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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.
A rectanglewith one pair of opposite sides of length zero degenerates to a line segment, with zero area. If both of the rectangle's pairs of opposite sides have length zero, the rectangle degenerates to a point. Hyperrectangle. [edit] A hyperrectangleis the n-dimensional analog of a rectangle.
Kramers' theorem. In quantum mechanics, the Kramers' degeneracy theorem states that for every energy eigenstate of a time-reversal symmetric system with half-integer total spin, there is another eigenstate with the same energy related by time-reversal. In other words, the degeneracy of every energy level is an even number if it has half-integer ...
Degenerate matter is usually modelled as an ideal Fermi gas, an ensemble of non-interacting fermions. In a quantum mechanical description, particles limited to a finite volume may take only a discrete set of energies, called quantum states. The Pauli exclusion principle prevents identical fermions from occupying the same quantum state.
An example is the neutral helium atom (He), which has two bound electrons, both of which can occupy the lowest-energy states by acquiring opposite spin; as spin is part of the quantum state of the electron, the two electrons are in different quantum states and do not violate the Pauli principle.
The degeneracy maps s i lengthen the sequence by inserting an identity morphism at position i. We can recover the poset S from the nerve NS and the category C from the nerve NC; in this sense simplicial sets generalize posets and categories. Another important class of examples of simplicial sets is given by the singular set SY of a topological ...
The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph. Degeneracy is also known as the k-core number, [1] width, [2] and linkage, [3] and is essentially the same as the coloring number [4] or Szekeres–Wilf number (named after Szekeres and Wilf ).
Degenerate conic. In geometry, a degenerate conic is a conic (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers (or more generally over an algebraically closed field) as the product of two linear ...