من ويكيبيديا، الموسوعة الحرة
في الرياضيات، يشير مصلح التوافقيات الكروية المتجهية إلى تعميم مفهوم التوافقيات الكروية على المتجهات (الأشعة) عوضا عن المقادير السلمية التي تستخدم عادة لحل معادلة لابلاس.
تعريف[عدل]
يوجد اصطلاحات متعددة سنستعرض احداها وحسب، تعرف التوافقيات الكروية المتجهية على النحو التالي :
![{\displaystyle \mathbf {Y} _{lm}=Y_{lm}{\hat {\mathbf {r} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2939e0959a607f7c2b51fbc83aba73a2f7cb3cb0)
![{\displaystyle \mathbf {\Psi } _{lm}=r\nabla Y_{lm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d0d4637bd38b7a4ddb397c2f8d43d6b133c655)
![{\displaystyle \mathbf {\Phi } _{lm}={\vec {\mathbf {r} }}\times \nabla Y_{lm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fdc6e53065defaf19544097af85ee31fc09b77d)
![{\displaystyle \mathbf {E} =\sum _{l=0}^{\infty }\sum _{m=-l}^{l}\left(E_{lm}^{r}(r)\mathbf {Y} _{lm}+E_{lm}^{(1)}(r)\mathbf {\Psi } _{lm}+E_{lm}^{(2)}(r)\mathbf {\Phi } _{lm}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9833d256febd27550f3ba31dc3088448577d377)
الخواص الأساسية[عدل]
![{\displaystyle \mathbf {Y} _{l,-m}=(-1)^{m}\mathbf {Y} _{lm}^{*}\qquad \mathbf {\Psi } _{l,-m}=(-1)^{m}\mathbf {\Psi } _{lm}^{*}\qquad \mathbf {\Phi } _{l,-m}=(-1)^{m}\mathbf {\Phi } _{lm}^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/895cf3299c7241106e962dfe9ab8c8ac1ad9c1af)
التعامد[عدل]
لنأخد منشور متعدد الأقطاب لحقل سلمي ما
![{\displaystyle \phi =\sum _{l=0}^{\infty }\sum _{m=-l}^{l}\phi _{lm}(r)Y_{lm}(\theta ,\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/475ee6c1a1ea8dbf6203ee07df3ae0c4521f214f)
يمكن التعبير عن التدرج باستخدام مفهوم التوافقيات الكروية المتجهية كما يلي
![{\displaystyle \nabla \phi =\sum _{l=0}^{\infty }\sum _{m=-l}^{l}\left({\frac {\mathrm {d} \phi _{lm}}{\mathrm {d} r}}\mathbf {Y} _{lm}+{\frac {\phi _{lm}}{r}}\mathbf {\Psi } _{lm}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58cbacb73c1f2429fb93f6f6b38142c8391b0c4b)
التفرق[عدل]
من أجل أي حقل متعدد الأقطاب لدينا :
![{\displaystyle \nabla \cdot \left(f(r)\mathbf {Y} _{lm}\right)=\left({\frac {\mathrm {d} f}{\mathrm {d} r}}+{\frac {2}{r}}f\right)Y_{lm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea375aea2717a694d0793e4ab8b863d1b708f2ae)
![{\displaystyle \nabla \cdot \left(f(r)\mathbf {\Psi } _{lm}\right)=-{\frac {l(l+1)}{r}}fY_{lm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/053504d374ee7d0054f0306f3f300022748b14ac)
![{\displaystyle \nabla \cdot \left(f(r)\mathbf {\Phi } _{lm}\right)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61494f370773f8887f131f2d62b22dced17d6eaf)
من خلال التركيب سنحصل على التفرق لأي حقل المتجهات كما يلي
![{\displaystyle \nabla \cdot \mathbf {E} =\sum _{l=0}^{\infty }\sum _{m=-l}^{l}\left({\frac {\mathrm {d} E_{lm}^{r}}{\mathrm {d} r}}+{\frac {2}{r}}E_{lm}^{r}-{\frac {l(l+1)}{r}}E_{lm}^{(1)}\right)Y_{lm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55d4de2254b3a875a5a076c7fa777e43956f6593)
الدوران[عدل]
![{\displaystyle \nabla \times \left(f(r)\mathbf {Y} _{lm}\right)=-{\frac {1}{r}}f\mathbf {\Phi } _{lm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/020c1b4f2e05a38baf76d20676f2fe3820ebd302)
![{\displaystyle \nabla \times \left(f(r)\mathbf {\Psi } _{lm}\right)=\left({\frac {\mathrm {d} f}{\mathrm {d} r}}+{\frac {1}{r}}f\right)\mathbf {\Phi } _{lm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d24b8eb273205d2ce539d3739ce19e206fcf087f)
![{\displaystyle \nabla \times \left(f(r)\mathbf {\Phi } _{lm}\right)=-{\frac {l(l+1)}{r}}f\mathbf {Y} _{lm}-\left({\frac {\mathrm {d} f}{\mathrm {d} r}}+{\frac {1}{r}}f\right)\mathbf {\Psi } _{lm}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56d01cdea4338bf785a1508712850486c423a68a)
أمثلة[عدل]
التوافقيات الكروية المتجهية الأولى[عدل]
![{\displaystyle l=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd86dca1efdb0e84aebc4066462bc5a50a9dde5)
![{\displaystyle \mathbf {Y} _{00}={\sqrt {\frac {1}{4\pi }}}{\hat {\mathbf {r} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee5688977fbadcf4d4187e8bc31aa88ecc3def2)
