Harmonic oscillator expression, quantum mechanical

To find q brationai for a diatomic molecule, we shall use the idealization of a harmonic oscillator. The quantum mechanical energy level expression for a harmonic oscillator developed earlier in Equahon 7.34 yields... 

I he bawis of tho alternate method mentioned in Sec. 7-3 is the use of polar rather than cartesian degenerate normal coordinates. It is shown in treatises on quantum mechanics that the solution of the wave equation for the doubly degenerate harmonic oscillator expressed in terms of coordinates p and (j> defined by... 

We will use the harmonic oscillator approximation to derive an equation for the vibrational partition function. The quantum mechanical expression gives the vibrational energies as... 

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

Note that if j = 1, (9.12) is formally identical with the classical expression (9.7) the classical multiple oscillator model, which will be discussed in Section 9.2, is even more closely analogous to (9.12). However, the interpretations of the terms in the quantum and classical expressions are quite different. Classically, o30 is the resonance frequency of the simple harmonic oscillator quantum mechanically 03 is the energy difference (divided by h) between the initial or ground state / and excited state j. Classically, y is a damping factor such as that caused by drag on an object moving in a viscous fluid quantum mechanically, y/... 

There are also situations when one is not in the classical limit, and so Equation (13) would not seem applicable, and instead one would like to approximate one of the quantum mechanical expressions for Ti by relating the relevant quantum time-correlation function to its classical analog. For the sake of definiteness, let us consider the case where the oscillator is harmonic and the oscillator-bath coupling is linear in q, as discussed above. In this case k 0 can be written as... 

Despite its utility at room temperature, simple Marcus theory cannot explain the DeVault and Chance experiment. All Marcus reactions have a conspicuous temperature dependence except in the region close to where AG = —A. Marcus theory does not predict that a temperature-dependent reaction will shift to a temperature-independent reaction as the temperature is lowered. Hopfield proposed a quantum enhancement of Marcus theory that would permit the behavior seen in the experiment [11]. He introduced a characteristic frequency of vibration hco) that is coupled to electron transfer, in other words, a vibration that distorts the nuclei of the reactant to resemble the product state. This quantum expression includes a hyperbolic cotangent (Coth) term that resembles the Marcus expression at higher temperatures, but becomes essentially temperature independent at lower temperatures. Other quantized expressions, such as a full quantum mechanical simple harmonic oscillator behavior [12] and that of Jortner [13], give analogous temperature behavior. 

Moreover, when the probability flux correlation function decays on a time scale shorter than the time scale of the solvent effects on the R motion, the solvent effects on the R motion can be neglected. In this case, Cf,(t) can be approximated by the standard analytical expression for the time correlation function of an undamped quantum mechanical harmonic oscillator [53] ... 

Before presenting the quantum mechanical description of a harmonic oscillator and selection rules, it is worthwhile presenting the energy level expressions that the reader is probably already familiar with. A vibrational mode v, with an equilibrium frequency of (in wavenumbers) has energy levels (also in... 

The Landau-Teller model considers a linear collision of a structureless particle A with a harmonic oscillator BC within an approach which by now is known as a semiclassical method the relative particle-oscillator motion (coordinate R) is described classically and the vibrational motion of the oscillator (coordinate x) by quantum mechanics the interaction between incoming particle A and the nearest end B of the oscillator BC is taken to be exponential, I/(/ g) c exp(-aR g). The expression for the transition probability in the near-adiabatic limit was found [4] to have the following generic form ... 

The above expressions were traditionally presented only for the harmonic Brownian oscillator in which a harmonic system couples with a harmonic bath [40]. However, the fact that Eq. (E.3) and Eq. (E.4) are equivalent is generally true. We shall show it here only via some basic quantum mechanics principles as follows. 

Diatomic molecules are particularly easy to treat quantum-mechanically because they are easily described in terms of the classical harmonic oscillator. For example, the expression... 

If the normal mode is acting as an ideal harmonic oscillator, then we can use the quantum-mechanical expressions that describe its energy. Recall that for an ideal harmonic oscillator,... 

The creation and annihilation operators provide alternative forms for many quantum mechanical expressions, and they are used widely for phonons (vibrational quanta) as well as photons. Eor example, the Hamiltonian operator for an harmonic oscillator can be written... 

The quantum mechanical information that follows from a normal mode analysis must reveal the same mechanical equivalence to a set of disconnected oscillators as the classical analysis. Each such oscillator (normal mode of vibration) can exist in any of the states possible for a one-dimensional harmonic oscillator. Each has its own contribution to the energy of the system, and thus, the Hamiltonian in Equation 7.35 corresponds to a quantum mechanical energy level expression... 

This Hamiltonian is that of a separable problem, and as we have already considered in Chapter 7, it is equivalent to a problem of 3N - 6-independent harmonic oscillators. The vibrational frequencies of the oscillators are the (o/s. The Schrodinger equation that develops from the quantum mechanical form of this Hamiltonian is also separable. The energy level expression comes from the sum of the eigenenergies of the separated harmonic oscillators or modes. For each, there is a quantum number, n,. 

Spreadsheet problem] (a) Evaluate cj imitonai ir Equation 11.46 for a quantum mechanical harmonic oscillator system for which the energies are E (kJ moh ) = 23.0 (n +1/2) at 20 temperatures from 10 to 1,000K. (b) Evaluate q it,raHonai the energy expression includes an anharmonidty term -0.1(n + 1/2). Plot the harmonic and anharmonic values of cimbranonai 3 function of temperature. 

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