在量子力學及量子場論等領域,外尔方程式(英語:Weyl Equation)為一相對論量子力學的波動方程式,用以描述無質量的自旋½粒子。其以德國數學家赫尔曼·外尔為名。
方程式[编辑]
外尔方程式的廣義形式可寫為:[1][2]
![{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fz2e4nDrEyqw3ytdBoqdCyqzDoAaOagi3njoNotiQaAnCzjeQygsO)
在SI單位中可寫為:
![{\displaystyle I_{2}{\frac {1}{c}}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DnDBCa2a1zgi0o2wOo2rEaAi4aAo4nDi5zDnBageQyjnAyjmPagzC)
其中
![{\displaystyle \sigma _{\mu }=(\sigma _{0},\sigma _{1},\sigma _{2},\sigma _{3})=(I_{2},\sigma _{x},\sigma _{y},\sigma _{z})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CngwNzgeNaNi0a2i5aNG3nji1ntCQaAwOaAhCzNe4ztm3ags3zga2)
為一向量。μ = 0分量為2 × 2 單位矩陣;μ = 1,2,3分量為包立矩陣。ψ則是波函數,為外尔旋量一例。
外尔旋量[编辑]
其組成有ψL與ψR,分別為左手(Left-handed)外尔旋量及右手(Right-handed)外尔旋量,各自有兩個分量。兩者皆有下列形式:
![{\displaystyle \psi ={\begin{pmatrix}\psi _{1}\\\psi _{2}\\\end{pmatrix}}=\chi e^{-i(\mathbf {k} \cdot \mathbf {r} -\omega t)}=\chi e^{-i(\mathbf {p} \cdot \mathbf {r} -Et)/\hbar }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eyqs2oNnBoNlFnDm0zDo2zqhEa2aNzNdCnDsQztzFajoQzjaOatoP)
其中
![{\displaystyle \chi ={\begin{pmatrix}\chi _{1}\\\chi _{2}\\\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84atlDztzAags4aqw4atnAaDdCntvEzArCaqvDyjBCaNG2yta5yto0)
為具有二常數分量之旋量。
既然粒子是不具質量的,亦即m = 0,動量p的範數為波向量k的簡單乘積,由德布羅伊關係所描述:
![{\displaystyle |\mathbf {p} |=\hbar |\mathbf {k} |=\hbar \omega /c\,\rightarrow \,|\mathbf {k} |=\omega /c}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PzNvEaqi1ygzEoAhCzDFBate5a2i0nqo5nDBEotJBzAsPotKPoqdD)
方程式可以左手及右手旋量來表示:
![{\displaystyle {\begin{aligned}&\sigma ^{\mu }\partial _{\mu }\psi _{R}=0\\&{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=0\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Bnja4aqnEzqeOo2o2o2w0ntrDzji2zNm3ztaNajiNaAhCajaOythF)
透過拉格朗日密度可得方程式:
![{\displaystyle {\mathcal {L}}=i\psi _{R}^{\dagger }\sigma ^{\mu }\partial _{\mu }\psi _{R}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aDvBaqi4zAnDaqs3oqdCoNGOzNCNnqnAoqw2aNa5oNhEatJAngvD)
![{\displaystyle {\mathcal {L}}=i\psi _{L}^{\dagger }{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OoArBzqi1a2w1ajvBnjCQzjs1z2iPatdDothFnDiPyqhEnAzFatFE)
將旋量及旋量的埃爾米特伴隨(以
標記)當作獨立變數處理,則可得外尔方程式。
相關條目[编辑]
參考資料[编辑]
- ^ Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
- ^ The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
延伸閱讀[编辑]
- Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
- Particle Physics (2nd Edition), B.R. Martin, G. Shaw, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7
- Supersymmetry P. Labelle, Demystified, McGraw-Hill (USA), 2010, ISBN 978-0-07-163641-4
- The Road to Reality, Roger Penrose, Vintage books, 2007, ISBN 0-679-77631-1
外部連結[编辑]