Energy of harmonic oscillator

Balmer s formula, but- also to establish one of the most important results of quantum mechanics the quantization of angular momentum in units of A/2tt. This result arises from the analysis in his paper it is not his starting point. Bohr s quantum postulate was based on Planck s assumption of the quantization of the energy of harmonic oscillators. By analogy with this, and arguing from a correspondence with classical physics, he set a restrictive condition on the mechanically possible electron orbits, and postulated that this limited set of orbits should be non-radiating. 

As an introduction, rewrite the expression for the molar energy of harmonic oscillators, cited above, to specify the average molar energy of a set of oscillators vibrating at a single frequency v ... 

The averaged values of kinetic and potential energies of harmonic oscillations for the period are equal to half the total energy, i.e.. 

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... 

Figure 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

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... 

To simplify the expression of the thermal average rate W, f given by Eq. (3.32), we shall assume that both potential energy surfaces of the two excited vibronic manifolds of the DA system consist of a collection of harmonic oscillators, that 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 ... 

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 ... 

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

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.

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. 

The QRRK model postulates that vibrational energy can freely flow (internally) from one vibrational mode in the molecule to another. This is a very significant assumption. For a collection of harmonic oscillators, energy in a particular vibrational mode will stay in that mode it cannot flow into other vibrational modes of the system. That is, a system of harmonic oscillators is uncoupled. 

The first derivatives of a potential energy function define the gradient of the potential and the second derivatives describe the curvature of the energy surface (Fig. 3.4). In most molecular mechanics programs the potential functions used are relatively simple and the derivatives are usually determined analytically. The second derivatives of harmonic oscillators correspond to the force constants. Thus, methods using the whole set of second derivatives result in some direct information on vibrational frequencies. 

The free BC oscillator is assumed to be harmonic with force constant k and equilibrium separation r the parameter e controls the coupling between the dissociation coordinate R and the vibrational coordinate r. For e = 0 (elastic limit) the equations of motion for (R, P) and (r, p) decouple and energy cannot flow from one degree of freedom to the other. As a consequence, the vibrational energy of the oscillator remains constant throughout the dissociation and the corresponding vibrational excitation function, which for zero initial momentum po is given by... 

We consider the sum of states, density of states, and energies for a set of harmonic oscillators. The Hamiltonian for s harmonic oscillators of unit mass is... 

The procedure, known as second quantization, developed as an essential first step in the formulation of quantum statistical mechanics, which, as in the Boltzmann version, is based on the interaction between particles. In the Schrodinger picture the only particle-like structures are associated with waves in 3N-dimensional configuration space. In the Heisenberg picture particles appear by assumption. Recall, that in order to substantiate the reality of photons, it was necessary to quantize the electromagnetic field as an infinite number of harmonic oscillators. By the same device, quantization of the scalar r/>-field, defined in configuration space, produces an equivalent description of an infinite number of particles in 3-dimensional space [35, 36]. The assumed symmetry of the sub-space in three dimensions decides whether these particles are bosons or fermions. The crucial point is that, with their number indeterminate, the particles cannot be considered individuals [37], but rather as intuitively understandable 3-dimensional waves - (Born) -with a continuous density of energy and momentum - (Heisenberg). 

---