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Pauli equation

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Pauli equation

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 with an external electromagnetic field. It is the non-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.[1]

Contents

  • Equation 1
  • Relationship with the Schrödinger equation and the Dirac equation 2
  • Relationship with Stern–Gerlach experiment 3
  • See also 4
  • References 5
  • External links 6

Equation

For a particle of mass m and charge q, in an electromagnetic field described by the vector potential A = (Ax, Ay, Az) and scalar electric potential ϕ, the Pauli equation reads:

Pauli equation (general)

\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

where σ = (σx, σy, σz) are the Pauli matrices collected into a vector for convenience, p = −∇ is the momentum operator wherein ∇ denotes the gradient operator, and

|\psi\rangle = \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix}

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

The Hamiltonian operator

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

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 for details of this classical case. The kinetic energy term for a free particle in the absence of an electromagnetic field is just p2/2m where p is the kinetic momentum, while in the presence of an EM field we have the minimal coupling p = P − qA, where P is the canonical momentum.

The Pauli matrices can be removed from the kinetic energy term, using the Pauli vector identity:

(\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)

to obtain[2]

Pauli equation (standard form)

\hat{H} |\psi\rangle = \left[\frac{1}{2m}\left[\left(\mathbf{p} - q \mathbf{A}\right)^2 - q \hbar \boldsymbol{\sigma}\cdot \mathbf{B}\right] + q \phi\right]|\psi\rangle = i \hbar \frac{\partial}{\partial t} |\psi\rangle

where B = ∇ × A is the magnetic field.

Relationship with 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:

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:

\left[ \frac{\mathbf{p}^2}{2m} + q \phi \right] \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix} = i \hbar \frac{\partial}{\partial t} \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix}.

Therefore, we can see that the spin of the particle only affects its motion in the presence of a magnetic field.

Relationship with Stern–Gerlach experiment

Both spinor components satisfy the Schrödinger equation. For a particle in an externally applied B field, the Pauli equation reads:

Pauli equation (B-field)

\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}

where

\hat 1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}

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

The 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

References


-- Module:Hatnote -- -- -- -- This module produces hatnote links and links to related articles. It -- -- implements the and meta-templates and includes -- -- helper functions for other Lua hatnote modules. --


local libraryUtil = require('libraryUtil') local checkType = libraryUtil.checkType local mArguments -- lazily initialise Module:Arguments local yesno -- lazily initialise Module:Yesno

local p = {}


-- Helper functions


local function getArgs(frame) -- Fetches the arguments from the parent frame. Whitespace is trimmed and -- blanks are removed. mArguments = require('Module:Arguments') return mArguments.getArgs(frame, {parentOnly = true}) end

local function removeInitialColon(s) -- Removes the initial colon from a string, if present. return s:match('^:?(.*)') end

function p.findNamespaceId(link, removeColon) -- Finds the namespace id (namespace number) of a link or a pagename. This -- function will not work if the link is enclosed in double brackets. Colons -- are trimmed from the start of the link by default. To skip colon -- trimming, set the removeColon parameter to true. checkType('findNamespaceId', 1, link, 'string') checkType('findNamespaceId', 2, removeColon, 'boolean', true) if removeColon ~= false then link = removeInitialColon(link) end local namespace = link:match('^(.-):') if namespace then local nsTable = mw.site.namespaces[namespace] if nsTable then return nsTable.id end end return 0 end

function p.formatPages(...) -- Formats a list of pages using formatLink and returns it as an array. Nil -- values are not allowed. local pages = {...} local ret = {} for i, page in ipairs(pages) do ret[i] = p._formatLink(page) end return ret end

function p.formatPageTables(...) -- Takes a list of page/display tables and returns it as a list of -- formatted links. Nil values are not allowed. local pages = {...} local links = {} for i, t in ipairs(pages) do checkType('formatPageTables', i, t, 'table') local link = t[1] local display = t[2] links[i] = p._formatLink(link, display) end return links end

