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This is the earliest known solvable system, which was discussed by Majorana in 1932. Among the other examples there are models of a pair of degenerate level crossing, [23] and the 1D quantum Ising chain in a linearly changing magnetic field. [24] [25] Landau–Zener transitions in infinite linear chains. [26]
Each Landau level is degenerate because of the second quantum number , which can take the values =, where is an integer. The allowed values of N {\displaystyle N} are further restricted by the condition that the center of force of the oscillator, x 0 {\displaystyle x_{0}} , must physically lie within the system, 0 ≤ x 0 < L x {\displaystyle 0 ...
In quantum physics and quantum chemistry, an avoided crossing (AC, sometimes called intended crossing, [1] non-crossing or anticrossing) is the phenomenon where two eigenvalues of a Hermitian matrix representing a quantum observable and depending on continuous real parameters cannot become equal in value ("cross") except on a manifold of dimension . [2]
Since , by increasing the magnetic field, the Landau levels move up in energy and the number of states in each level grow, so fewer electrons occupy the top level until it becomes empty. If the magnetic field keeps increasing, eventually, all electrons will be in the lowest Landau level ( ν < 1 {\displaystyle \nu <1} ) and this is called the ...
This is because Landau theory is a mean field theory, and does not include long-range correlations. This theory does not explain non-analyticity at the critical point, but when applied to superfluid and superconductor phase transition, Landau's theory provided inspiration for another theory, the Ginzburg–Landau theory of superconductivity.
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The result of zero resistivity between the contacts 2 and 3 is a consequence of the electrons being mobile only in the edge channels of the sample. The situation would be different if a Landau level came close to the Fermi energy E F. Any electrons in that level would become mobile as their energy approaches the Fermi energy E F.
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