Eric Bach - Revision history

en>Rjwilmsi: Added 1 doi to a journal cite using AWB (10213)

2014-05-24T14:00:04Z

Added 1 doi to a journal cite using AWB (10213)

← Older revision		Revision as of 16:00, 24 May 2014
Line 1:		Line 1:
	~~{{Quantum mechanics\|cTopic=Equations}}~~		Claude is her name and she completely digs that name. Delaware has always been my living place and will by no means move. Interviewing is what I do in my day occupation. Bottle tops collecting is the only pastime his spouse doesn't approve of.<br><br>Feel free to visit my site ... extended auto warranty ([http://Mwarrenassociates.com/UserProfile/tabid/253/userId/228462/Default.aspx visit the next post])

	In [[quantum mechanics]], the '''Pauli equation''' or '''Schrödinger–Pauli equation''' is the formulation of the [[Schrödinger equation]] for [[spin-½]] particles, which takes into account the interaction of the particle's [[spin (physics)\|spin]] with an external [[electromagnetic field]]. It is ~~the non-[[special relativity\|relativistic]] limit of the [[Dirac equation]]~~ and ~~can be used where particles are moving at speeds much less than the [[speed of light]], so~~ that ~~relativistic effects can be neglected~~. ~~It was formulated~~ by ~~[[Wolfgang Pauli]] in 1927~~.~~<ref>Wolfgang Pauli (1927) ''Zur Quantenmechanik des magnetischen Elektrons'' ''Zeitschrift für Physik'' (43) 601-623</ref>~~

	~~== Equation ==~~

	For a particle of mass ''m'' and charge ''q'', in an [[electromagnetic field]] described by the [[vector potential]] '''A''' = (''A<sub>x</sub>'', ''A<sub>y</sub>'', ''A<sub>z</sub>'') and [[scalar potential\|scalar]] [[electric potential]] ''ϕ'', the Pauli equation reads:

	~~{{Equation box 1~~
	~~\|title='''Pauli equation''' ''(General)''~~
	~~\|indent =:~~
	~~\|equation = <math>\left[ \frac{1}{2m}(\boldsymbol{\sigma}\cdot(\mathbf{p} - q \mathbf{A}))^2 + q \phi \right] \|\psi\rangle = i \hbar \frac{\partial}{\partial t} \|\psi\rangle </math>~~
	~~\|cellpadding~~
	~~\|border~~
	~~\|border colour = #50C878~~
	~~\|background colour = #ECFCF4}}~~

	~~where '''σ''' = (''σ<sub>x</sub>'', ''σ<sub>y</sub>'', ''σ<sub>z</sub>'') are the [[Pauli matrices]] collected into a vector for convenience, '''p''' = −''iħ''∇~~ is ~~the [[momentum operator]] wherein ∇ denotes the [[gradient operator]], and~~

	~~:<math> \|\psi\rangle = \begin{pmatrix}~~
	~~\psi_+ \\~~
	~~\psi_-~~
	~~\end{pmatrix}</math>~~

	~~is the two-component [[spinor]] [[wavefunction]], a [[column vector]] written~~ in ~~[[Dirac notation]].~~

	~~The [[Hamiltonian (quantum mechanics)\|Hamiltonian operator]]~~

	~~:<math>\hat{H} = \frac{1}{2m}(\boldsymbol{\sigma}\cdot(\mathbf{p} - q \mathbf{A}))^2 + q \phi</math>~~

	~~is a 2 × 2 matrix operator, because of the Pauli matrices. Substitution into the [[Schrödinger equation]] gives the Pauli equation~~. ~~This Hamiltonian~~ is ~~similar to~~ the classical Hamiltonian for a charged particle interacting with an electromagnetic field, see [[Lorentz_force#Lorentz_force_and_analytical_mechanics\|Lorentz force]] for details of this classical case. The [[kinetic energy]] term for a free particle in the absence of an electromagnetic field is just '~~''p'''<sup>2</sup>/2''m'' where '''p''' is the [[Momentum#Particle in a field\|''kinetic'' momentum]], while in the presence~~ of ~~an EM field we have the [[minimal coupling]] '''p''' = '''P''' − q'''A''', where '''P''' is the [[canonical momentum]]~~.

