equations:dirac_equation
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equations:dirac_equation [2018/03/26 17:16] jakobadmin |
equations:dirac_equation [2023/04/02 03:11] (current) edi [Concrete] |
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| - | ====== Dirac Equation: $ \quad (i\gamma_\mu \partial^\mu - m ) \Psi =0 $ ====== | + | <WRAP lag>$ (i\gamma_\mu \partial^\mu - m ) \Psi =0 $</WRAP> |
| + | |||
| + | ====== Dirac Equation ====== | ||
| <tabbox Intuitive> | <tabbox Intuitive> | ||
| + | The Dirac equation describes how the state of a relativistic (= fast moving) quantum system with half-integer spin changes in time. | ||
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| + | The analogous equation for systems without spin is the [[equations:klein-gordon_equation|Klein-Gordon equation]]. | ||
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| + | If the system only moves slowly, the Dirac equation becomes the [[equations:pauli_equation|Pauli equation]]. | ||
| - | <note tip> | ||
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | ||
| - | </note> | ||
| - | | ||
| <tabbox Concrete> | <tabbox Concrete> | ||
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| * For a nice description, see section 7.4.3 "Diracology" in the book The Conceptual Framework of Quantum Field Theory by Duncan | * For a nice description, see section 7.4.3 "Diracology" in the book The Conceptual Framework of Quantum Field Theory by Duncan | ||
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| + | ---- | ||
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| + | **Graphical Summary** | ||
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| + | The diagram below shows the Dirac equation and its Lagrangian in various forms. For a more detailed explanation see [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#dirac|Fun with Symmetry]]. | ||
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| + | {{:equations:dirac.jpg?nolink}} | ||
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| <tabbox Abstract> | <tabbox Abstract> | ||
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| * $\partial _{\mu} $ denotes the partial derivative and $ \gamma_{\mu} \partial^{\mu}$ stands for a sum using the Einstein sum convention, i.e. $\gamma_{\mu} \partial ^{\mu} = \gamma_0 \partial^0 - \gamma_1 \partial^1 -\gamma_2 \partial^2 -\gamma_3 \partial^3$, | * $\partial _{\mu} $ denotes the partial derivative and $ \gamma_{\mu} \partial^{\mu}$ stands for a sum using the Einstein sum convention, i.e. $\gamma_{\mu} \partial ^{\mu} = \gamma_0 \partial^0 - \gamma_1 \partial^1 -\gamma_2 \partial^2 -\gamma_3 \partial^3$, | ||
| * $m$ denotes the mass of the particle, | * $m$ denotes the mass of the particle, | ||
| - | * $\Psi$ is either the wave function of the spin $1/2$ particle if we use the Dirac equation in a particle theory, or describes the spin $1/2$ field if we work in a field theory, | + | * $\Psi$ is either the wave function of the spin $1/2$ particle if we use the Dirac equation in a particle theory, or describes the spin $1/2$ field if we work in a field theory. In any case, $\Psi$ is not a vector but a [[advanced_tools:spinors|spinor]]. |
| * $\gamma_\mu$ are the Dirac gamma matrices. | * $\gamma_\mu$ are the Dirac gamma matrices. | ||
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equations/dirac_equation.1522077387.txt.gz · Last modified: 2018/03/26 15:16 (external edit)
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