Elektrodynamikvorlesung von Prof. Dr. E. Schöll, PhD
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Der Artikel TCP- Invarianz basiert auf der Vorlesungsmitschrift von Franz- Josef Schmitt des 3.Kapitels (Abschnitt 1) der Elektrodynamikvorlesung von Prof. Dr. E. Schöll, PhD.
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{{#set:Urheber=Prof. Dr. E. Schöll, PhD|Inhaltstyp=Script|Kapitel=3|Abschnitt=1}}
Kategorie:Elektrodynamik
__SHOWFACTBOX__
- Zeitumkehr T
- t → t´=-t
- Ladungsumkehr / Konjugation
- C : Q → Q´= - Q
- Paritätsumkehr P
- r → r´= -r (für den Ortsvektor)
Die Zeitumkehr- Transformation[edit | edit source]
![{\displaystyle {\begin{aligned}&{{T}_{g}}:=\left\{T-in\operatorname {var} iante\ ObservableA:TA=A\right\}\\&=\left\{{\bar {r}},d{\bar {r}},a:={\frac {{{d}^{2}}{\bar {r}}}{d{{t}^{2}}}},m,q,\rho :={\begin{matrix}\lim \\\Delta V\to 0\\\end{matrix}}{\frac {\Delta q}{\Delta V}},{\bar {F}}=m{\bar {a}},{\bar {E}}={\frac {\bar {F}}{q}},\Phi ...\right\}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e65aa100808f87f419d7cf96b9f3b6545864268)
Diese Observablen sind "gerade" unter T
Daneben gibt es auch Observablen, die "ungerade" unter T sind:
![{\displaystyle {{T}_{u}}:=\left\{A:TA=-A\right\}=\left\{{\bar {v}}:={\frac {d{\bar {r}}}{dt}},{\bar {j}}=\rho {\bar {v}},{\bar {B}},{\bar {A}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86c8a0feedea5a130c59eaad5b829c90af96952e)
Denn:
![{\displaystyle {\begin{aligned}&{\bar {F}}=q{\bar {v}}\times {\bar {B}}\\&{\bar {F}}\in {{T}_{g}},{\bar {v}}\in {{T}_{u}},q\in {{T}_{g}}\Rightarrow {\bar {B}}\in {{T}_{u}}\\&{\bar {B}}=\nabla \times {\bar {A}},\nabla \in {{T}_{g}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f000e8d8d4879857ac07867ce07710e40022a525)
Somit folgt jedoch vollständige T- Invarianz der elektromagnetischen Grundgleichungen:
![{\displaystyle {\begin{aligned}&T:\left\{{{\nabla }_{r}}\times {\bar {E}}=0\right\}\to \left\{{{\nabla }_{r}}\times {\bar {E}}=0\right\}\\&T:\left\{{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=\rho \right\}\to \left\{{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=\rho \right\}\\&T:\left\{{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\to \left\{-{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\Leftrightarrow \left\{{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\\&T:\left\{\nabla \times {\bar {B}}={{\mu }_{0}}{\bar {j}}\right\}\to \left\{-\nabla \times {\bar {B}}=-{{\mu }_{0}}{\bar {j}}\right\}\\&\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/773e6246a65f92490883b3e979d490a39e0ef17c)
Kontinuitätsgleichung{{#set:Fachbegriff=Kontinuitätsgleichung|Index=Kontinuitätsgleichung}}:
![{\displaystyle T:\left\{{\frac {\partial }{\partial t}}\rho +{{\nabla }_{r}}\cdot {\bar {j}}=0\right\}\to \left\{-{\frac {\partial }{\partial t}}\rho -{{\nabla }_{r}}\cdot {\bar {j}}=0\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/184b49b4ea508c09b0b372e65bf0ab822f9c38bd)
Die Gleichungen sind forminvariant{{#set:Fachbegriff=forminvariant|Index=forminvariant}}!
Ladungsumkehr (Konjugation)[edit | edit source]
![{\displaystyle {\begin{aligned}&{{C}_{g}}:=\left\{C-in\operatorname {var} iante\ ObservableA:CA=A\right\}\\&{{C}_{g}}=\left\{{\bar {F}},m,{\bar {r}},{\bar {v}},{\bar {a}}\right\}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5fb03eeba8b8d03e89e7af58b32a79d154bf334)
sind gerade unter C
Ungerade unter c sind:
![{\displaystyle {\begin{aligned}&{{C}_{u}}:=\left\{A:CA=-A\right\}=\left\{{\bar {E}}={\frac {1}{q}}{\bar {F}},{\bar {B}},{\bar {j}},\rho \right\}\\&{\bar {F}}=q{\bar {v}}\times {\bar {B}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ec4662bf580a24cdd56a1b0b21a34279d00ecf)
- C- Invarianz der Elektro- Magnetostatik:
![{\displaystyle {\begin{aligned}&C:\left\{{{\nabla }_{r}}\times {\bar {E}}=0\right\}\to \left\{-{{\nabla }_{r}}\times {\bar {E}}=0\right\}\\&C:\left\{{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=\rho \right\}\to \left\{-{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=-\rho \right\}\\&C:\left\{{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\to \left\{-{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\\&C:\left\{\nabla \times {\bar {B}}={{\mu }_{0}}{\bar {j}}\right\}\to \left\{-\nabla \times {\bar {B}}=-{{\mu }_{0}}{\bar {j}}\right\}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28daf9edbec62d6ec64340e492793b9e98b801d1)
![{\displaystyle C:\left\{{\frac {\partial }{\partial t}}\rho +{{\nabla }_{r}}\cdot {\bar {j}}=0\right\}\to \left\{-{\frac {\partial }{\partial t}}\rho -{{\nabla }_{r}}\cdot {\bar {j}}=0\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8346cc0cd9fa0a9d9804157b637810b9d775f701)
Paritätsumkehr: Räumliche Spiegelung/ Inversion[edit | edit source]
Vertauschung: rechts ↔ links
Man unterscheidet:
![{\displaystyle P{\bar {r}}=-{\bar {r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba88e8119b2cf238bad30fb2387d2d7a2769a865)
→ polarer Vektor
und
![{\displaystyle P\left({\bar {a}}\times {\bar {b}}\right)=\left(-{\bar {a}}\times -{\bar {b}}\right)=\left({\bar {a}}\times {\bar {b}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/869594f56d340d77989b8b63ef3b4e1e457ab6a4)
P- invariant = " axialer Vektor", sogenannter Pseudovektor!!
