Harmonic oscillator quantum energy levels

The quantum-mechanical description of a polyatomic system may be extrapolated from the treatment of the diatomic molecule. Starting again with the harmonic oscillator, the energy levels for the entire system (E) can be given in terms of the characteristic frequencies (v,) and quantum numbers ( ,) of a series of independent harmonic oscillators ... 

Fig. 1.12. The harmonic oscillator. The energy levels, wavefunctions, and probability functions for the quantum mechanical oscillator are illustrated along with the energy function and probability functions for the classical oscillator.

When the electron transfer process is coupled to classical reorientation modes and to only one harmonic oscillator whose energy quantum h( is high enough for only the ground vibrational level to be populated, the expression of the electron transfer rate is given by [4, 9] ... 

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. 

With the assumption of harmonic oscillators, the molecule s quantum energy levels are... 

In the electronic ground state, acetylene is a linear molecule. By using the harmonic approximation, vibrational energy levels for small amplitude oscillations can be labeled as (Uj, V2, v, v, v ) where is the quantum... 

Figurell.8 The parabola shows the harmonic oscillator potential. The horizontal lines show the energy levels, which are equally spaced for the quantum harmonic oscillator. The lowest level has energy liv/2. This is called the zero-point energy. The curve on each horizontal line shows tp x), the particle distribution probability for each energy level.

For a diatomic molecule, modeled as a harmonic oscillator, quantum mechanics reveals that the vibronic molecular energy levels are restricted to discrete values given by Equation 2.1... 

The atomic harmonic oscillator follows the same frequency equation that the classical harmonic oscillator does. The difference is that the classical harmonic oscillator can have any amplitude of oscillation leading to a continuum of energy whereas the quantum harmonic oscillator can have only certain specific amplitudes of oscillation leading to a discrete set of allowed energy levels. 

Experimental. The vibrational spectrum of an ideal harmonic oscillator would consist of one line at frequency v corresponding to A = hv, where A is the distance between levels on the vertical energy axis in Fig. 10-la. In the harmonic oscillator, AE is the same for a transition from one energy level to an adjacent level. A selection rule An = 1, where n is the vibrational quantum number, requires that the transition be to an adjacent level. 

In practice, the harmonic oscillator has limits. In the ideal case, the two atoms can approach and recede with no change in the attractive force and without any repulsive force between electron clouds. In reality, the two atoms will dissociate when far enough apart, and will be repulsed by van der Waal s forces as they come closer. The net effect is the varying attraction between the two in the bond. When using a quantum model, the energy levels would be evenly spaced, making the overtones forbidden. 

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