Ngā Whārite Whakauru i roto i te Ahupūngao
He taputapu pāngarau kaha ngā whārite whakauru i roto i te ahupūngao, e whakamahia ana hei ako i te whānuitanga o ngā āhuatanga taiao. He tikanga ēnei e whakamahi ana i ngā whakauru hei kimi otinga mō ngā momo raruraru, pērā i te tohatoha o ngā āpure i te wāhi, i te wā rānei. I roto i tēnei tuhinga, ka matapakihia e mātou te ariā me ngā tono o ngā whārite whakauru i roto i te ahupūngao, me te whakarato i ētahi tauira e whakaatu ana i te whakamahinga o tēnei tikanga i roto i ngā momo mara ahupūngao.
1. Whakataki ki ngā Whārite Whakauru
Ko te whārite taupū he kīanga pāngarau e whakauru ana i tētahi mahi kāore i te mōhiotia, i hangaia i roto i te āhua taupū. He mea nui ngā whārite taupū nā te mea he maha ngā raruraru ahupūngao tūturu e ngāwari ake ana, e mārama noa ana rānei te whakapuaki i roto i te āhua taupū i te āhua rerekētanga.
Ko ngā āhua whānui e rua o ngā whārite taupū ko:
– Whārite Whakakotahi o Fredholm
– Whārite Whakauru Volterra
He rerekē ngā momo whārite e rua i runga i ngā here whakauru, e pā ana ki te kimi otinga me ngā āhuatanga o aua otinga. He here whakauru pumau tō te Whārite Whakauru Fredholm, engari he rerekē ngā here whakauru i roto i te Whārite Whakauru Volterra i runga i te taurangi motuhake.
2. Te Aukumetanga Hiko me ngā Whārite Whakauru
I roto i te aukume hiko, ka whakamahia pinepinetia ngā whārite whakauru hei whakatau i te āpure e pā ana ki te tohatoha o ngā utu hiko, o ngā iahiko rānei. Hei tauira, ka taea te whakatakoto i te ture a Coulomb mō te āpure hiko \( E \) i roto i te āhua whakauru penei:
\[
\mathbf{E}(\mathbf{r}) = \frac{1}{4 \pi \epsilon_0} \int_{\mathcal{V}} \frac{\rho(\mathbf{r}') (\mathbf{r} – \mathbf{r}')}{|\mathbf'|3}
\]
I konei, ko \(\rho(\mathbf{r}')\) te tohatoha utu i roto i te rōrahi \( \mathcal{V} \), ko \(\mathbf{r}\) te tūranga o te pūwāhi e tatauhia ana te mara, ā, ko \(\epsilon_0\) te āheinga korehau. Ka tatauhia e tēnei whakaurunga te takoha o te mara hiko i te pūwāhi \(\mathbf{r}\) mai i ngā huānga rōrahi katoa i roto i te tohatoha utu.
He mahi nui anō hoki tā ngā whārite whakauru i roto i ngā tikanga pūmanawa whārite mō ngā mara hikohiko, tae atu ki te whakatakotoranga o ngā Whārite a Maxwell.
3. Ngā Āhuahira Irahiko me ngā Whārite Whakauru
I roto i te hangarau matūriki, ko tētahi o ngā whakamahinga nui o ngā whārite whakauru ko te whakatakotoranga whakauru ara i whakaurua mai e Richard Feynman. Mā tēnei whakaaturanga ka puta he huarahi hou ki te whakatakoto i te ariā matūriki he rerekē i ngā huarahi a Schrödinger, a Heisenberg rānei.
Ka puta anō hoki ngā whārite whakauru i te āhua o te whārite whakauru Lippmann-Schwinger, he āhua whakauru tēnei o te whārite Schrödinger mō ngā āhua marara. E whakamahia ana hei ako i ngā tukanga marara i roto i te miihini irahiko:
\[
\psi(\mathbf{r}) = \psi_0(\mathbf{r}) + \int G(\mathbf{r}, \mathbf{r}') V(\mathbf{r}') \psi(\mathbf{r}') \, d^3r'
\]
I konei, ko \( \psi \) te mahi ngaru katoa, ko \( \psi_0 \) te mahi ngaru kore utu, ko \( V \) te pūmanawa, ā, ko \( G \) te kaiwhakatipu, te mahi rānei a Green e tohu ana i te horapa o te raruraru mai i te pūmanawa \( V \) puta noa i te wāhi.
