillator wave function for the strong molecular bond ( =i in fig. 10.7), is the rotational quantum number associated with the rotational wave function (not included in the one-dimensional picture of fig. 10.7), and is the quantum number associated with the vibrational wave function for the van der Waals bond ( Pv in fig. 10.7). A Morse potential is assumed for this latter interaction potential. For the final state, the wave function is f) = 4f) 7) t)> where f) is the quantum number associated with the final state of the strong bond ( I>v"=o fig- 10-7), the rotational quantum number, J, represents the rotational wave function (now a free... 

The ground-state wave function for a proton in the double well shown in fig. 9.12(a) can be approximately represented by a linear combination of two ground-state harmonic oscillator wave functions and centred about the points A and B respectively. Thus + (9.63)... 

In order to quantify the intensity of an electronic band the oscillator strength f is used. This dimensionless quantity relates the observed integrated intensity to that of a calculated integrated intensity according to a simple model where the excited electron is attracted to the center of the molecule with a Hooke s law type force, i.e., using harmonic oscillator wave functions, f is the ratio of observed and calculated integrated intensities and amounts to... 

FIGURE 4.3 Harmonic-oscillator wave functions.Hie same scale is used for all graphs. The points marked on the x axes are for = 2. 

The polynomial factors in the harmonic-oscillator wave functions are well known in mathematics and are called Hermite polynomials, after a French mathematician. (See Problem 4.19.)... 

Verify the normalization factors for the t = 1 and v = 2 harmonic-oscillator wave functions. 

Use the recursion relation (4.48) to find the w = 3 normalized harmonic-oscillator wave function. 

Point out the similarities and differences between the one-dimensional particle-in-a-box and the harmonic-oscillator wave functions and energies. 

Verify this identity for n = 0,1, and 2. (c) The normalized harmonic-oscillator wave functions can be written as (Pauling and Wilson, pages 79-80)... 

Use the normalized Numerov-method harmonic-oscillator wave functions found by going from —5 to 5 in steps of 0.1 to estimate the probability of being in the classically forbidden region for the v = 0 and v = l states. 

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