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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.
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]
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} :
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).
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 ...
Examples of such ground states are superconductors, ferromagnets, Jahn–Teller distortions and spin density waves. The state occupancy of fermions like electrons is governed by Fermi–Dirac statistics so at finite temperatures the Fermi surface is accordingly broadened. In principle all fermion energy level populations are bound by a Fermi ...