	~~The Pauli matrices can be removed from the kinetic energy term, using the [[Pauli matrices#Relation to dot and cross product\|Pauli vector identity]]:~~

	:<~~math~~>~~(\boldsymbol{\sigma}\cdot \mathbf{a})(\boldsymbol{\sigma}\cdot \mathbf{b}) = \mathbf{a}\cdot\mathbf{b} + i\boldsymbol{\sigma}\cdot \left(\mathbf{a} \times \mathbf{b}\right)~~<~~/math~~>

	to ~~obtain<ref>{{Cite book\|title=Physics of Atoms and Molecules\|author=Bransden, BH\|coauthors=Joachain, CJ\|year=1983\|publisher=Prentice Hall\|edition=1st\|page=638-638\|isbn=0-582-44401-2}}</ref>~~

	~~:<math>\hat{H} = \frac{1}{2m}\left[\left(\mathbf{p} - q \mathbf{A}\right)^2 - \hbar q \boldsymbol{\sigma}\cdot \mathbf{B}\right] + q \phi</math>~~

	~~where '''B''' = ∇ × '''A''' is the [[magnetic field]]~~.

	~~== Relationship to the Schrödinger equation and the Dirac equation ==~~

	~~The Pauli equation is non-relativistic, but it does predict spin~~. ~~As such, it can be thought of as occupying the middle ground between:~~
	* The familiar Schrödinger equation (on a complex scalar [[wavefunction]]), which is non-relativistic and does not predict spin.
	* The Dirac equation (~~on a~~ [~~[dirac spinor\|complex four-component spinor]]), which is fully [[special relativity\|relativistic]] (with respect to [[special relativity]]) and predicts spin.~~

	Note that because of the properties of the Pauli matrices, if the magnetic vector potential '''A''' is equal to zero, then the equation reduces to the familiar Schrödinger equation for a particle in a purely electric potential ''ϕ'', except that it operates on a two component spinor. Therefore, we can see that the spin of the particle only affects its motion in the presence of a magnetic field.

	~~== Special cases ==~~

	~~Both spinor components satisfy the Schrödinger equation. <!---What does this mean~~:~~"This means that the system is degenerated as to the additional degree of freedom~~.~~" ???---> For a particle in an externally applied '''B''' field, the Pauli equation reads:~~

	~~{{Equation box 1~~
	~~\|title='''Pauli equation''' ''(B-field)''~~
	~~\|indent =:~~
	~~\|equation = <math>~~
	\underbrace{i \hbar \frac{\partial}{\partial t} \|\psi_\pm\rangle = \left( \frac{( \mathbf{p} -q \bold A)^2}{2 m} + q \phi \right) \hat 1 \bold \|\psi\rangle }_\mathrm{Schr\ddot{o}dinger~equation} - \underbrace{\frac{q \hbar}{2m}\boldsymbol{\sigma} \cdot \bold B \bold \|\psi\rangle }_\mathrm{Stern \, Gerlach \, term}</~~math>~~
	~~\|cellpadding~~
	~~\|border~~
	~~\|border colour = #0073CF~~
	~~\|background colour=#F5FFFA}}~~

	~~where~~

	~~:<math> \hat 1 = \begin{pmatrix}~~
	~~1 & 0 \\~~
	~~0 & 1 \\~~
	~~\end{pmatrix} <~~/~~math>~~

	~~is the 2 × 2 [[identity matrix]], which acts as an [[identity operator]]~~.

	~~The [[Stern–Gerlach experiment\|Stern–Gerlach term]] can obtain~~ the ~~spin orientation of atoms with one [[valence electron~~]~~], e.g. silver atoms which flow through an inhomogeneous magnetic field.~~

	~~Analogously, the term is responsible for the splitting of spectral lines (corresponding to energy levels~~) ~~in a magnetic field as can be viewed in the [[anomalous Zeeman effect]].~~

	~~==See also==~~

	* [[Semiclassical physics]]
	* [[Atomic, molecular, and optical physics]]

	~~== References ==~~

	~~{{reflist}}~~

	* {{cite book \| author=Schwabl, Franz\| title=Quantenmechanik I \| publisher=Springer \|year=2004 \|id=ISBN 978-3540431060}}
	* {{cite book \| author=Schwabl, Franz\| title=Quantenmechanik für Fortgeschrittene \| publisher=Springer \|year=2005 \|id=ISBN 978-3540259046}}
	* {{cite book \| author=Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe\| title= Quantum Mechanics 2\| publisher=Wiley, J \|year=2006 \|id=ISBN 978-0471569527}}

