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| {{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]) |
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| 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>
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| == Equation ==
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| 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:
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| {{Equation box 1
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| |title='''Pauli equation''' ''(General)''
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| |indent =:
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| |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>
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| 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
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| :<math> |\psi\rangle = \begin{pmatrix}
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| \psi_+ \\
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| \psi_-
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| \end{pmatrix}</math>
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| is the two-component [[spinor]] [[wavefunction]], a [[column vector]] written in [[Dirac notation]].
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| The [[Hamiltonian (quantum mechanics)|Hamiltonian operator]]
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| :<math>\hat{H} = \frac{1}{2m}(\boldsymbol{\sigma}\cdot(\mathbf{p} - q \mathbf{A}))^2 + q \phi</math>
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| 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]].
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| The Pauli matrices can be removed from the kinetic energy term, using the [[Pauli matrices#Relation to dot and cross product|Pauli vector identity]]:
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| :<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>
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| 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> | |
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| :<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>
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| where '''B''' = ∇ × '''A''' is the [[magnetic field]].
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| == Relationship to the Schrödinger equation and the Dirac equation ==
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| The Pauli equation is non-relativistic, but it does predict spin. As such, it can be thought of as occupying the middle ground between:
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| * The familiar Schrödinger equation (on a complex scalar [[wavefunction]]), which is non-relativistic and does not predict spin.
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| * 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.
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| 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.
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| == Special cases ==
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| 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:
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| {{Equation box 1
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| |title='''Pauli equation''' ''(B-field)''
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| |indent =:
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| |equation = <math>
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| \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>
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| |background colour=#F5FFFA}}
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| where
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| :<math> \hat 1 = \begin{pmatrix}
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| 1 & 0 \\
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| 0 & 1 \\
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| \end{pmatrix} </math>
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| is the 2 × 2 [[identity matrix]], which acts as an [[identity operator]].
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| 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.
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| 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]].
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| ==See also==
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| * [[Semiclassical physics]]
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| * [[Atomic, molecular, and optical physics]]
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| == References ==
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| {{reflist}}
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| * {{cite book | author=Schwabl, Franz| title=Quantenmechanik I | publisher=Springer |year=2004 |id=ISBN 978-3540431060}}
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| * {{cite book | author=Schwabl, Franz| title=Quantenmechanik für Fortgeschrittene | publisher=Springer |year=2005 |id=ISBN 978-3540259046}}
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| * {{cite book | author=Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe| title= Quantum Mechanics 2| publisher=Wiley, J |year=2006 |id=ISBN 978-0471569527}}
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| == External links ==
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| {{DEFAULTSORT:Pauli Equation}}
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| [[Category:Quantum mechanics]]
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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.
Feel free to visit my site ... extended auto warranty (visit the next post)