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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.
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} :
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]
This force is balanced by the electron degeneracy pressure keeping the star stable. [4] In metals, the positive nuclei are partly ionized and spaced by normal interatomic distances. Gravity has negligible effect; the positive ion cores are attracted to the negatively charged electron gas. This force is balanced by the electron degeneracy pressure.
Degeneracy can also occur even with non-zero covariance. For example, when scalar X is symmetrically distributed about 0 and Y is exactly given by Y = X 2, all possible points (x, y) fall on the parabola y = x 2, which is a one-dimensional subset of the two-dimensional space. [citation needed]
The change in entropy in the entropy of mixing example may be viewed as an example of a non-extensive entropy resulting from the distinguishability of the two types of particles being mixed. Quantum particles are either bosons (following Bose–Einstein statistics ) or fermions (subject to the Pauli exclusion principle , following instead Fermi ...