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| In quantum mechanics, the '''Kramers degeneracy theorem''' states that for every energy eigenstate of a [[T-symmetry|time-reversal symmetric]] system with half-integer total spin, there is at least one more eigenstate with the same energy. In other words, every energy level is at least doubly degenerate if it has half-integer spin.
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| In theoretical physics, the [[T-symmetry|time reversal symmetry]] is the symmetry of physical laws under a time reversal transformation:
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| :<math> T: t \mapsto -t.</math>
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| If the Hamiltonian operator commutes with the time-reversal operator, that is
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| :<math>[H,\Theta]=0,</math>
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| then for every energy eigenstate <math>|n\rangle</math>, the time reversed state <math>\Theta|n\rangle</math> is also an eigenstate with the same energy. Of course, this time reversed state might be identical to the original state, but that is not possible in a half-integer spin system since time reversal reverses all angular momenta, and reversing a half-integer spin cannot yield the same state (the [[magnetic quantum number]] is never zero).
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| For instance, the [[energy level]]s of a system with an odd total number of fermions (such as [[electron]]s, [[proton]]s and [[neutron]]s) remain at least doubly [[degenerate energy level|degenerate]] in the presence of purely [[electric field]]s (i.e. no [[magnetic field]]s). It was first discovered in 1930 by [[Hendrik Anthony Kramers|H. A. Kramers]]<ref>Kramers, H. A., Proc. Amsterdam Acad. 33, 959 (1930)</ref> as a consequence of [[Breit equation]].
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| As shown by [[Eugene Wigner]] in 1932,<ref>E. Wigner, Über die Operation der Zeitumkehr in der Quantenmechanik, Nachr. Akad. Ges. Wiss. Göttingen 31, 546–559 (1932) http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002509032</ref> it is a consequence of the [[time reversal invariance]] of [[electric field]]s, and follows from an application of the [[antiunitary]] T-operator to the wavefunction of an odd number of fermions. The theorem is valid for any configuration of static or time-varying electric fields.
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| For example: the [[hydrogen]] (H) atom contains one proton and one electron, so that the Kramers theorem does not apply. The lowest (hyperfine) energy level of H is nondegenerate. The [[deuterium]] (D) isotope on the other hand contains an extra neutron, so that the total number of fermions is three, and the theorem does apply. The ground state of D contains two hyperfine components, which are twofold and fourfold degenerate.
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| ==See also==
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| *[[Degenerate energy levels|Degeneracy]]
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| *[[Hendrik Anthony Kramers]]
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| *[[T-symmetry#Kramers.27_theorem|T-symmetry]]
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| Degeneracy g of a proton is 2
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| ==References==
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| {{reflist|2}}
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| {{DEFAULTSORT:Kramers Theorem}}
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| [[Category:Physics theorems]]
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| [[Category:Atomic physics]]
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| {{Atomic-physics-stub}}
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Hello and welcome. My title is Numbers Wunder. To do aerobics is a thing that I'm totally addicted to. He used to be unemployed but now he is a pc operator but his promotion never arrives. Years ago we moved to North Dakota and I love each day living right here.
Feel free to surf to my web page - http://www.blaze16.com