Harmonic probability distribution

The leading term P0(u) is the harmonic probability distribution, af = 1, 2, or 3, and Z)ai... Dtr is the rth partial derivative operator dr/(duxi. .. duXr). The Einstein convention of summation over repeated indices is implied. 

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 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). 

A trivariate normal distribution describes the probability distribution for anisotropic harmonic motion in three-dimensional space. In tensor notation (see appendix A for the notation, and appendix B for the treatment of symmetry and symmetry restrictions of tensor elements), with j and k (= 1, 3) indicating the axial directions,... 

The probability distribution of Eq. (2.21) was derived assuming rectilinear motion in a harmonic potential. The true potential in a crystal is often more complex, especially in the upper parts of the potential surface, which are of importance at higher temperatures. 

For a distribution expanded around the equilibrium position, the first derivative is zero, and may be omitted, while the second derivatives are redundant as they merely modify the harmonic distribution. Since P0(u) is a Gaussian distribution, Eq. (2.28) can be simplified by use of the Tchebycheff-Hermite polynomials, often referred to simply as Hermite polynomials,3. , related to the derivatives of the three-dimensional Gaussian probability distribution by... 

As Eq. (2.31) shows, the Gram-Charlier temperature factor is a power-series expansion about the harmonic temperature factor, with real even terms, and imaginary odd terms. This is an expected result, as the even-order Hermite polynomials in the probability distribution of Eq. (2.30) are symmetric, and the odd-order polynomials are antisymmetric with respect to the center of the distribution. 

Let us now consider a NESS where the trap is moved at constant velocity, x t) = vt. It is not possible to solve the Fokker-Planck equation to find the probability distribution in the steady state for arbitrary potentials. Only for harmonic potentials, U x) = ioc /2, can the Fokker-Planck equation be solved exactly. The result is... 

Fig. 3.6 The probability distribution functions for a harmonic oscillator in the n = 0 and n =10 levels, each plotted at the height corresponding to its energy, with the curve showing the potential energy function. The points where the energy equals the potential energy are the classical turning points, corresponding to the maximum possible displacement of a classical particle with the same energy.

Figure 1-5 Wave functions (left) and probability distributions (right) of the harmonic oscillator.

Harmonic vibration A vibration that occurs under the influence of a force directly proportional to the displacement from equilibrium. The range of the vibration extends equal distances in either direction from an equilibrium position (the origin), and the acceleration is always toward the origin and directly proportional to the distance from it. For a Boltzmann distribution of particle energies, harmonic vibration leads to a probability distribution for the positions of the vibrating particles that is a Gaussian function of the displacements. 

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. 

Without even solving the equation in the radial variable, just knowmg the spherical harmonics gives us a lot of information about the structure of the atom. The probability distribution of the electron is usually pictured as a cloud, where the density of the cloud denotes greater probability. Nodes of the spherical functions are choices of 9 and where the solution is zero. For example,... 

Figure 10. Simulation of the EPR state preparation in an optical lattice with 25 sites, at three consecutive times. First row shows the joint probability distribution in x representation, the second one in p representation, (ol) and (a2) initially (t = 0), the atoms are cooled down to the external harmonic potential ground state, whereas the LIDDI is off. (61) and (62) at t = 1.4 x 10-4 s LIDDI and the repulsive linear potential (with the slope 0.04 Erec per lattice site) are on, whereas the harmonic potential is off. The diatoms are moving through the lattice very slowly in comparison to the single atoms, (cl) and (c2) at t = 2.16 x 10 4 s single atoms are ejected out of the lattice and discarded and the diatoms are separated out.

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... 

Figure 13. Joint phase probability distribution P(0a,0t) of the fundamental and second-harmonic modes for various evolution time x. In the last two figures the phase windows for 0 and 0t are shifted by tl...

Fiq. 11-2.—The wave function o( ) for the normal state of the harmonic oscillator (left), and the corresponding probability distribution function [ / <>( )Is (right). The classical distribution function for an oscillator with the same total energy is shown by the dashed curve. 

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. 

The harmonic oscillator provides another example of the correspondence principle. The same effects mentioned above are observed. We see from Figure 9.26 of the text that probability distributions for large values on n approach the classical picture of the motion. (Look at the graph for v = 20.)... 

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