Pauli equation

From Wikipedia, the free encyclopedia

Please help improve this article or section by expanding it.
Further information might be found on the talk page or at requests for expansion.
This article has been tagged since February 2007.

This article or section is in need of attention from an expert on the subject.
WikiProject Physics or the Physics Portal may be able to help recruit one.
If a more appropriate WikiProject or portal exists, please adjust this template accordingly.

Quantum physics

Quantum mechanics
Introduction to... Mathematical formulation of...
Fundamental concepts
Decoherence · Interference Uncertainty · Exclusion Transformation theory Ehrenfest theorem · Measurement
Experiments
Double-slit experiment Davisson-Germer experiment Stern–Gerlach experiment EPR paradox · Popper's experiment Schrödinger's cat
Equations
Schrödinger equation Pauli equation Klein-Gordon equation Dirac equation
Advanced theories
Quantum field theory Quantum electrodynamics Quantum chromodynamics Quantum gravity Feynman diagram
Interpretations
Copenhagen · Quantum logic Hidden variables · Transactional Many-worlds · Ensemble Consistent histories · Relational Consciousness causes collapse Orchestrated objective reduction
Scientists
Planck · Schrödinger Heisenberg · Bohr · Pauli Dirac · Bohm · Born de Broglie · von Neumann Einstein · Feynman Everett · Others
This box: view • talk • edit

The Pauli equation is a Schrödinger equation which describes the time evolution of spin 1/2 particles (eg. electrons). It is the non-relativistic border case of the Dirac equation and can be used where particles are slow enough that relativistic effects can be neglected.

The Pauli equation was formulated by Wolfgang Pauli.

1 Details
2 Derivation of the Pauli equation
3 Examples
4 References
5 External links

The time dependent linear Pauli equation :

$(\vec{\sigma}\cdot \vec{p} - c)|\psi\rangle=0$

where

$\vec{p}$ is the momentum

c is the speed of light

$\vec{\sigma}$ are the Pauli matrices

$|\psi\rangle := \begin{pmatrix} |\psi_+\rangle \\ |\psi_-\rangle \end{pmatrix}$ is the Pauli-Spinor

Both spinor components satisfy the Schrödinger-Equation. This means that the system is as to the additional degree of freedom, degenerated.

With an external electromagnetic field the full Pauli equation reads:

$\underbrace{i \hbar \partial_t \vec \varphi_\pm = \left( \frac{(\underline{\vec p}-q \cdot \vec A)^2}{2 m} + q \phi \right) \hat 1 \vec \varphi_\pm}_{\rm{Schr}\ddot o\rm{dingerequation}} - \underbrace{\frac{q \hbar}{2m}\vec{\hat \sigma} \cdot \vec B \vec \varphi_\pm}_{\rm Stern Gerlach term}$ .

where

φ

is the skalar electric potenial

A

das Vector potential

$\vec \varphi_\pm$ bzw. in Dirac-Notation $|\psi\rangle :=\begin{pmatrix} |\varphi_+\rangle \\ |\varphi_-\rangle \end{pmatrix}$ are the Pauli-Spinors

$\vec{\hat \sigma}$ are the Pauli matrices

$\vec B$ is the external magnetic field

$\hat 1$ two dimensional Identity matrix

With the Stern Gerlach term it is possible to comprehend the obtaing of spin orientation of atoms with one valence electron e.g. silver atoms which flow through an inhomogenous magntic field.

Analogous the term is resonsible for the energetic disperment in a magnetic field as can be viewed in the anomal Zeeman effect.

Starting from the Dirac equation for weak electromagnetic interactions :

$i \hbar \partial_t \left( \begin{array}{c} \vec \varphi_1\\\vec \varphi_2\end{array} \right) = c \left( \begin{array}{c} \vec{\hat \sigma} \vec \pi \vec \varphi_2\\\vec{\hat \sigma} \vec \pi \vec \varphi_1\end{array} \right)+q \phi \left( \begin{array}{c} \vec \varphi_1\\\vec \varphi_2\end{array} \right) + mc^2 \left( \begin{array}{c} \vec \varphi_1 \\-\vec \varphi_2\end{array} \right)$

with $\vec \pi = \vec p - q \vec A$

using the following approximatations :

Simplification of the equation through following ansatz

$\left( \begin{array}{c} \vec \varphi_1 \\ \vec \varphi_2 \end{array} \right) = e^{-i \frac{mc^2t}{\hbar}} \left( \begin{array}{c} \vec{\tilde \varphi_1} \\ \vec{\tilde \varphi_2} \end{array} \right)$

Eliminating the rest energy through an Ansatz with slow time dependence

$\partial_t \vec \varphi_i \ll \frac{mc^2}{\hbar} \vec \varphi_i$

weak coupling of the electric potential

$q \phi \ll mc^2$

Schwabl, Franz (2004). Quantemechanik I. Springer. ISBN 978-3540431060.
Schwabl, Franz (2005). Quantemechanik für Fortgeschrittene. Springer. ISBN 978-3540259046.
Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe (2006). Quantum Mechanics 2. Wiley, J. ISBN 978-0471569527.

This physics-related article is a stub. You can help Wikipedia by expanding it.

Retrieved from "http://en.wikipedia.org../../../p/a/u/Pauli_equation.html"

Categories: Articles to be expanded since February 2007 | All articles to be expanded | Physics articles needing expert attention | Physics stubs

Searching reference...

Reference

Sources

Pauli equation

From Wikipedia, the free encyclopedia

Contents

Top Matching Results

Sponsored Links
This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Searching reference...

Reference

Sources

Pauli equation

From Wikipedia, the free encyclopedia

Contents

Top Matching Results

Sponsored Links This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Sponsored Links
This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.