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Examples of two-state systems in which the degeneracy in energy states is broken by the presence of off-diagonal terms in the Hamiltonian resulting from an internal interaction due to an inherent property of the system include: Benzene, with two possible dispositions of the three double bonds between neighbouring Carbon atoms.
Examples of degeneracy are found in the genetic code, when many different nucleotide sequences encode the same polypeptide; in protein folding, when different polypeptides fold to be structurally and functionally equivalent; in protein functions, when overlapping binding functions and similar catalytic specificities are observed; in metabolism, when multiple, parallel biosynthetic and ...
In quantum mechanics terminology, the degeneracy is said to be "lifted" by the presence of the magnetic field. In the presence of more than one unpaired electron, the electrons mutually interact to give rise to two or more energy states. Zero-field splitting refers to this lifting of degeneracy even in the absence of a magnetic field.
To complete Kramers degeneracy theorem, we just need to prove that the time-reversal operator acting on a half-odd-integer spin Hilbert space satisfies =. This follows from the fact that the spin operator S {\textstyle \mathbf {S} } represents a type of angular momentum , and, as such, should reverse direction under T {\displaystyle T} :
While degeneracy pressure usually dominates at extremely high densities, it is the ratio between degenerate pressure and thermal pressure which determines degeneracy. Given a sufficiently drastic increase in temperature (such as during a red giant star's helium flash ), matter can become non-degenerate without reducing its density.
He derived equations for the line intensities which were a decided improvement over Kramers's results obtained by the old quantum theory. While the first-order-perturbation (linear) Stark effect in hydrogen is in agreement with both the old Bohr–Sommerfeld model and the quantum-mechanical theory of the atom, higher-order corrections are not. [9]
For example, two particles may have different momenta, in which case they are distinguishable from each other, yet they can still have the same energy. The value of g i {\displaystyle g_{i}} associated with level i {\displaystyle i} is called the "degeneracy" of that energy level.
For example, NO 2 − is a strong-field ligand and produces a large Δ. The octahedral ion [Fe(NO 2 ) 6 ] 3− , which has 5 d -electrons, would have the octahedral splitting diagram shown at right with all five electrons in the t 2 g level.