Potential energy, of a harmonic oscillator

Thus, the potential energy of a harmonic oscillator is given by... 

The potential energy of a harmonic oscillator is proportional to the square of the displacement from the equilibrium position. The restoring force exerted by the two atoms of a molecule on each other when they are displaced from their equilibrium position (Te) is approximately proportional to the change of internuclear distance... 

To simplify our presentation, we ignore the vector characteristic of the position and focus on its magnitude, x. Because x is squared in the expression for V, negative values of x don t need to be treated in any special fashion. The resulting working equation for the potential energy of a harmonic oscillator is more simply written as... 

Obtain a formula for the expectation value of the potential energy of a harmonic oscillator in the u = 1 state. How does this relate to the total energy of the harmonic oscillator in this state ... 

The potential energy function for a chemical bond is far more complex than a harmonic potential at high energies, as discussed in Chapter 3. However, near the bottom of the well, the potential does not look much different from the potential for a harmonic oscillator we can then define an effective force constant for the chemical bond. This turns out to be another problem that can be solved exactly by Schrodinger s equation. Vibrational energy is also quantized the correct formula for the allowed energies of a harmonic oscillator turns out to be ... 

The potential energies of two harmonic oscillators with frequencies oi.(. and

The second term on the right-hand side of equation (10.50) is the energy of a harmonic oscillator. Since the factor in equation (10.51a) depends on the third and fourth derivatives of the intemuclear potential at R, the third term in equation (10.50) gives the change in energy due to the anharmonicity of that potential. The fourth term is the energy of a rigid rotor with moment of inertia... 

Just as we corrected the expressions for the rigid rotor to allow for the centrifugal effect and an interaction with the vibration, we also must adjust the expression for the harmonic oscillator to account for the anharmonicity in the oscillation. The potential energy surface for the molecule is not symmetrical (Fig. 25.2). The parabola (dotted figure) represents the potential energy of the harmonic oscillator. The correct potential energy is shown by the full lines the vibration is anharmonic. The vibrational energy levels for such a system can be approximated by a series ... 

In addition, Dirac noted that Eq. (3.48) is identical to the classical expression for the energy of a harmonic oscillator with unit mass (Eq. 2.28). The first term in the braces corresponds formally to the kinetic energy of the oscillator the second, to the potential energy. It follows that if we replace Pj and Qj by momentum and position operators Pj and Q, respectively, the eigenstates of the Schrodinger equation for electromagnetic radiation will be the same as those for harmonic oscillators. In particular, each oscillation mode will have a ladder of states with wavefunctions r y and energies... 

The statistical treatment of the vibrational degrees of freedom of crystals is far more difficult compared to gases. Let us initially consider a monoatomic crystal. An atom in a crystal vibrates about its equilibrium lattice position. In the simplest approach, three non-interacting superimposed linear harmonic oscillators represent the vibrations of each atom. The total energy, given by the sum of the kinetic and potential energies for the harmonic oscillators, is... 

The discrete energy levels sketched as horizontal lines on each potential curve of Figure 5.10 are consistent with the quantized energy levels (phonon levels) of a harmonic oscillator. For each harmonic oscillator at frequency 12, the permitted phonon energies are given by... 

Our discussion of vibrations has all been within the context of the harmonic approximation. When using this approach, each vibrational mode can be thought of as being defined by a harmonic oscillator. The potential energy of a one-dimensional harmonic oscillator is... 

So far we have illustrated the classic and quantum mechanical treatment of the harmonic oscillator. The potential energy of a vibrator changes periodically as the distance between the masses fluctuates. In terms of qualitative considerations, however, this description of molecular vibration appears imperfect. For example, as two atoms approach one another, Coulombic repulsion between the two nuclei adds to the bond force thus, potential energy can be expected to increase more rapidly than predicted by harmonic approximation. At the other extreme of oscillation, a decrease in restoring force, and thus potential energy, occurs as interatomic distance approaches that at which the bonds dissociate. 

The potential energy of such an oscillator can be plotted as a function of the separation r, or, for a normal mode in a polyatomic molecule, as a function of a parameter characterizing the phase of the oscillation. For a simple harmonic oscillator, the potential energy function is parabolic, but for a molecule its shape is that indicated in Figure 2.6. The true curve is close to a parabola at the bottom, and it is for this reason that the assumption of simple harmonic motion is justified for vibrations of low amplitude. 

Even if there is no electromagnetic field present, the vector potential exhibits fluctuations A = (4 ) + 84, so that even if there is only the vacuum, physics still involves this fluctuation. This is also seen in the zero-point energy of the harmonic oscillator expansion of the fields. So an electron will interact with virtual photons. If we represent all of these interactions as a blob coupled to the path of an electron, this blob may be expanded into a sum of diagrams where the electron interacts with photons. Each term is an order expansion and contributes... 

Chapter 3. This predicts that for a small enough distortion, the potential energy function looks essentially like that of a harmonic oscillator ... 

Although the vibrational motion of a diatomic molecule conforms quite closely to that of a harmonic oscillator, in practice the anharmonic deviations are quite significant and must be taken into account if vibrational energy levels are to be modelled accurately. A general form of the potential fimction V in equation (2.157) was proposed by Dunham... 

Valence-Force Model. The simplest harmonic oscillator model for the potential energy (/of a tetrahedral molecule such as CCL can be written as... 

The simplest model of a vibrating diatomic molecule is a harmonic oscillator, for which the potential energy depends quadratically on the change in intemuclear distance. The allowed energy levels of a harmonic oscillator, as calculated from quantum mechanics, are... 

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