function p.makeWikitextError(msg, helpLink, addTrackingCategory) -- Formats an error message to be returned to wikitext. If -- addTrackingCategory is not false after being returned from -- Module:Yesno, and if we are not on a talk page, a tracking category -- is added. checkType('makeWikitextError', 1, msg, 'string') checkType('makeWikitextError', 2, helpLink, 'string', true) yesno = require('Module:Yesno') local title = mw.title.getCurrentTitle() -- Make the help link text. local helpText if helpLink then helpText = ' (help)' else helpText = end -- Make the category text. local category if not title.isTalkPage and yesno(addTrackingCategory) ~= false then category = 'Hatnote templates with errors' category = string.format( '%s:%s', mw.site.namespaces[14].name, category ) else category = end return string.format( '%s', msg, helpText, category ) end


-- Format link -- -- Makes a wikilink from the given link and display values. Links are escaped -- with colons if necessary, and links to sections are detected and displayed -- with " § " as a separator rather than the standard MediaWiki "#". Used in -- the template.


function p.formatLink(frame) local args = getArgs(frame) local link = args[1] local display = args[2] if not link then return p.makeWikitextError( 'no link specified', 'Template:Format hatnote link#Errors', args.category ) end return p._formatLink(link, display) end

function p._formatLink(link, display) -- Find whether we need to use the colon trick or not. We need to use the -- colon trick for categories and files, as otherwise category links -- categorise the page and file links display the file. checkType('_formatLink', 1, link, 'string') checkType('_formatLink', 2, display, 'string', true) link = removeInitialColon(link) local namespace = p.findNamespaceId(link, false) local colon if namespace == 6 or namespace == 14 then colon = ':' else colon = end -- Find whether a faux display value has been added with the | magic -- word. if not display then local prePipe, postPipe = link:match('^(.-)|(.*)$') link = prePipe or link display = postPipe end -- Find the display value. if not display then local page, section = link:match('^(.-)#(.*)$') if page then display = page .. ' § ' .. section end end -- Assemble the link. if display then return string.format('%s', colon, link, display) else return string.format('%s%s', colon, link) end end


-- Hatnote -- -- Produces standard hatnote text. Implements the template.


function p.hatnote(frame) local args = getArgs(frame) local s = args[1] local options = {} if not s then return p.makeWikitextError( 'no text specified', 'Template:Hatnote#Errors', args.category ) end options.extraclasses = args.extraclasses options.selfref = args.selfref return p._hatnote(s, options) end

function p._hatnote(s, options) checkType('_hatnote', 1, s, 'string') checkType('_hatnote', 2, options, 'table', true) local classes = {'hatnote'} local extraclasses = options.extraclasses local selfref = options.selfref if type(extraclasses) == 'string' then classes[#classes + 1] = extraclasses end if selfref then classes[#classes + 1] = 'selfref' end return string.format( '
%s
', table.concat(classes, ' '), s )

end

return p-------------------------------------------------------------------------------- -- Module:Hatnote -- -- -- -- This module produces hatnote links and links to related articles. It -- -- implements the and meta-templates and includes -- -- helper functions for other Lua hatnote modules. --


local libraryUtil = require('libraryUtil') local checkType = libraryUtil.checkType local mArguments -- lazily initialise Module:Arguments local yesno -- lazily initialise Module:Yesno

local p = {}


-- Helper functions


local function getArgs(frame) -- Fetches the arguments from the parent frame. Whitespace is trimmed and -- blanks are removed. mArguments = require('Module:Arguments') return mArguments.getArgs(frame, {parentOnly = true}) end

local function removeInitialColon(s) -- Removes the initial colon from a string, if present. return s:match('^:?(.*)') end

