Quantum distributions harmonic oscillators

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). 

In the classical high-temperature limit, kBT hv, where kB is the Boltzmann constant, and hv is the spacing of the quantum-mechanical harmonic oscillator energy levels. If this condition is fulfilled, the energy levels may be considered as continuous, and Boltzmann statistics apply. The corresponding distribution is... 

Figure 21.5 The Bell tunnel model (a) Quantum mechanical harmonic oscillator with its ground state wavefunctions. (b) Inverted harmonic oscillator potential, (c) A stream of particles with a Boltzmann distribution of energies hits the barrier. Classical only those particles with W>Vq can pass the barrier.

This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /iLorentzian distribution of the system s normal modes. In the absence of friction (2.27) describes thermally activated as well as tunneling processes when < 1, or fhcoo > 1, respectively. At first glance it may seem surprising... 

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.

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

D. J. Tannor To understand the role of dissipation in quantum mechanics, it is useful to consider the density operator in the Wigner phase-space representation. Energy relaxation in a harmonic oscillator looks as shown in Fig. 1, whereas phase relaxation looks as shown in Fig. 2 that is, in pure dephasing the density spreads out over the energy shell (i.e., spreads in angle) while not changing its radial distribution... 

Slater s theory assumes that the normal modes behave as harmonic oscillators, which requires that there be no flow of energy between the normal modes once the molecule is suitably activated, and so the energy distribution remains fixed between collisions. But spectroscopy shows that energy can flow around a molecule, and allowing for such a flow between collisions vastly improves the theory. Like Kassel s theory a fully quantum theory would be superior. 

According to quantum mechanics, only those transitions involving Ad = 1 are allowed for a harmonic oscillator. If the vibration is anhar-monic, however, transitions involving Au = 2, 3,. .. (overtones) are also weakly allowed by selection rules. Among many Au = 1 transitions, that of u = 0 <-> 1 (fundamental) appears most strongly both in IR and Raman spectra. This is expected from the Maxwell-Boltzmann distribution law, which states that the population ratio of the u = 1 and u = 0 states is given by... 

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. 

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. 

We wish to compare the quantum probability distributions with those obtained from the classical treatment of the harmonic oscillator at the same energies. The classical probability density P y) as a function of the reduced distance y(—l y l)is given by equation (4.10) and is shown in Figure... 

In light of these claims, it is useful to commence our study of the thermodynamic properties of the harmonic oscillator from the statistical mechanical perspective. In particular, our task is to consider a single harmonic oscillator in presumed contact with a heat reservoir and to seek the various properties of this system, such as its mean energy and specific heat. As we found in the section on quantum mechanics, such an oscillator is characterized by a series of equally spaced energy levels of the form E = (n + )hu>. From the point of view of the previous section on the formalism tied to the canonical distribution, we see that the consideration of this problem amounts to deriving the partition function. In this case it is given by... 

Fiu. 49-1.—The probability values Pn for system-part a in a system of five coupled harmonic oscillators with total quantum number n 10 (closed circles), and values calculated by the Boltzmann distribution law (open circles). 

Figure 1. Position probability distribution for the state n = 10 of a harmonic oscillator (solid curve) and a classical oscillator of the same total energy (dashed curve). (From L. I. Shiff, Quantum Mechanics, 3rd ed., McGraw-Hill, New York, 1968).

The microcanonical [Eq. (9.14)] and the canonical [Eq. (9.15)] translational energy distributions are compared graphically in figure 9.1 for the case of three molecules with 3,5, and 25 classical harmonic oscillators at a constant product temperature, k T = 400 cm (ca. 580 K). Because of the different product heat capacities, the excess energy varies for the three different reactions. It is evident that the largest molecule with 25 product oscillators results in microcanonical and canonical distributions that are nearly indistinguishable. As the size of the molecule decreases, the discrepancy between the two distributions increases. The same trends would be evident had we used the more correct quantum vibrational density of states for the microcanonical translational PEDs. 

The Fourier component of interaction force F(t) on the transition frequency (2-184) characterizes the level of resonance. Matrix elements for harmonic oscillators m y n) are non-zero only for one-quantum transitions, n = m . The W relaxation probability of the one-quantum exchange as a function of translational temperature To can be found by averaging the probability over Maxwelhan distribution ... 

The GED approach uses a mixed QC ansatz to construct a separable model for the electronuclear problem. As it corresponds to the case of a set of identical fermions, the electronic part is fully quantized in the GED scheme. The PCB is endowed with a mass distribution allowing for a correspondence with the external potential created by nuclei. The harmonic oscillator model of nuclear dynamics then follows as a quantum extension of the latter model, one where mass fluctuations are described by normal modes related... 

These concepts may be quantified as follows. A quantum of lattice vibration is termed a phonon, and the mean deviation of an atom from its lattice position is the mean-square displacement (u ). Phonons are detected by vibrational spectroscopy by absorption peaks below 500 cm . According to the Debye model, atoms vibrate as harmonic oscillators with a distribution of frequencies, the highest of which is COD- then the Debye temperature 9d is defined as... 

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