Of harmonic oscillator

A 10,0-g mass connected by a spi itig to a statiotiaiy poitit executes exactly 4 complete cycles of harmonic oscillation in 1,00 s. What are the period of oscillation, the frequency, and the angular frequency What is the force constant of the spring ... 

Let us consider an example, that of the derivative operator in the orthonormal basis of Harmonic Oscillator functions. The fact that the solutions of the quantum Harmonic... 

Of particular interest is the model of a bath as a set of harmonic oscillators qj with frequencies cOj, which are linearly coupled to the tunneling coordinate... 

Finally, we have applied equation (10.14) to a collection of harmonic oscillators. But it can be applied to any collection of energy levels and units of energy with one modification. Equation (10.14) assumes that each level has an equal probability (as in a harmonic oscillator), and this is true only if g, the degeneracy, is one. The quantity g, is also known as the statistical weight factor. If it is greater than one, equation (10.14) must be multiplied by the g, for each... 

Finally, it is a weU-known result of quantum mechanics" that the wavefunctions of harmonic oscillators extend outside of the bounds dictated by classical energy barriers, as shown schematically in Figure 10.1. Thus, in situations with narrow barriers it can... 

Although Eqs. (4-1) and (4-2) have identical expressions as that of the classical rate constant, there is no variational upper bound in the QTST rate constant because the quantum transmission coefficient Yq may be either greater than or less than one. There is no practical procedure to compute the quantum transmission coefficient Yq- For a model reaction with a parabolic barrier along the reaction coordinate coupled to a bath of harmonic oscillators, the quantum transmission... 

In order to obtain an explicit expression for a(oo), we shall consider a system consisting of a collection of harmonic oscillators, that is,... 

The reason that AG/ = Ef — ) is due to the fact that our system consists of a collection of harmonic oscillators with displaced surfaces between the initial and final electronic states. In this case, the vibrational partition functions are the same between the initial and final electronic states. 

That is, we consider a system oscillator embedded in a bath of a collection of harmonic oscillators. The interaction between the system and heat bath is... 

To solve this equation, an appropriate basis set ( >.,( / ) is required for the nuclear functions. These could be a set of harmonic oscillator functions if the motion to be described takes place in a potential well. For general problems, a discrete variable representation (DVR) [100,101] is more suited. These functions have mathematical properties that allow both the kinetic and potential energy... 

The first term is the intrinsic electronic energy of the adsorbate eo is the energy required to take away an electron from the atom. The second term is the potential energy part of the ensemble of harmonic oscillators we do not need the kinetic part since we are interested in static properties only. The last term denotes the interaction of the adsorbate with the solvent the are the coupling constants. By a transformation of coordinates the last two terms can be combined into the same form that was used in Chapter 6 in the theory of electron-transfer reactions. 

Consider an ensemble of harmonic oscillators interacting linearly with an ion of charge number z, so that the potential energy of the system is given by ... 

This is not as useful as Eq. (19.4) because products of different coordinates appear in the second term. However, the symmetry properties of this term ensure the existence of a coordinate system in which the cross-terms can be eliminated and the nuclear Hamiltonian reduces to a sum of harmonic oscillator terms ... 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

In the harmonic approximation the functions Xi and Xf are products of harmonic oscillator functions. We therefore specify the initial state by a set of quantum numbers n — (ni, ri2,..., n/v), and those for the final state by m = (mi,m2,..., tun)- So the nuclear wavefunctions are henceforth denoted by Xi,n and Xf,m- Equation (19.21) tells us how to calculate the rate of transition from one particular initial quantum mode n to a final quantum state m. This is more than we want to know. All we are interested in is the total rate from any initial state to any final state. The ensemble of reactants is in thermal equilibrium therefore... 

Now it becomes apparent why it was useful to replace the delta function by its Fourier transform. The wavefUnctions Xin are products of harmonic oscillator functions, the Hamiltonians Hi and H/ are sums of harmonic oscillator terms. Therefore the terms in the brackets factorize in the form ... 

Fig. 6.2. Energy curve for Hooke s law versus Quantum Model of harmonic oscillator.

The equations rely on the assumption of harmonic oscillators and equal force constants in both the reactant and product states. Deviations from... 

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). 

In a first model, these motions are represented by harmonic vibrations, and the functions (Q) and Xbw (Q) are then replaced by products of harmonic oscillator-like wavefunctions. The solutions of Eqs. (9) take this particular form when the T jJ are negligible and when and H b can be expanded in terms of normal coordinates ... 

The strategy, usually adopted to achieve a theoretical description of this complex dynamics, is to describe the influence of the solvent environment on the electron-transfer reaction within linear response theory [5, 26, 196, 197] as linear coupling to a bath of harmonic oscillators. Within this model, all properties of the bath enter through a single function called the spectral density [5, 168]... 

The first attempts (G. Klein and I. Prigogine, 1953, MSN.5,6,7) were very timid and not very conclusive. They were devoted to a chain of harmonic oscillators. In spite of a tendency to homogenization of the phases, there was no intrinsic irreversibility here, because an essential ingredient is lacking in this model the interaction among normal modes. The latter were introduced as a small perturbation in the fourth paper of the series (MSN.8). 

Figure 3.1 Energy levels and wave functions of harmonic oscillator. Heavy line bounding potential (3.2). Light solid lines quantum-mechanic probability density distributions for various quantum vibrational numbers see section 1.16.1). Dashed lines classical probability distribution maximum classical probability is observed in the zone of inversion of motion where velocity is zero. From McMillan (1985). Reprinted with permission of The Mineralogical Society of America.

The quantum mechanical treatment of harmonic oscillators is described in essentially all books on quantum mechanics. Several good examples are ... 

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