function p.findNamespaceId(link, removeColon) -- Finds the namespace id (namespace number) of a link or a pagename. This -- function will not work if the link is enclosed in double brackets. Colons -- are trimmed from the start of the link by default. To skip colon -- trimming, set the removeColon parameter to true. checkType('findNamespaceId', 1, link, 'string') checkType('findNamespaceId', 2, removeColon, 'boolean', true) if removeColon ~= false then link = removeInitialColon(link) end local namespace = link:match('^(.-):') if namespace then local nsTable = mw.site.namespaces[namespace] if nsTable then return nsTable.id end end return 0 end

function p.formatPages(...) -- Formats a list of pages using formatLink and returns it as an array. Nil -- values are not allowed. local pages = {...} local ret = {} for i, page in ipairs(pages) do ret[i] = p._formatLink(page) end return ret end

function p.formatPageTables(...) -- Takes a list of page/display tables and returns it as a list of -- formatted links. Nil values are not allowed. local pages = {...} local links = {} for i, t in ipairs(pages) do checkType('formatPageTables', i, t, 'table') local link = t[1] local display = t[2] links[i] = p._formatLink(link, display) end return links end

function p.makeWikitextError(msg, helpLink, addTrackingCategory) -- Formats an error message to be returned to wikitext. If -- addTrackingCategory is not false after being returned from -- Module:Yesno, and if we are not on a talk page, a tracking category -- is added. checkType('makeWikitextError', 1, msg, 'string') checkType('makeWikitextError', 2, helpLink, 'string', true) yesno = require('Module:Yesno') local title = mw.title.getCurrentTitle() -- Make the help link text. local helpText if helpLink then helpText = ' (help)' else helpText = end -- Make the category text. local category if not title.isTalkPage and yesno(addTrackingCategory) ~= false then category = 'Hatnote templates with errors' category = string.format( '%s:%s', mw.site.namespaces[14].name, category ) else category = end return string.format( '%s', msg, helpText, category ) end


-- Format link -- -- Makes a wikilink from the given link and display values. Links are escaped -- with colons if necessary, and links to sections are detected and displayed -- with " § " as a separator rather than the standard MediaWiki "#". Used in -- the template.


function p.formatLink(frame) local args = getArgs(frame) local link = args[1] local display = args[2] if not link then return p.makeWikitextError( 'no link specified', 'Template:Format hatnote link#Errors', args.category ) end return p._formatLink(link, display) end

function p._formatLink(link, display) -- Find whether we need to use the colon trick or not. We need to use the -- colon trick for categories and files, as otherwise category links -- categorise the page and file links display the file. checkType('_formatLink', 1, link, 'string') checkType('_formatLink', 2, display, 'string', true) link = removeInitialColon(link) local namespace = p.findNamespaceId(link, false) local colon if namespace == 6 or namespace == 14 then colon = ':' else colon = end -- Find whether a faux display value has been added with the | magic -- word. if not display then local prePipe, postPipe = link:match('^(.-)|(.*)$') link = prePipe or link display = postPipe end -- Find the display value. if not display then local page, section = link:match('^(.-)#(.*)$') if page then display = page .. ' § ' .. section end end -- Assemble the link. if display then return string.format('%s', colon, link, display) else return string.format('%s%s', colon, link) end end


-- Hatnote -- -- Produces standard hatnote text. Implements the template.


function p.hatnote(frame) local args = getArgs(frame) local s = args[1] local options = {} if not s then return p.makeWikitextError( 'no text specified', 'Template:Hatnote#Errors', args.category ) end options.extraclasses = args.extraclasses options.selfref = args.selfref return p._hatnote(s, options) end

function p._hatnote(s, options) checkType('_hatnote', 1, s, 'string') checkType('_hatnote', 2, options, 'table', true) local classes = {'hatnote'} local extraclasses = options.extraclasses local selfref = options.selfref if type(extraclasses) == 'string' then classes[#classes + 1] = extraclasses end if selfref then classes[#classes + 1] = 'selfref' end return string.format( '
%s
', table.concat(classes, ' '), s )

end

return p
  1. ^ Wolfgang Pauli (1927) Zur Quantenmechanik des magnetischen Elektrons Zeitschrift für Physik (43) 601-623
  2. ^

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