Harmonic oscillator energy

It has already been noted that the new quantum theory and the Schrodinger equation were introduced in 1926. This theory led to a solution for the hydrogen atom energy levels which agrees with Bohr theory. It also led to harmonic oscillator energy levels which differ from those of the older quantum mechanics by including a zero-point energy term. The developments of M. Born and J. R. Oppenheimer followed soon thereafter referred to as the Born-Oppenheimer approximation, these developments are the cornerstone of most modern considerations of isotope effects. 

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... 

The harmonic oscillator energies and wavefunctions comprise the simplest reasonable model for vibrational motion. Vibrations of a polyatomic molecule are often characterized in terms of individual bond-stretching and angle-bending motions each of which is, in turn, approximated harmonically. This results in a total vibrational wavefunction that is written as a product of functions one for each of the vibrational coordinates. 

Explain how the conclusion is "obvious", how for J = 0, k = R, and A = 0, we obtain the usual harmonic oscillator energy levels. Describe how the energy levels would be expected to vary as J increases from zero and explain how these changes arise from changes in k and re. Explain in terms of physical forces involved in the rotating-vibrating molecule why re and k are changed by rotation. 

The QRRK model postulates that vibrational energy can freely flow (internally) from one vibrational mode in the molecule to another. This is a very significant assumption. For a collection of harmonic oscillators, energy in a particular vibrational mode will stay in that mode it cannot flow into other vibrational modes of the system. That is, a system of harmonic oscillators is uncoupled. 

For the c = 0 and v= 1 vibrational levels of CO, calculate the maximum departure of each nucleus from its equilibrium position in the principal-axis coordinate system if it is assumed the nuclei move classically. Assume harmonic-oscillator energy levels. 

Fig. 5. Schematic representation of various energy curves due to weak (w) and strong (s) interactions between two harmonic oscillators. Energy in arbitrary units with the following values (k/2)% = 3, E = 0 for strong coupling, U = 7, for weak coupling, U = 5. The dotted curves indicate the intermediate states for weak coupling case, [W (w)].

Fig. 2.26. The harmonic oscillator energy levels and wave functions. The potential is bounded by the curve V = /ikiArf (heavy solid line). The quantum-mechanical probability density functions 4> M are shown as light solid lines for each energy level, while the corresponding classical probabilities are shown as dashed lines (after McMillan, 1985 reproduced with the publisher s permission).

With A. defined in Eq (5.10), we find the general formula for harmonic-oscillator energy eigenvalues... 

The first order perturbation correction to the harmonic oscillator energy level, calculated... 

Reconsider Ex. 3.7. Devise an electric circuit for harmonically oscillating energy generation. What is the analog solution of this problem ... 

The harmonic-oscillator wave functions are given by a Hermite polynomial times an exponential factor (Problem 4.19b). By virtue of the expansion postulate, any well-behaved function/(x) can be expanded as a linear combination of harmonic-oscillator energy eigenfunctions ... 

If Vkj(R) were a parabola (as it is for the harmonic oscUlator), the system would never acquire the energy corresponding to the bottom of the parabola because the harmonic oscillator energy levels (cf., p. 190) correspond to higher energy. The same pertains to Vu of a more complex shape. 

True or false (a) All harmonic-oscillator wave functions with v an odd integer must have a node at the origin, (b) The w = 10 harmonic-oscillator wave function has 10 interior nodes, (c) The w = 1 harmonic-oscillator wave function must be negative for a < 0. (d) The harmonic-oscillator energy levels are equally spaced, (e) The one-dimensional harmonic-oscillator energy levels are nondegenerate. 

Figure 8.2 The harmonic oscillator energy levels and wavefunctions. = 1600 cm m = 1 amu.

This expression includes the total energy E of the harmonic oscillator. Energy is an important observable, so let us detour to consider it. In order for the wavefunction to be noninfinite, the energy of the harmonic oscillator, when combined with the other terms like a, n, m, and h, must have only those values that satisfy the above equation. Therefore, we can solve for what the values of the energy must be. Substituting also for a = 2TTvmlh, we find a simple conclusion ... 

Unlike the harmonic oscillator energy levels, these are no longer equally spaced. One result is that hot bands (v = 2 = 1, etc.) will no longer have... 

The photon energy matches the spacing between harmonic oscillator energy levels, i.e. the photon frequency equals the classical frequency of vibration. This gives a transition involving only a unit change of quantum number m = n l. 

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