Diferencia entre revisiones de «Usuario:Jorgealda/OscilacionesNeutras»

sin resumen de edición
En 1968, [[Bruno Pontecorvo]] demostró que si se supone que los neutrinos tienen masa, los {{SubatomicParticle|Electron Neutrino}} producidos en el Sol se pueden transformar en otras especies ({{SubatomicParticle|Muon Neutrino}} o {{SubatomicParticle|Tau Neutrino}}), que no serían detectadas por el experimento de Homestake. La confirmación final a esta solución del problema de los neutrinos solares la proporcionó el [[Sudbury Neutrino Observatory|SNO]] en abril de 2002, midiendo tanto el flujo de {{SubatomicParticle|Electron Neutrino}} como el flujo total de neutrinos.<ref>{{Cite journal |last=Ahmad |first=Q. R. |display-authors=etal |collaboration=[[SNO Collaboration]] |date=2002 |title=Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory |journal=[[Physical Review Letters]] |volume=89 |issue= |pages=011301 |arxiv=nucl-ex/0204008 |bibcode=2002PhRvL..89a1301A |doi=10.1103/PhysRevLett.89.011301 |doi-access=free |pmid=12097025}}</ref>
 
== Descripción comoen unla sistemamecánica de dos estadoscuántica ==
 
=== CasoSistema especial:de solodos mezclaestados ===
 
==== Caso especial: solo mezcla ====
Sea <math>{H_{0}}</math> el [[Hamiltoniano (mecánica cuántica)|hamiltoniano]] del sistema de dos estados, y <math>\left| 1 \right\rangle </math> y <math>\left| 2 \right\rangle </math> sus dos [[Vector propio y valor propio|autoestados]] ortogonales con autovalores <math>{E_ 1 } </math> y <math>{E_ 2 } </math> respectivamente. En la base <math>\left\{ {\left| 1 \right\rangle ,\left| 2 \right\rangle } \right\}</math>, <math>{H_{0}}</math> es diagonal. Esto es,
 
De la expresión de <math>{{P}_{21}}(t)</math> se puede inferir que la oscilación solo puede existir en caso de que exista el término de acoplamiento, <math>{{\left| {{W}_{12}} \right|}^{2}}\ne 0</math>. La oscilación tampoco se produce si los autovalores del hamiltoniano <math>H</math> son degenerados, <math>{{E}_{+}}={{E}_{-}}</math>. Pero esto es un caso trivial, ya que en esta situación el acoplamiento se debe anular y el hamiltoniano es diagonal, por lo que se trata del caso inicial.
 
==== TheCaso general case: considering mixingmezcla andy decaydesintegración ====
IfSi thelas particle(s)partículas underpueden consideration undergoes decaydesintegrarse, then theel Hamiltonianhamiltoniano describingque thedescribe systemla isoscilación no longeres Hermitianhermítico.<ref name=":2">{{Cite web |last=Dighe |first=A. |date=26 July 2011 |title=B physics and CP violation: An introduction |url=http://theory.tifr.res.in/~amol/talks/B-notes.pdf |publisher=[[Tata Institute of Fundamental Research]] |accessdate=2016-08-12}}</ref> SinceCualquier anymatriz matrixse canpuede beescribir writtencomo asla asuma sumde ofsus itspartes Hermitianhermítica and anti-Hermitiany partsantihermítica, por lo que <math>H</math> can be written as,es
 
<math>H=M-\frac{i}{2}\Gamma =\left( \begin{matrix}
 
{| class="wikitable collapsible collapsed"
! wheredonde,
|-
| <math>M=\left( \begin{matrix}
{{M}_{11}} & {{M}_{12}} \\
{{M}_{21}} & {{M}_{22}} \\
\end{matrix} \right)</math> andy,
 
<math>\Gamma =\left( \begin{matrix}
{{\Gamma }_{11}} & {{\Gamma }_{12}} \\
\end{matrix} \right)</math>
 
<math>M</math> andy <math>\Gamma</math> areson Hermitian. Hencehermíticas, por lo que
 
<math>{{M}_{21}}={{M}_{12}}^{*}</math> and <math>{{\Gamma}_{21}}={{\Gamma}_{12}}^{*}
</math>
 
La [[simetría CPT]] implica que
[[CPT symmetry|CPT conservation (symmetry)]] implies,
 
<math>{{M}_{22}}={{M}_{11}}</math> andy <math>{{\Gamma}_{22}}={{\Gamma}_{11}}</math>
 
