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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. | |||
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}} | |||
Revision as of 18:53, 8 January 2014
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 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 time reversal symmetry is the symmetry of physical laws under a time reversal transformation:
If the Hamiltonian operator commutes with the time-reversal operator, that is
![{\displaystyle [H,\Theta ]=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70bb239916c86871b24ded9d34e1b9b959cb43bd)
then for every energy eigenstate , the time reversed state 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 levels of a system with an odd total number of fermions (such as electrons, protons and neutrons) remain at least doubly degenerate in the presence of purely electric fields (i.e. no magnetic fields). It was first discovered in 1930 by H. A. Kramers[1] as a consequence of Breit equation.
As shown by Eugene Wigner in 1932,[2] it is a consequence of the time reversal invariance of electric fields, 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
Degeneracy g of a proton is 2
References
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- ↑ Kramers, H. A., Proc. Amsterdam Acad. 33, 959 (1930)
- ↑ 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