Quantum degeneracy

The Sun will eventually lose its gaseous perfection because the stellar core is not immune to sclerosis. Quantum degeneracy is the root of this change in low-mass stars when they have completely burnt the hydrogen reserves at their centre. They are then well on the way to becoming white dwarfs. Let us now analyse this corpse-like stiffness. 

We note that, of the above, a phase-space density increase is concomitant only with the processes of sympathetic and evaporative cooling. In all other cases, phase-space density remains at most constant. A drop in phase-space density takes the gaseous sample further away from quantum degeneracy. 

Fig. 8.6 Emergence of quantum degeneracy, as seen in the energy of a trapped Fermi gas (Reprinted with courtesy and permission of AAAS (USA) from DeMarco and Jin 1999.)...

The one-dimensional cases discussed above illustrate many of die qualitative features of quantum mechanics, and their relative simplicity makes them quite easy to study. Motion in more than one dimension and (especially) that of more than one particle is considerably more complicated, but many of the general features of these systems can be understood from simple considerations. Wliile one relatively connnon feature of multidimensional problems in quantum mechanics is degeneracy, it turns out that the ground state must be non-degenerate. To prove this, simply assume the opposite to be true, i.e. 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

The rotational states are characterized by a quantum number J = 0, 1, 2,. .. are degenerate with degeneracy (2J + 1) and have energy t r = ) where 1 is the molecular moment of inertia. Thus... 

Eqs. (D.5)-(D.7). However, when perturbations occur due to anharmonicity, the wave functions in Eqs. (D.11)-(D.13) will provide the conect zeroth-order ones. The quantum numbers and v h are therefore not physically significant, while V2 arid or V2 and I2 = m, are. It should also be pointed out that the degeneracy in the vibrational levels will be split due to anharmonicity [28]. 

Each set of p orbitals has three distinct directions or three different angular momentum m-quantum numbers as discussed in Appendix G. Each set of d orbitals has five distinct directions or m-quantum numbers, etc s orbitals are unidirectional in that they are spherically symmetric, and have only m = 0. Note that the degeneracy of an orbital (21+1), which is the number of distinct spatial orientations or the number of m-values. 

The sum in the denominator relates to the quantum states. The formula is often written in terms of energy levels rather than quantum states in the case that some of the energy levels are degenerate, with degeneracy factors gi then the formula can be modified to refer to energy-level populations directly ... 

BalKSl Balasubramanian, K. Symmetry simplifications of spare types in configuration int by orbital degeneracy. Internat. J. Quantum Chem. 20 (1981) 1255-1271. 

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