{| class="wikitable collapsible collapsed"
! Demostración
! Proof
|-
| Let,Sea <math>\Theta =CPT</math>. <math>\Theta</math> changescambia auna particlepartícula topor itssu antiparticle.antipartícula, Thatesto is,es
 
<math>\Theta \left| 1 \right\rangle =\left| 2 \right\rangle</math> andy <math>\Theta \left| 2 \right\rangle =\left| 1 \right\rangle</math>
 
CPTLa conservationconservación impliesde thatCPT theimplica Hamiltonianque el hamiltoniano <math>H</math> andy en henceconsecuencia <math>M</math> andy <math>\Gamma</math> areson invariantinvariantes underbajo thela followingsiguiente transformationtransformación:
 
<math>{{\Theta }^{-1}}M\Theta=M</math> andy <math>{{\Theta }^{-1}}\Gamma \Theta =\Gamma</math>
 
<math>\Theta</math> ises anun [[Antiunitaryoperador operator|anti-Unitary operatorantiunitario]]<ref>{{Cite book |last=Sakurai |first=J. J. |last2=Napolitano |first2=J. J. |year=2010 |title=Modern Quantum Mechanics |pages= |edition=Second |publisher=[[Addison-Wesley]] |isbn=978-0-805-38291-4}}</ref> andque satisfiessatisface thela relationrelación
 
<math>{{\Theta }^{\dagger }}\Theta =I</math>
 
Por lo que
Hence,
 
<math>{{M}_{22}}=\left\langle 2 \right|M\left| 2 \right\rangle =\left\langle 2 \right|{{\Theta }^{-1}}M\Theta \left| 2 \right\rangle =\left\langle 2 \right|{{\Theta }^{\dagger }}M\Theta \left| 2 \right\rangle =\left\langle 1 \right|M\left| 1 \right\rangle ={{M}_{11}}</math>
 
ande similarlyigualmente forpara thelos diagonalelementos elements ofde <math>\Gamma</math>.
|}
 
HermiticityLa ofhermiticidad de <math>M</math> andy <math>\Gamma</math> alsotambién impliesimplica thatque theirlos diagonalelementos elementsde la diagonal areson realreales.
|}
 
TheLos eigenvaluesautovalores de of <math>H</math> are,son
 
{{Equation box 1 |equation = <math>{{\mu }_{H}}={{M}_{11}}-\frac{i}{2}{{\Gamma }_{11}}+\frac{1}{2}\left( \Delta m-\frac{i}{2}\Delta \Gamma \right)</math> andy,
 
<math>{{\mu }_{L}}={{M}_{11}}-\frac{i}{2}{{\Gamma }_{11}}-\frac{1}{2}\left( \Delta m-\frac{i}{2}\Delta \Gamma \right)</math> |ref=8}}
 
{| class="wikitable collapsible collapsed"
! wheredonde,
|-
| <math>\Delta m</math> andy <math>\Delta \Gamma</math> satisfy,satisfacen
 
<math>{{\left( \Delta m \right)}^{2}}-{{\left( \frac{\Delta \Gamma }{2} \right)}^{2}}=4{{\left| {{M}_{12}} \right|}^{2}}-{{\left| {{\Gamma }_{12}} \right|}^{2}}</math> andy,
 
<math>\Delta m\Delta \Gamma =4\operatorname{Re}\left( {{M}_{12}}{{\Gamma }_{12}}^{*} \right)</math>
|}
TheLos suffixessubíndices standprovienen forde ''Heavy'' and(pesado) y ''Light'' respectively (by convention)ligero, andlo thisque impliessupone thatque <math>\Delta m</math> ises positivepositivo.
 
TheLos normalizedautoestados eigenstatesnormalizados correspondingcorrespondientes toa <math>{{\mu }_{L}}</math> andy <math>{{\mu }_{H}}</math> respectivelyrespectivamente, inen thela [[Standardbase basis|natural basis]] <math>\left\{ \left| P \right\rangle ,\left| {\bar{P}} \right\rangle \right\}\equiv \left\{ \left( 1,0 \right),\left( 0,1 \right) \right\}</math> areson,
 
{{Equation box 1 |equation = <math>\left| {{P}_{L}} \right\rangle =p\left| P \right\rangle +q\left| \bar{P} \right\rangle</math> andy,
 
<math>\left| {{P}_{H}} \right\rangle =p\left| P \right\rangle -q\left| \bar{P} \right\rangle</math> |ref=9}}
 
{| class="wikitable collapsible collapsed"
! wheredonde,
|-
| <math>{{\left| p \right|}^{2}}+{{\left| q \right|}^{2}}=1</math> and,
<math>{{\left( \frac{p}{q} \right)}^{2}}=\frac{{{M}_{12}}^{*}-\frac{i}{2}{{\Gamma }_{12}}^{*}}{{{M}_{12}}-\frac{i}{2}{{\Gamma }_{12}}}</math>
|}
<math>p</math> andy <math>q</math> areson thelos mixingtérminos terms.de Notemezcla. thatLos thedos eigenstatesautoestados areahora no longerson orthogonalortogonales.
 
