Relaxation (approximation) - Revision history

en>Makyen: /* References */Fix various citation "unknown parameter", ISBN, and URL errors/issues. Run gen fixes and typos if found. using AWB

2014-04-12T10:00:51Z

References: Fix various citation "unknown parameter", ISBN, and URL errors/issues. Run gen fixes and typos if found. using AWB

← Older revision		Revision as of 12:00, 12 April 2014
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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~~.		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.<br><br>Feel free to surf to my web page - [http://www.blaze16.com/blog/257588 http://www.blaze16.com]

	~~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}}~~

83.110.16.21 at 16:53, 8 January 2014

2014-01-08T16:53:27Z

← Older revision		Revision as of 18:53, 8 January 2014
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	~~They contact me Emilia~~. ~~Managing people~~ is ~~what I do~~ and ~~over~~ the ~~counter std test~~ ([http://www.~~lankaclipstv~~.~~com~~/~~blog~~/~~165665 visit~~ the ~~next post~~]) ~~salary has been really fulfilling~~. ~~California~~ is ~~our beginning place~~. ~~To gather coins~~ is ~~what her family~~ and ~~her enjoy~~.		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}}

en>Venuur: /* Properties */ Corrects typo in previous edit.

2012-03-12T16:57:19Z

Properties: Corrects typo in previous edit.

New page

They contact me Emilia. Managing people is what I do and over the counter std test ([http://www.lankaclipstv.com/blog/165665 visit the next post]) salary has been really fulfilling. California is our beginning place. To gather coins is what her family and her enjoy.