![{\displaystyle \mathbf {\Psi } _{00}=\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/eac8cc0823e8b4dfa0bf44ba6ec811ec085d0be3)
![{\displaystyle \mathbf {\Phi } _{00}=\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/074f597297801eb0a63a079298502443a7cd61d5)
![{\displaystyle l=1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b14df8eeb61146502640651dddf307f725052c8)
![{\displaystyle \mathbf {Y} _{10}={\sqrt {\frac {3}{4\pi }}}\cos \theta \,{\hat {\mathbf {r} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48c75287c77773a3fb49fb5436d531c17fe8825a)
![{\displaystyle \mathbf {Y} _{11}=-{\sqrt {\frac {3}{8\pi }}}\mathrm {e} ^{\mathrm {i} \varphi }\sin \theta \,{\hat {\mathbf {r} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1422e3467e5df602a7d9337caf9db54d2c980812)
![{\displaystyle \mathbf {\Psi } _{10}=-{\sqrt {\frac {3}{4\pi }}}\sin \theta \,{\hat {\mathbf {\theta } }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2ed34f8b60bbcbe393980316787f7624dced48)
![{\displaystyle \mathbf {\Psi } _{11}=-{\sqrt {\frac {3}{8\pi }}}\mathrm {e} ^{\mathrm {i} \varphi }\left(\cos \theta \,{\hat {\mathbf {\theta } }}+\mathrm {i} \,{\hat {\mathbf {\varphi } }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3965debfcb76264944fcdd3c77d6f4b04dc3553d)
![{\displaystyle \mathbf {\Phi } _{10}=-{\sqrt {\frac {3}{4\pi }}}\sin \theta \,{\hat {\mathbf {\varphi } }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/863f5b81ab8de242d79f4eaa5c1517e10cb020b4)
![{\displaystyle \mathbf {\Phi } _{11}={\sqrt {\frac {3}{8\pi }}}\mathrm {e} ^{\mathrm {i} \varphi }\left(\mathrm {i} \,{\hat {\mathbf {\theta } }}-\cos \theta \,{\hat {\mathbf {\varphi } }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc1209b0af1a75bcb62bdaee2afa863804d278a)
![{\displaystyle l=2\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08aab9012f2351a5e8b7fd8ff53a5b793b597f46)
![{\displaystyle \mathbf {Y} _{20}={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}\,(3\cos ^{2}\theta -1)\,{\hat {\mathbf {r} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/717ccb3436bc5e13b750ff58ecdd6fdc701c6b7e)
![{\displaystyle \mathbf {Y} _{21}=-{\sqrt {\frac {15}{8\pi }}}\,\sin \theta \,\cos \theta \,e^{i\varphi }\,{\hat {\mathbf {r} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e843c989226c7acb18c169bdd403a14e54f03c9d)
![{\displaystyle \mathbf {Y} _{22}={\frac {1}{4}}{\sqrt {\frac {15}{2\pi }}}\,\sin ^{2}\theta \,e^{2i\varphi }\,{\hat {\mathbf {r} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7aeb05ac0f0789682f3c0857ce32696f9f6629)
![{\displaystyle \mathbf {\Psi } _{20}=-{\frac {3}{2}}{\sqrt {\frac {5}{\pi }}}\,\sin \theta \,\cos \theta \,{\hat {\mathbf {\theta } }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e5b78912030e63b34f8a1fbb689b3a4f76db36)
![{\displaystyle \mathbf {\Psi } _{21}=-{\sqrt {\frac {15}{8\pi }}}\,e^{i\varphi }\,\left(\cos 2\theta \,{\hat {\mathbf {\theta } }}+\mathrm {i} \cos \theta \,{\hat {\mathbf {\varphi } }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df01ccaf051968eed48f8bdcb2d0703aa5f6b51a)
![{\displaystyle \mathbf {\Psi } _{22}={\sqrt {\frac {15}{8\pi }}}\,\sin \theta \,e^{2i\varphi }\,\left(\cos \theta \,{\hat {\mathbf {\theta } }}+\mathrm {i} \,{\hat {\mathbf {\varphi } }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64ae424f4206074058b9413fcf4691ed9796306b)
![{\displaystyle \mathbf {\Phi } _{20}=-{\frac {3}{2}}{\sqrt {\frac {5}{\pi }}}\sin \theta \,\cos \theta \,{\hat {\mathbf {\phi } }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1645a2048afe77c280ee599eb098e95859438ed)
![{\displaystyle \mathbf {\Phi } _{21}={\sqrt {\frac {15}{8\pi }}}\,e^{i\varphi }\,\left(\mathrm {i} \cos \theta \,{\hat {\mathbf {\theta } }}-\cos 2\theta \,{\hat {\mathbf {\varphi } }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/609973877278f0ec6740eb84f804b8bf506f498e)
![{\displaystyle \mathbf {\Phi } _{22}={\sqrt {\frac {15}{8\pi }}}\,\sin \theta \,e^{2i\varphi }\,\left(-\mathrm {i} \,{\hat {\mathbf {\theta } }}+\cos \theta \,{\hat {\mathbf {\varphi } }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b41b6550ea33ba848d2d36fa7058ea0952692a2d)