The Classical Harmonic Oscillator

The classical harmonic oscillator is a repetitive motion that follows Hooke s law. For some mass m, Hooke s law states that for a one-dimensional displacement x from some equilibrium position, the force F acting against the displacement (that is, the force that is acting to return the mass to the equilibrium point) is proportional to the displacement  

The potential energy, denoted V, of a Hooke s-law harmonic oscillator is related to the force by a simple integral. The relationship and final result are 

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 

Classically, the behavior of the ideal harmonic oscillator is well known. The position of the oscillator versus time, x t), is 

It takes a certain time, t seconds, for the oscillator to complete one full cycle. Therefore, in 1 second, there will be 1/t oscillations. In a sinusoidal motion, one cycle corresponds to an angular change of 27T. The frequency of the oscillator in number of cycles per second or simply 1/second (s another Sl-approved name for s is hertz, or Hz) is defined as v (Greek nu) and is equal to 

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The atomic harmonic oscillator follows the same frequency equation that the classical harmonic oscillator does. The difference is that the classical harmonic oscillator can have any amplitude of oscillation leading to a continuum of energy whereas the quantum harmonic oscillator can have only certain specific amplitudes of oscillation leading to a discrete set of allowed energy levels. 

A simple eigenvalue problem can be demonstrated by the example of two coupled oscillators. The system is illustrated in fug. 2. It should be compared with the classical harmonic oscillator that was treated in Section 5.2.2. Here also, the system will be assumed to be harmonic, namely, that both springs obey Hooke s law. The potential energy can then be written in the form... 

It should be evident that the expressions for the Laplace transforms of derivatives of functions can facilitate the solution of differential equations. A trivial example is that of the classical harmonic oscillator. Its equation of motion is given by Eq. (5-33), namely,... 

The classical harmonic oscillator in one dimension was illustrated in Seetfon 5.2.2 by the simple pendulum. Hooke s law was employed in the fSfin / = —kx where / is the force acting on the mass and k is the force constant The force can also be expressed as the negative gradient of a scalar potential function, V(jc) = for the problem in one dimension [Eq. (4-88)]. Similarly, the three-dimensional harmonic oscillator in Cartesian coordinates can be represented by the potential function... 

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). 

The inner-sphere component of the reorganization energy represents the minimum energy required to change the internal structure of the redox center to its nuclear transition state configuration. Equation (2.3) is derived from the classical harmonic oscillator model and is an expression of the free energy associated with... 

The Nose-Hoover thermostat exhibits non-ergodicity problems for some systems, e.g. the classical harmonic oscillator. These problems can be solved by using a chain... 

First, in contrast with the classical harmonic oscillator, the quantum harmonic oscillator in its ground state is most likely to be found at its equilibrium position. A classical harmonic oscillator spends most of its time at the classical turning points, the positions where it slows down, stops, and reverses directions. (It might... 

Particle-in-a-box models and the qnantnm harmonic oscillator illustrate a number of important features of quantum mechanics. The energy level structure depends on the natnre of the potential in the particle in a box, E n, whereas for the harmonic oscillator, E n. The probability distributions in both cases are different than for the classical analogs. The most probable location for the particle-in-a-box model in its gronnd state is the center of the box, rather than uniform over the box as predicted by classical mechanics. The most probable position for the quantum harmonic oscillator in the ground state is at its equilibrium position, whereas the classical harmonic oscillator is most likely to be fonnd at the two classical turning points. Normalization ensures that the probabilities of finding the particle or the oscillator at all positions add np to one. Finally, for large values of n, the probability distribution looks mnch more classical, in accordance with the correspondence principle. 

Fio. 11-3.—The wave functions n( ), n = 1 to 6, for the harmonic oscillator. For each case the heavy horizontal line indicates the region traversed by the classical harmonic oscillator with the same total energy. 

Pio. 11-4.—The probability distribution function ( io( )]2 for the state n 10 of the harmonic oscillator. Note how closely the function approximates in its average value the probability distribution function for the classical harmonic oscillator with the same total energy, represented by the dashed curve. 

Consider the loss of an atom from a polyatomic molecule (e.g. C4H9I —> C4H9+I) for which r = 3 so that the rotational density of states is just a constant. The translational. density of states is also a constant because of the two translational degrees of freedom. In order to solve the problem analytically, we use the classical harmonic oscillator density of states which is... 

Expressing this distribution in terms of the classical harmonic oscillator density of states permits a comparison of this translational PED with the prior PED of Eq. (9.14). This classical translational PED, based on the transition state energy partitioning, is given by... 

Goldhaber and Teller who suggested that the origin of the resonance was the classical harmonic oscillation of the protons in the nucleus with respect to the neutrons. Numerical estimates give approximately the correct position of the peak of the giant resonance. Similar but more elaborate calculations have been made Steinwedel and Jensen and by Danos. ... 

The probability function for the harmonic oscillator with v= 10, showing how the potential energy for the quantum mechanical harmonic oscillator approaches that of the classical harmonic oscillator for very large values of v. [ Michael D Payer, Elements of Quantum Mechanics, 2001, by permission of Oxford University Press, USA.]... 

Figure 14.12a demonstrates the Rietveld refinement pattern for time-of-flight (TOF) neutron diffraction data measured at room temperature for LiFeP04. Fitting was satisfactory (/ p = 2.66%, Rf - 0.46%, 5=1.34) with accurately refined atomic positions as well as anisotropic atomic displacement parameters for all atoms under the classical harmonic oscillation model. 

Comparison between (2.38) and the high barrier limit excitation energy equation (2.34) leads first to the satisfying conclusion that the angular oscillator problem excitation energy does indeed approach the classical harmonic oscillator value for very high barriers, i e., for... 

FIGURE 4.1 The classical harmonic oscillator. is the equilibrium bond length. 

The vibrational motion of the nuclei in a molecule is approximately described by Equation 4.30, the equation for the classical harmonic oscillator. The bond length is oscillating between Xq + A and Xq - A. The restoring force is proportional to the deviation from the equilibrium, according to Equation 4.28. The energy of the oscillator may take arbitrary values. In reality, the classical description is only valid as a limit case, but this limit case is the basis for classical molecular dynamics. 

The classical harmonic oscillator model or normal mode is often illustrated using a potential energy curve (Figure 1.14). The larger the mass, the lower the frequency of the molecular vibration, yet the potential energy curve does not change. The relationship for a simple diatomic molecule model is... 

Let us calculate and compare the time-averaged potential and kinetic energies for the classical harmonic oscillator. When the particle is at some instantaneous displacement x its potential energy is... 

Comparing Eqs. (3-12) and (3-13) we see that, f = kL -/A. On the average, then, the classical harmonic oscillator stores half of its total energy as potential energy and half as kinetic energy. 

We showed in Section 3-2 that the classical harmonic oscillator stores, on the average, half of its energy as kinetic energy, and half as potential. We now make the analogous comparison in the quantum-mechanical system for the ground (n = 0) state. 

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... 

The vibrational entropy is calculated from the vibrational frequencies by employing the Classical Harmonic Oscillator approximation... 

From the earlier analysis of the classical harmonic oscillator we have the Hamiltonian function. 

Because p increases with the mass of the particle and with the spring constant as in Equation 7.31, a heavier mass or a stiffer spring makes the value of the Gaussian fimction smaller for any given displacement in x. For the classical harmonic oscillator, we expect a heavier mass or a stiffer spring to allow for smaller excursions from equilibrium. In the quantum mechanical picture, the wavefunction spreads less with a heavier mass or stiffer spring. 

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