equations:dirac_equation
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equations:dirac_equation [2017/10/22 18:37] jakobadmin [Student] |
equations:dirac_equation [2021/05/30 18:16] edi [Concrete] |
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| + | <WRAP lag>$ (i\gamma_\mu \partial^\mu - m ) \Psi =0 $</WRAP> | ||
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| ====== Dirac Equation ====== | ====== Dirac Equation ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | <note tip>It's the correct equation of motion that describes free [[basic_notions:spin|spin]] $1/2$ particles. | + | <tabbox Intuitive> |
| - | </note> | + | The Dirac equation describes how the state of a relativistic (= fast moving) quantum system with half-integer spin changes in time. |
| + | The analogous equation for systems without spin is the [[equations:klein-gordon_equation|Klein-Gordon equation]]. | ||
| - | <blockquote>In fact, Dirac's equation for the electron must be rated, alongside the [[equations:maxwell_equations|Maxwell]] and [[equations:einstein_equation|Einstein equations]], as one of the Great Field Equations of physics.<cite>page 289 in "The Emperors new Mind" by Penrose</cite></blockquote> | + | If the system only moves slowly, the Dirac equation becomes the [[equations:pauli_equation|Pauli equation]]. |
| - | <tabbox Layman> | + | |
| - | <note tip> | + | <tabbox Concrete> |
| - | 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 Student> | + | |
| + | * A nice discussion can be found in [[http://www.quantumfieldtheory.info/website_Chap04.pdf |chapter 4 of Klauber's Student Friendly QFT book]] | ||
| * For an elementary derivation and a discussion of what the solution of the Dirac equation mean, see Physics from Symmetry by Schwichtenberg | * For an elementary derivation and a discussion of what the solution of the Dirac equation mean, see Physics from Symmetry by Schwichtenberg | ||
| * Alternative, two possible derivations can be found at page 100ff in Relativistic Quantum Mechanics by Paul Strange | * Alternative, two possible derivations can be found at page 100ff in Relativistic Quantum Mechanics by Paul Strange | ||
| Line 24: | Line 22: | ||
| ** Gamma Gymnastics:** | ** Gamma Gymnastics:** | ||
| - | * See in the book The Conceptual Framework of Quantum Field Theory by Duncan | + | There are many important rules for the $\gamma$ matrices that appear in the Dirac equation. These rules are important for many practical calculations. |
| - | <tabbox Researcher> | + | |
| + | * For a nice description, see section 7.4.3 "Diracology" in the book The Conceptual Framework of Quantum Field Theory by Duncan | ||
| + | |||
| + | ---- | ||
| + | |||
| + | **Graphical Summary** | ||
| + | |||
| + | The diagram below shows the Dirac equation and its Lagrangian in various forms. For a more detailed explanation see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. | ||
| + | |||
| + | {{:equations:dirac.jpg?nolink}} | ||
| + | |||
| + | <tabbox Abstract> | ||
| <note tip> | <note tip> | ||
| Line 31: | Line 40: | ||
| </note> | </note> | ||
| - | --> Common Question 1# | + | <tabbox Why is it interesting?> |
| - | + | The Dirac equation is the correct equation of motion that describes free [[basic_notions:spin|spin]] $1/2$ particles. | |
| - | <-- | + | |
| - | --> Common Question 2# | ||
| - | |||
| - | <-- | ||
| - | | ||
| - | <tabbox Examples> | ||
| - | --> Example1# | + | <blockquote>In fact, Dirac's equation for the electron must be rated, alongside the [[equations:maxwell_equations|Maxwell]] and [[equations:einstein_equation|Einstein equations]], as one of the Great Field Equations of physics.<cite>page 289 in "The Emperors new Mind" by Penrose</cite></blockquote> |
| - | + | <tabbox Definitions> | |
| - | <-- | + | |
| - | --> Example2:# | ||
| - | + | * $\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, |
| + | * $\Psi$ 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. | ||
| | | ||
| <tabbox History> | <tabbox History> | ||
| Line 59: | Line 62: | ||
| be added, however, that it was Dirac who found most of the additional | be added, however, that it was Dirac who found most of the additional | ||
| insights.” <cite>Weisskopf on Dirac</cite></blockquote> | insights.” <cite>Weisskopf on Dirac</cite></blockquote> | ||
| + | |||
| + | <blockquote>Niels Bohr: “What are you working on Mr. Dirac?” | ||
| + | Paul Dirac: “I’m trying to take the square root of something” </blockquote> | ||
| </tabbox> | </tabbox> | ||
equations/dirac_equation.txt · Last modified: 2023/04/02 03:11 by edi
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