Degeneracy harmonic oscillator

Finally, we have applied equation (10.14) to a collection of harmonic oscillators. But it can be applied to any collection of energy levels and units of energy with one modification. Equation (10.14) assumes that each level has an equal probability (as in a harmonic oscillator), and this is true only if g, the degeneracy, is one. The quantity g, is also known as the statistical weight factor. If it is greater than one, equation (10.14) must be multiplied by the g, for each... 

Derive the result that the degeneracy of the energy level E for an isotropic three-dimensional harmonic oscillator is (n + l)(n + 2)/2. 

Fig. 3 Bneigy levels of the three-dimensional harmonic oscillator. The degree of degeneracy of each level is shown in parenthesis.

Consider the degeneracies of the vibrational levels in the harmonic-oscillator approximation. For the ground level there is only one possible set of vibrational quantum numbers (00- ()) hence the ground vibrational level is always nondegenerate. If none of the normal modes are... 

The addition of the spin-orbit term to the nuclear harmonic oscillator potential causes a separation or removal of the degeneracy of the energy levels according to their total angular momentum (j = l + s). In the nuclear case, the states with... 

The Boltzmann distribution is illustrated in Fig. 1.2.1 for the vibrational states of a one-dimensional harmonic oscillator with the frequency uj = 2itis, where the energy levels are given by Eq. (1.18), and in Fig. 1.2.2 for the rotational states of a linear molecule with the moment of inertia I, where the energy levels are given by Eq. (1.20) with the degeneracy u>j = 2 J + 1. 

Fig. 1.2.1 The Boltzmann distribution for a system with equally-spaced energy levels En and identical degeneracy ton of all levels (T > 0). This figure gives the population of states at the temperature T for a harmonic oscillator.

In order to evaluate the vibrational partition function we consider a single harmonic oscillator. The energy levels, with the zero of energy as the zero-point level, are En = hi/sn, with n = 0,1,..., and degeneracy ojn = 1. The partition function takes the form... 

It would be of interest to apply the method of March and Murray [12] to convert C, the electron density for non-degenerate electrons, into results applicable to intermediate degeneracy governed by Fermi-Dirac statistics. Unfortunately, without switching on the model potential F(r), this is already difficult to handle by purely analytically methods, as can be seen from the case of complete degeneracy for the harmonic oscillator alone. No doubt, numerical procedures will eventually enable present results to be transformed according to the route established in [12]. 

Fia. 15-1.—Energy levels, degrees of degeneracy, and quantum numbers for the three-dimensional isotropic harmonic oscillator. 

First, let us consider a system of weakly coupled harmonic oscillators of frequencies v, v, each having gi, g2, , gm degeneracies, respec-... 

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