	~~== External links ==~~

	~~{{DEFAULTSORT:Pauli Equation}}~~
	~~[[Category:Quantum mechanics]]~~

en>Waacstats: add persondata short description using AWB (8759)

2013-03-31T15:36:38Z

add persondata short description using AWB (8759)

← Older revision		Revision as of 17:36, 31 March 2013
Line 1:		Line 1:
	~~Alyson~~ is the title ~~individuals use to call me~~ and ~~I think it sounds quite great when you say it~~. ~~Playing badminton~~ is a ~~factor that he~~ is ~~totally addicted~~ to. He is an ~~purchase clerk~~ and it'~~s some thing he really appreciate. Some time in~~ the ~~past he selected~~ to ~~reside~~ in ~~North Carolina and he doesn~~'~~t plan~~ on ~~altering it~~.<br><br>~~My blog~~ - ~~love psychic readings; [http~~:/~~/brazil~~.~~amor-amore~~.~~com/irboothe brazil~~.~~amor-amore~~.~~com~~],		{{Quantum mechanics\|cTopic=Equations}}

			In [[quantum mechanics]], the '''Pauli equation''' or '''Schrödinger–Pauli equation''' is the formulation of the [[Schrödinger equation]] for [[spin-½]] particles, which takes into account the interaction of the particle's [[spin (physics)\|spin]] with an external [[electromagnetic field]]. It is the non-[[special relativity\|relativistic]] limit of the [[Dirac equation]] and can be used where particles are moving at speeds much less than the [[speed of light]], so that relativistic effects can be neglected. It was formulated by [[Wolfgang Pauli]] in 1927.<ref>Wolfgang Pauli (1927) ''Zur Quantenmechanik des magnetischen Elektrons'' ''Zeitschrift für Physik'' (43) 601-623</ref>

			== Equation ==

			For a particle of mass ''m'' and charge ''q'', in an [[electromagnetic field]] described by the [[vector potential]] '''A''' = (''A<sub>x</sub>'', ''A<sub>y</sub>'', ''A<sub>z</sub>'') and [[scalar potential\|scalar]] [[electric potential]] ''ϕ'', the Pauli equation reads:

			{{Equation box 1
			\|title='''Pauli equation''' ''(General)''
			\|indent =:
			\|equation = <math>\left[ \frac{1}{2m}(\boldsymbol{\sigma}\cdot(\mathbf{p} - q \mathbf{A}))^2 + q \phi \right] \|\psi\rangle = i \hbar \frac{\partial}{\partial t} \|\psi\rangle </math>
			\|cellpadding
			\|border
			\|border colour = #50C878
			\|background colour = #ECFCF4}}

			where '''σ''' = (''σ<sub>x</sub>'', ''σ<sub>y</sub>'', ''σ<sub>z</sub>'') are the [[Pauli matrices]] collected into a vector for convenience, '''p''' = −''iħ''∇ is the [[momentum operator]] wherein ∇ denotes the [[gradient operator]], and

			:<math> \|\psi\rangle = \begin{pmatrix}
			\psi_+ \\
			\psi_-
			\end{pmatrix}</math>

			is the two-component [[spinor]] [[wavefunction]], a [[column vector]] written in [[Dirac notation]].

			The [[Hamiltonian (quantum mechanics)\|Hamiltonian operator]]

			:<math>\hat{H} = \frac{1}{2m}(\boldsymbol{\sigma}\cdot(\mathbf{p} - q \mathbf{A}))^2 + q \phi</math>

			is a 2 × 2 matrix operator, because of the Pauli matrices. Substitution into the [[Schrödinger equation]] gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field, see [[Lorentz_force#Lorentz_force_and_analytical_mechanics\|Lorentz force]] for details of this classical case. The [[kinetic energy]] term for a free particle in the absence of an electromagnetic field is just '''p'''<sup>2</sup>/2''m'' where '''p''' is the [[Momentum#Particle in a field\|''kinetic'' momentum]], while in the presence of an EM field we have the [[minimal coupling]] '''p''' = '''P''' − q'''A''', where '''P''' is the [[canonical momentum]].

