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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.
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.
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.
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} :
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]
Examples include carbonyl sulfide, OCS, with μ = 0.71521 ± 0.00020 debye. However, because the splitting depends on μ 2, the orientation of the dipole must be deduced from quantum mechanical considerations. [31] A similar removal of degeneracy will occur when a paramagnetic molecule is placed in a magnetic field, an instance of the Zeeman ...