LetSi theel systemsistema startinicialmente inse theencuentra stateen el estado <math>\left| P \right\rangle</math>. That,esto ises,
 
<math>\left| P \left( 0 \right) \right\rangle =\left| P \right\rangle =\frac{1}{2p}\left( \left| {{P}_{L}} \right\rangle +\left| {{P}_{H}} \right\rangle \right)</math>
 
Bajo evolución temporal se obtiene
Under time evolution we then get,
 
<math>
 
{| class="wikitable collapsible collapsed"
! wheredonde,
|-
| <math>{{g}_{\pm }}\left( t \right)=\frac{1}{2}\left( {{e}^{-\frac{i}{\hbar }\left( {{m}_{H}}-\frac{i}{2}{{\gamma }_{H}} \right)t}}\pm {{e}^{-\frac{i}{\hbar }\left( {{m}_{L}}-\frac{i}{2}{{\gamma }_{L}} \right)t}} \right)</math>
|}
Similarly,De ifun themodo systemsimilar, startssi inel theestado stateinicialmente es <math>\left| {\bar{P}} \right\rangle</math>, underbajo timeevolución evolutiontemporal wese obtain,obtiene
 
<math>
\left| \bar{P}(t) \right\rangle =\frac{1}{2q}\left( \left| {{P}_{L}} \right\rangle {{e}^{-\frac{i}{\hbar }\left( {{m}_{L}}-\frac{i}{2}{{\gamma }_{L}} \right)t}}-\left| {{P}_{H}} \right\rangle {{e}^{-\frac{i}{\hbar }\left( {{m}_{H}}-\frac{i}{2}{{\gamma }_{H}} \right)t}} \right)=-\frac{p}{q}{{g}_{-}}\left( t \right)\left| P \right\rangle +{{g}_{+}}\left( t \right)\left| {\bar{P}} \right\rangle</math>.
 
=== The mixing matrix - a brief introduction ===
{{Main|Cabibbo–Kobayashi–Maskawa matrix|Pontecorvo–Maki–Nakagawa–Sakata matrix}}
 
If the system is a three state system (for example, three species of neutrinos {{SubatomicParticle|Electron Neutrino}}–{{SubatomicParticle|Muon Neutrino}}–{{SubatomicParticle|Tau Neutrino}}, three species of quarks {{SubatomicParticle|down quark}}–{{SubatomicParticle|strange quark}}–{{SubatomicParticle|bottom quark}}), then, just like in the two state system, the flavor eigenstates (say <math>
\left| {{\varphi }_{\alpha }} \right\rangle</math>, <math>
\left| {{\varphi }_{\beta}} \right\rangle</math>, <math>
\left| {{\varphi }_{\gamma}} \right\rangle</math>) are written as a linear combination of the energy (mass) eigenstates (say <math>
\left| {{\psi }_{1}} \right\rangle</math>, <math>
\left| {{\psi }_{2}} \right\rangle</math>, <math>
\left| {{\psi }_{3}} \right\rangle</math>). That is,
 
<math>
\left( \begin{matrix}
\left| {{\varphi }_{\alpha }} \right\rangle \\
\left| {{\varphi }_{\beta }} \right\rangle \\
\left| {{\varphi }_{\gamma }} \right\rangle \\
\end{matrix} \right)=\left( \begin{matrix}
{{\Omega }_{\alpha 1}} & {{\Omega }_{\alpha 2}} & {{\Omega }_{\alpha 3}} \\
{{\Omega }_{\beta 1}} & {{\Omega }_{\beta 2}} & {{\Omega }_{\beta 3}} \\
{{\Omega }_{\gamma 1}} & {{\Omega }_{\gamma 2}} & {{\Omega }_{\gamma 3}} \\
\end{matrix} \right)\left( \begin{matrix}
\left| {{\psi }_{1}} \right\rangle \\
\left| {{\psi }_{2}} \right\rangle \\
\left| {{\psi }_{3}} \right\rangle \\
\end{matrix} \right)</math>.
 