4. Te Ariā Horoho me ngā Whārite Whakauru
Ko ngā āhuatanga horapa, ahakoa i roto i te horopaki o te ahupūngao matū whakapoto, te koiora rānei, ka whakaatuhia pinepinetia mā roto i ngā whārite whakauru. Hei tauira, ka taea te whakatakoto i te whārite horapa i roto i te āhua whakauru mā te whakamahi i te kākano horapa, e whakaahua ana i te horapa o ngā matūriki mai i tētahi pūtake pūwāhi.
Tauira o te whārite horapa:
\[
C(\mathbf{r}, t) = \int_{\mathcal{V}} G(\mathbf{r}, \mathbf{r}', t) C(\mathbf{r}', 0) \, d^3r'
\]
Anei ko \( C(\mathbf{r}, t) \) te kukū matūriki i te tūranga \(\mathbf{r}\) me te wā \(t\), \( G(\mathbf{r}, \mathbf{r}', t) \) ko te pūtake horapa e whakaahua ana i te tūponotanga o te noho o te matūriki i \(\mathbf{r}\) i te wā \(t\) i muri i te tīmatanga mai i \(\mathbf{r}'\) i te wā \(t = 0\).
5. Te Ariā o te Whakawhanaungatanga me ngā Whārite Whakauru
I roto i te ariā whānui o te whanaungatanga, he maha ngā wā ka tātarihia ngā mara ā-papa mā te whakamahi i ngā tikanga whakauru. Hei tauira, he māmā ake te mārama ki ngā otinga i roto i te āhua whakauru. Ko te pūmanawa ā-papa me te inenga wāhi-wā, e pā ana ki ngā ara o te māramatanga me ngā mea e neke ana, ka taea te whakatakoto mā roto i ngā whakauru, me te whakanui i te takoha o te tohatoha katoa o te papatipu me te pūngao i roto i te ao whānui.
6. Ngā Tikanga Tau me ngā Otinga o ngā Whārite Whakauru
I roto i te mahi, he tino uaua ki te whakaoti ma te tātari ngā whārite taupū i roto i te ahupūngao. Nō reira, ka whakamahia ngā tikanga tau hei kimi i ngā otinga tata. Ko ētahi o ngā tikanga tau e whakamahia whānuitia ana ko ngā tikanga Monte Carlo, ngā tikanga tāruarua, me ngā tikanga wehewehe pērā i te tikanga huānga mutunga kore me te tikanga matūriki.
Hei tauira, i roto i ngā tono rorohiko hou pērā i te whakatauira i ngā āpure hikohiko i roto i ngā rauemi uaua, te tātari rānei i te tohatoha wera i roto i ngā rauemi, ka whakaratohia e ngā tikanga tau mō ngā whārite taupū he whakatau tata me ngā otinga tino whai hua ki ngā raruraru tūturu.
Whakamutunga
He taputapu pāngarau nui ngā whārite whakauru i roto i te ahupūngao. He huarahi kaha ēnei hei tātari me te mārama ki te whānuitanga o ngā āhuatanga taiao mā roto i ngā whakatakotoranga he maha ngā wā he taiao ake i ngā whārite rerekētanga. Mai i te hikohiko me te miihini irahiko ki te horapa me te whanaungatanga whānui, he whānui, he hohonu hoki ngā tono o ngā whārite whakauru.
Hei mārama me te whakamahi pai i ngā whārite taupū, me mārama te tangata ki ngā ariā pāngarau taketake me te pūkenga ki ngā tikanga tau. Heoi anō, ko ngā painga o te whakamahi i ēnei hei whakarato i ngā otinga huatau me te whānui ake ki ngā raruraru ahupūngao ka whai hua ai tā rātou ako.
I te haere tonu o te whanaketanga o te hangarau rorohiko me tō tātou māramatanga ki te ao whānui, tera pea ka whānui haere tonu ngā tono o ngā whārite taupū, ka huaki te tatau ki ngā kitenga hou i ngā peka katoa o te ahupūngao.