Seien:
![{\displaystyle {\bar {a}},{\bar {b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d37052edc16cc9654cecd7b733d8f03cba37db70)
polar,
![{\displaystyle {\bar {w}},{\bar {\sigma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9abde8ec409cab22ae5163010a61e095c2ee2aee)
axial
Dann ist
![{\displaystyle {\begin{aligned}&{\bar {a}}\times {\bar {w}}\quad polar\\&{\bar {a}}\times {\bar {b}},{\bar {w}}\times {\bar {\sigma }}\quad axial\\&{\bar {a}}{\bar {b}}\ skalar:P({\bar {a}}{\bar {b}})={\bar {a}}{\bar {b}}\\&{\bar {w}}{\bar {\sigma }}\ pseudoskalarP({\bar {w}}{\bar {\sigma }})=-{\bar {w}}{\bar {\sigma }}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4ee30f62d7a6688439220fe1f9a6ff9a37a5f4)
![{\displaystyle {\begin{aligned}&{{C}_{g}}:=\left\{C-in\operatorname {var} iante\ ObservableA:CA=A\right\}\\&{{C}_{g}}=\left\{{\bar {F}},m,{\bar {r}},{\bar {v}},{\bar {a}}\right\}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5fb03eeba8b8d03e89e7af58b32a79d154bf334)
Wegen
![{\displaystyle {\begin{aligned}&{\bar {F}}=q{\bar {v}}\times {\bar {B}}\\&{\bar {F}}\in {{P}_{u}}\\&q\in {{P}_{g}}\\&{\bar {v}}\in {{P}_{u}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a14b3ed1f10e1c6aa394d4578e00b6cb5efff804)
ungerade Parität dagegen:
![{\displaystyle {{P}_{u}}=\left\{polareVektoren,{\bar {r}},d{\bar {r}},{\bar {v}},{\bar {a}},{\bar {F}},{\bar {E}}={\frac {1}{q}}{\bar {F}},{\bar {j}}=\rho {\bar {v}},{\bar {A}},Pseudoskalare\quad \nabla \cdot {\bar {B}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/375dffead224aa90989f78bf9a87ad79c41baf1c)
Wegen
![{\displaystyle {\begin{aligned}&{\bar {B}}=\nabla \times {\bar {A}}\\&\nabla \in {{P}_{u}}\\&{\bar {B}}\in {{P}_{g}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48aaeebf6ca3433913df3e28089d9f274548387b)
P- Invarianz der Elektro- / Magnetostatik:
![{\displaystyle {\begin{aligned}&P:\left\{{{\nabla }_{r}}\times {\bar {E}}=0\right\}\to \left\{{{\nabla }_{r}}\times {\bar {E}}=0\right\}\\&P:\left\{{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=\rho \right\}\to \left\{{{\varepsilon }_{0}}{{\nabla }_{r}}\cdot {\bar {E}}=\rho \right\}\\&P:\left\{{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\to \left\{-{{\nabla }_{r}}\cdot {\bar {B}}=0\right\}\\&P:\left\{\nabla \times {\bar {B}}={{\mu }_{0}}{\bar {j}}\right\}\to \left\{-\nabla \times {\bar {B}}=-{{\mu }_{0}}{\bar {j}}\right\}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e76cff196a1f0fb45a4fe96533e3ca91d1891ddd)
![{\displaystyle P:\left\{{\frac {\partial }{\partial t}}\rho +{{\nabla }_{r}}\cdot {\bar {j}}=0\right\}\to \left\{{\frac {\partial }{\partial t}}\rho +{{\nabla }_{r}}\cdot {\bar {j}}=0\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57b612aadacfed25e86ee703c281d7d2fc8a4a0f)
Nebenbemerkung: Gäbe es magnetische Ladungen, dann wären sie pseudoskalare
Außerdem (Weinberg e.a.) : Schwache Wechselwirkung verletzt die Paritätserhaltung!