In case of leptons (neutrinos for example) the transformation matrix is the [[Pontecorvo–Maki–Nakagawa–Sakata matrix|PMNS matrix]], and for quarks it is the [[Cabibbo–Kobayashi–Maskawa matrix|CKM matrix]].<ref>{{cite book |last=Griffiths |first=D. J. |year=2008 |title=Elementary Particles |pages=397 |edition=Second, Revised |publisher=[[Wiley-VCH]] |isbn=978-3-527-40601-2}}</ref>
 
N.B. The three familiar neutrino species {{SubatomicParticle|Electron Neutrino}}–{{SubatomicParticle|Muon Neutrino}}–{{SubatomicParticle|Tau Neutrino}} are flavor eigenstates, whereas the three familiar quarks species {{SubatomicParticle|down quark}}–{{SubatomicParticle|strange quark}}–{{SubatomicParticle|bottom quark}} are energy eigenstates.
 
The off diagonal terms of the transformation matrix represent coupling, and unequal diagonal terms imply mixing between the three states.
 
The transformation matrix is unitary and appropriate parameterization (depending on whether it is the CKM or PMNS matrix) is done and the values of the parameters determined experimentally.
 
== CP violation as a consequence ==
}}
 
== The mixing matrix - a brief introduction ==
{{Main|Cabibbo–Kobayashi–Maskawa matrix|Pontecorvo–Maki–Nakagawa–Sakata matrix}}
 
If the system is a three state system (for example, three species of neutrinos {{SubatomicParticle|Electron Neutrino}}–{{SubatomicParticle|Muon Neutrino}}–{{SubatomicParticle|Tau Neutrino}}, three species of quarks {{SubatomicParticle|down quark}}–{{SubatomicParticle|strange quark}}–{{SubatomicParticle|bottom quark}}), then, just like in the two state system, the flavor eigenstates (say <math>
\left| {{\varphi }_{\alpha }} \right\rangle</math>, <math>
\left| {{\varphi }_{\beta}} \right\rangle</math>, <math>
\left| {{\varphi }_{\gamma}} \right\rangle</math>) are written as a linear combination of the energy (mass) eigenstates (say <math>
\left| {{\psi }_{1}} \right\rangle</math>, <math>
\left| {{\psi }_{2}} \right\rangle</math>, <math>
\left| {{\psi }_{3}} \right\rangle</math>). That is,
 
<math>
\left( \begin{matrix}
\left| {{\varphi }_{\alpha }} \right\rangle \\
\left| {{\varphi }_{\beta }} \right\rangle \\
\left| {{\varphi }_{\gamma }} \right\rangle \\
\end{matrix} \right)=\left( \begin{matrix}
{{\Omega }_{\alpha 1}} & {{\Omega }_{\alpha 2}} & {{\Omega }_{\alpha 3}} \\
{{\Omega }_{\beta 1}} & {{\Omega }_{\beta 2}} & {{\Omega }_{\beta 3}} \\
{{\Omega }_{\gamma 1}} & {{\Omega }_{\gamma 2}} & {{\Omega }_{\gamma 3}} \\
\end{matrix} \right)\left( \begin{matrix}
\left| {{\psi }_{1}} \right\rangle \\
\left| {{\psi }_{2}} \right\rangle \\
\left| {{\psi }_{3}} \right\rangle \\
\end{matrix} \right)</math>.
 
In case of leptons (neutrinos for example) the transformation matrix is the [[Pontecorvo–Maki–Nakagawa–Sakata matrix|PMNS matrix]], and for quarks it is the [[Cabibbo–Kobayashi–Maskawa matrix|CKM matrix]].<ref>{{cite book |last=Griffiths |first=D. J. |year=2008 |title=Elementary Particles |pages=397 |edition=Second, Revised |publisher=[[Wiley-VCH]] |isbn=978-3-527-40601-2}}</ref>
 
N.B. The three familiar neutrino species {{SubatomicParticle|Electron Neutrino}}–{{SubatomicParticle|Muon Neutrino}}–{{SubatomicParticle|Tau Neutrino}} are flavor eigenstates, whereas the three familiar quarks species {{SubatomicParticle|down quark}}–{{SubatomicParticle|strange quark}}–{{SubatomicParticle|bottom quark}} are energy eigenstates.
 
The off diagonal terms of the transformation matrix represent coupling, and unequal diagonal terms imply mixing between the three states.
 
The transformation matrix is unitary and appropriate parameterization (depending on whether it is the CKM or PMNS matrix) is done and the values of the parameters determined experimentally.
 
== Véase también ==
514

ediciones