Quantum mechanics classical harmonic oscillator

The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applications in quantum physics and held theory. 

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 wave functions for u = 0 to 4 are plotted in figure 6.20 the point where the function crosses through zero is called a node, and we note that the wave function for level v has v nodes. The probability density distribution for each vibrational level is shown in figure 6.21, and the difference between quantum and classical behaviour is a notable feature of this diagram. For example, in the v = 0 level the probability is a maximum at y = 0, whereas for a classical harmonic oscillator it would be a minimum at v = 0, with maxima at the classical turning points. Furthermore the probability density is small but finite outside the classical region, a phenomenon known as quantum mechanical tunnelling. 

For a classical harmonic oscillator, the particle cannot go beyond the points where the total energy equals the potential energy. Identify these points for a quantum-mechanical harmonic oscillator in its ground state. Write an integral giving the probability that the particle will go beyond these classically-allowed points. (You need not evaluate the integral.)... 

Because of their importance to nucleation kinetics, there have been a number of attempts to calculate free energies of formation of clusters theoretically. The most important approaches for the current discussion are harmonic models, " Monte Carlo studies, and molecular dynamics calcula-tions. In the harmonic model the cluster is assumed to be composed of constituent atoms with harmonic intermolecular forces. The most recent calculations, which use the harmonic model, have taken the geometries of the clusters to be those determined by the minimum in the two-body additive Lennard-Jones potential surface. The oscillator frequencies have been obtained by diagonalizing the Lennard-Jones force constant matrix. In the harmonic model the translational and rotational modes of the clusters are treated classically, and the vibrational modes are treated quantum mechanically. The harmonic models work best at low temjjeratures where anharmonic-ity effects are least important and the system is dominated by a single structure. 

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. 

Let us note that both Bethe and Davydov splittings are not consequences of specific quantum-mechanical properties of molecules. They appear also in the crystals where the molecules are modeled, for example, by classical harmonic oscillators. 

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.]... 

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... 

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. 

Consider a classical harmonic oscillator. What is the size, E, of the phase space that is occupied by the oscillator when its energy, E, is constant Considering quantum mechanical uncertainty and the consequent discretization of phase space, what is the number of microscopic states S2 What are the entropy, enthalpy, pressure, temperature, and Gibbs free energy of the oscillator ... 

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

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... 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

Quantization (the idea of quantums, photons, phonons, gravitons) is postulated in Quantum Mechanics, while the Theory of Relativity does not derive quantization from geometric considerations. In the case of the established phenomenon the quantized nature of portioned energy transfer stems directly from the mechanisms of the process and has a precise mathematical description. The quasi-harmonic oscillator obeys the classical laws to a greater extent than any other system. A number of problems, related to quasi-harmonic oscillators, have the same solution in classical and quantum mechanics. 

The decrease in the heat capacity at low temperatures was not explained until 1907, when Einstein demonstrated that the temperature dependence of the heat capacity arose from quantum mechanical effects [1], Einstein also assumed that all atoms in a solid vibrate independently of each other and that they behave like harmonic oscillators. The motion of a single atom is again seen as the sum of three linear oscillators along three perpendicular axes, and one mole of atoms is treated by looking at 3L identical linear harmonic oscillators. Whereas the harmonic oscillator can take any energy in the classical limit, quantum theory allows the energy of the harmonic oscillator (en) to have only certain discrete values ( ) ... 

The motion in the classical domain corresponds to a harmonic oscillator of frequency v, with the displacement from equilibrium varying sinusoidally with time. The transcription of this problem into quantum mechanics is simple and straightforward it is a standard problem in introductory quantum mechanics texts. The energy levels of the quantum system are given by... 

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