Linear harmonics

Consider for the nth time step the linear harmonic oscillator... 

Another option is a q,p) = p and b q,p) = VU q). This guarantees that we are discretizing a pure index-2 DAE for which A is well-defined. But for this choice we observed severe difficulties with Newton s method, where a step-size smaller even than what is required by explicit methods is needed to obtain convergence. In fact, it can be shown that when the linear harmonic oscillator is cast into such a projected DAE, the linearized problem can easily become unstable for k > . Another way is to check the conditions of the Newton-Kantorovich Theorem, which guarantees convergence of the Newton method. These conditions are also found to be satisfied only for a very small step size k, if is small. 

Show that the wave functions A (y) in momentum space corresponding to 0 ( ) in equation (4.40) for a linear harmonic oscillator are... 

The Sehrodinger equation for the linear harmonic oscillator leads to the differential equation (4.17)... 

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 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 ( ) ... 

Another frequently used model system is the Frenkel-Kontorova (FK) model, in which a linear harmonic chain is embedded in an external potential. For a review, we direct the interested reader to Ref. 62. The potential energy in the FK model reads as follows ... 

Molecular structure enters into the rotational entropy component, and vibrational frequencies into the vibrational entropy component. The translational entropy component cancels in a (mass) balanced reaction, and the electronic component is most commonly zero. Note that the vibrational contribution to the entropy goes to oo as v goes to 0. This is a consequence of the linear harmonic oscillator approximation used to derive equation 7, and is inappropriate. Vibrational entropy contributions from frequencies below 300 cm should be treated with caution. 

PROBLEM 2.16.6. Solve by Laplace transform methods the classical linear harmonic oscillator differential equation mdzy/cHz= —kHy(t), where kH is the Hooke s law force constant, with the initial condition dy/dt = 0 at t = 0. Note Use p for the Laplace transform variable, to not confuse it with the Hooke s law force constant kH ... 

There is one, admittedly elementary, example where an exact differential equation has been derived by Lawes and March.127 This is for N particles moving in a one-dimensional harmonic oscillator potential. The motivation of their argument was to study the functional derivative dtx/6p appearing in the Euler equation (49). Adapted to the linear harmonic oscillator, this reads, with suitable choice of units... 

In fact this result is exact for the linear harmonic oscillator for an arbitrary number of particles N, because of the result187... 

The spin-related section mle can be proved by elastic neutron scattering measurements. In order to establish the specific fingerprint of the spin correlation, the scattering functions for the linear harmonic oscillator, for the double-well minimum function, and for pairs of coupled oscillators have been calculated in Ref. 119. 

Figure 12. Schematic representation of two coupled linear harmonic oscillators. Equilibrium positions are at q 0. cri and Q2 are relative displacements.

Figure 13. Schematic representation of the symmetric (top) and antisymmetric (bottom) normal coordinates for two coupled linear harmonic oscillators with a center of symmetry.

In the treatment of a linear harmonic oscillator, we assume that a mass m, attached to a spring with a spring constant k is freely vibrating without loss of energy in the vertical (z) direction (see Fig. 1.8). 

Each equation is now a total differential equation in one variable, Q, This is the linear harmonic oscillator equation in terms of the normal coordinate Q. The solution is then expandable as the product of harmonic oscillator functions, one for each normal mode, and the total energy corresponds to the sum of the energies of the 3A atom 6 oscillators. 

We proceed to illustrate the fundamental ideas of matrix mechanics by means of an example, namely, the linear harmonic oscillator. We start from the classical expression for the energy,... 

Here again, therefore, we obtain for our term scheme an equidistant succession of energy levels, as in Bohr s theory. The sole difference lies in the fact that the whole term diagram of quantum mechanics is displaced relative to that of Bohr s theory by half a quantum of energy. Although this difference does not manifest itself in the spectrum, it plays a part in statistical problems. In any case it is important to note that the linear harmonic oscillator possesses energy hv in. the lowest state, the so-called zem-jpoint energy. 

In this section we shall obtain the solution of the wave equation for the linear harmonic oscillator. The equation is... 

After these preliminary remarks on the radiation field we now pass on to the equation giving the vibrations of the linear harmonic oscillator. If the oscillator is capable of vibrating only in the r-direction, this equation is... 

The equation for P(p) may be treated by the same general method as was employed for the equation of the linear harmonic oscillator in Section llo. The first step is to obtain an asymptotic solution for large values of p, in which region Equation 17-13 becomes approximately... 

However, just as in the case of the linear harmonic oscillator, the infinite series so obtained is not a satisfactory wave function for general values of X, because its value increases so rapidly with increasing as to cause the total wave function to become infinite as increases without limit. In order to secure an acceptable wave function it is necessary to cause the scries to break off after a finite number of terms. The condition that the series break off at the term an " + ml, where n is an even integer, is obtained from 17-22c by putting n + 2 in place of v and equating the coefficient of a , to zero. This yields the result... 

As an example, let us determine the time-dependent position of a mass on a spring, relative to its equilibrium position, after release from an initial displacement A. This is, of course, the linear harmonic oscillator problem. Let y(t) be the displacement and k be the spring s force constant. The force on the mass is then... 

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