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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.
The theory of hyperfine structure comes directly from electromagnetism, consisting of the interaction of the nuclear multipole moments (excluding the electric monopole) with internally generated fields. The theory is derived first for the atomic case, but can be applied to each nucleus in a molecule. Following this there is a discussion of the ...
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]
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.
In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j) as follows: =, where g j is the degeneracy factor, or number of quantum states s that have the same energy level defined by E j = E s.
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} :
In inorganic chemistry, crystal field theory (CFT) describes the breaking of degeneracies of electron orbital states, usually d or f orbitals, due to a static electric field produced by a surrounding charge distribution (anion neighbors).
This degeneracy is lifted when spin–orbit interaction is treated to higher order in perturbation theory, but still states with same |M S | are degenerate in a non-rotating molecule. We can speak of a 5 Σ 2 substate, a 5 Σ 1 substate or a 5 Σ 0 substate. Except for the case Ω = 0, these substates have a degeneracy of 2.