Energy levels harmonic oscillator

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... 

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. 

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. 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).

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

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.

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. 

The vibrational energy levels of polyatomic molecules are the sum of harmonic oscillator energy level expressions, as in Eq. (22.4-19). 

Solution Since the populations are probability times the number of molecules, the ratio of the population of one state to the ground state population is the same as the corresponding ratio of the probabilities. To obtain that ratio, we use Equation 1.10 because the degeneracies of the harmonic oscillator energy levels all happen to be 1 (i.e., nondegenerate). All that is needed to complete the problem is the energy level expression for harmonic oscillators, and that is Equation 1.2. 

Redo Example 11.2 and include the next two terms before truncating the power series expansion of the exponential. Using a harmonic oscillator energy level expression of E (kJ mol ) = 23.0 (n + 1/2), estimate the percentage error in U from truncating the expansion. 

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