formulasearchengine - User contributions [en]

Position and momentum space

2014-11-26T11:45:04Z

83.110.223.141:

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Power-flow study

2014-02-17T13:16:10Z

83.110.73.175: /* Load flow */

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Relaxation (approximation)

2014-01-08T16:53:27Z

83.110.16.21:

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.

In theoretical physics, the [[T-symmetry|time reversal symmetry]] is the symmetry of physical laws under a time reversal transformation:
:<math> T: t \mapsto -t.</math>
If the Hamiltonian operator commutes with the time-reversal operator, that is
:<math>[H,\Theta]=0,</math>
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).

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]].

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.

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.

==See also==
*[[Degenerate energy levels|Degeneracy]]
*[[Hendrik Anthony Kramers]]
*[[T-symmetry#Kramers.27_theorem|T-symmetry]]
Degeneracy g of a proton is 2

==References==
{{reflist|2}}

{{DEFAULTSORT:Kramers Theorem}}
[[Category:Physics theorems]]
[[Category:Atomic physics]]

{{Atomic-physics-stub}}