			The Pauli matrices can be removed from the kinetic energy term, using the [[Pauli matrices#Relation to dot and cross product\|Pauli vector identity]]:

			:<math>(\boldsymbol{\sigma}\cdot \mathbf{a})(\boldsymbol{\sigma}\cdot \mathbf{b}) = \mathbf{a}\cdot\mathbf{b} + i\boldsymbol{\sigma}\cdot \left(\mathbf{a} \times \mathbf{b}\right)</math>

			to obtain<ref>{{Cite book\|title=Physics of Atoms and Molecules\|author=Bransden, BH\|coauthors=Joachain, CJ\|year=1983\|publisher=Prentice Hall\|edition=1st\|page=638-638\|isbn=0-582-44401-2}}</ref>

			:<math>\hat{H} = \frac{1}{2m}\left[\left(\mathbf{p} - q \mathbf{A}\right)^2 - \hbar q \boldsymbol{\sigma}\cdot \mathbf{B}\right] + q \phi</math>

			where '''B''' = ∇ × '''A''' is the [[magnetic field]].

			== Relationship to the Schrödinger equation and the Dirac equation ==

			The Pauli equation is non-relativistic, but it does predict spin. As such, it can be thought of as occupying the middle ground between:
			* The familiar Schrödinger equation (on a complex scalar [[wavefunction]]), which is non-relativistic and does not predict spin.
			* The Dirac equation (on a [[dirac spinor\|complex four-component spinor]]), which is fully [[special relativity\|relativistic]] (with respect to [[special relativity]]) and predicts spin.

			Note that because of the properties of the Pauli matrices, if the magnetic vector potential '''A''' is equal to zero, then the equation reduces to the familiar Schrödinger equation for a particle in a purely electric potential ''ϕ'', except that it operates on a two component spinor. Therefore, we can see that the spin of the particle only affects its motion in the presence of a magnetic field.

			== Special cases ==

			Both spinor components satisfy the Schrödinger equation. <!---What does this mean:"This means that the system is degenerated as to the additional degree of freedom." ???---> For a particle in an externally applied '''B''' field, the Pauli equation reads:

			{{Equation box 1
			\|title='''Pauli equation''' ''(B-field)''
			\|indent =:
			\|equation = <math>
			\underbrace{i \hbar \frac{\partial}{\partial t} \|\psi_\pm\rangle = \left( \frac{( \mathbf{p} -q \bold A)^2}{2 m} + q \phi \right) \hat 1 \bold \|\psi\rangle }_\mathrm{Schr\ddot{o}dinger~equation} - \underbrace{\frac{q \hbar}{2m}\boldsymbol{\sigma} \cdot \bold B \bold \|\psi\rangle }_\mathrm{Stern \, Gerlach \, term}</math>
			\|cellpadding
			\|border
			\|border colour = #0073CF
			\|background colour=#F5FFFA}}

			where

			:<math> \hat 1 = \begin{pmatrix}
			1 & 0 \\
			0 & 1 \\
			\end{pmatrix} </math>

			is the 2 × 2 [[identity matrix]], which acts as an [[identity operator]].

			The [[Stern–Gerlach experiment\|Stern–Gerlach term]] can obtain the spin orientation of atoms with one [[valence electron]], e.g. silver atoms which flow through an inhomogeneous magnetic field.

			Analogously, the term is responsible for the splitting of spectral lines (corresponding to energy levels) in a magnetic field as can be viewed in the [[anomalous Zeeman effect]].

			==See also==

			* [[Semiclassical physics]]
			* [[Atomic, molecular, and optical physics]]

			== References ==

			{{reflist}}

			* {{cite book \| author=Schwabl, Franz\| title=Quantenmechanik I \| publisher=Springer \|year=2004 \|id=ISBN 978-3540431060}}
			* {{cite book \| author=Schwabl, Franz\| title=Quantenmechanik für Fortgeschrittene \| publisher=Springer \|year=2005 \|id=ISBN 978-3540259046}}
			* {{cite book \| author=Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe\| title= Quantum Mechanics 2\| publisher=Wiley, J \|year=2006 \|id=ISBN 978-0471569527}}

			== External links ==

			{{DEFAULTSORT:Pauli Equation}}
			[[Category:Quantum mechanics]]

en>Closedmouth: Reverted edits by Supagirl88 (talk) to last version by Δ

2011-11-13T14:12:18Z

Reverted edits by Supagirl88 (talk) to last version by Δ

New page

Alyson is the title individuals use to call me and I think it sounds quite great when you say it. Playing badminton is a factor that he is totally addicted to. He is an purchase clerk and it's some thing he really appreciate. Some time in the past he selected to reside in North Carolina and he doesn't plan on altering it.<br><br>My blog - love psychic readings; [http://brazil.amor-amore.com/irboothe brazil.amor-amore.com],