Wynik zapytania API MediaWiki
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{
"compare": {
"fromid": 1,
"fromrevid": 1,
"fromns": 0,
"fromtitle": "Strona g\u0142\u00f3wna",
"toid": 2,
"torevid": 2,
"tons": 0,
"totitle": "Mechanika kwantowa/Podstawy mechaniki kwantowej",
"*": "<tr><td colspan=\"2\" class=\"diff-lineno\" id=\"mw-diff-left-l1\">Linia 1:</td>\n<td colspan=\"2\" class=\"diff-lineno\">Linia 1:</td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><<del class=\"diffchange diffchange-inline\">strong</del>><del class=\"diffchange diffchange-inline\">Instalacja MediaWiki si\u0119 powiod\u0142a.</del></<del class=\"diffchange diffchange-inline\">strong</del>></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><<ins class=\"diffchange diffchange-inline\">noinclude</ins>><ins class=\"diffchange diffchange-inline\"><!--</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">-->{{SkomplikowanaStronaStart<!--</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\u00a0 \u00a0 \u00a0 -->| stopka strony = {{Kreska nawigacja|{{AktualnaKsi\u0105\u017cka}}|{{Nast\u0119pnyArtyku\u0142}}|{{PoprzedniArtyku\u0142}}}}<!--</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">-->}}</ins></<ins class=\"diffchange diffchange-inline\">noinclude</ins>></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Zapoznaj si\u0119 z [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents Podr\u0119cznikiem u\u017cytkownika] zawieraj\u0105cym informacje o </del>tym <del class=\"diffchange diffchange-inline\">jak korzysta\u0107 z oprogramowania wiki</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">W </ins>tym <ins class=\"diffchange diffchange-inline\">rozdziale przedstawimy podstawowe prawa mechaniki kwantowej b\u0119d\u0105ce podwalinami tej\u017ce teorii</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>== <del class=\"diffchange diffchange-inline\">Na pocz\u0105tek </del>==</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>== <ins class=\"diffchange diffchange-inline\">Zasada Huygensa ==</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* <del class=\"diffchange diffchange-inline\">[https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">Configuration_settings Lista ustawie\u0144 konfiguracyjnych]</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Rysunek|Refraction - Huygens-Fresnel principle.svg|i1|Za\u0142amanie fali na granicy dw\u00f3ch o\u015brodk\u00f3w.}}</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* <del class=\"diffchange diffchange-inline\">[https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">FAQ MediaWiki FAQ</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Rysunek|Refraction on an aperture - Huygens-Fresnel principle.svg|i2|Dyfrakcja falowa wed\u0142ug '''zasady Huygensa'''.}}</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* </del>[<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">postorius</del>/<del class=\"diffchange diffchange-inline\">lists</del>/<del class=\"diffchange diffchange-inline\">mediawiki</del>-<del class=\"diffchange diffchange-inline\">announce</del>.<del class=\"diffchange diffchange-inline\">lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.<del class=\"diffchange diffchange-inline\">org</del>/ <del class=\"diffchange diffchange-inline\">Komunikaty </del>o <del class=\"diffchange diffchange-inline\">nowych wersjach MediaWiki </del>(<del class=\"diffchange diffchange-inline\">lista dyskusyjna</del>)<del class=\"diffchange diffchange-inline\">]</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* [https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Localisation#Translation_resources Przet\u0142umacz MediaWiki </del>na <del class=\"diffchange diffchange-inline\">sw\u00f3j j\u0119zyk</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">''' Zasada Huygensa ''' m\u00f3wi, i\u017c punkt (\u017c\u00f3\u0142te kropki), do kt\u00f3rego dotar\u0142a rozchodz\u0105ca si\u0119 fala, jest zn\u00f3w \u017ar\u00f3d\u0142em nowych fal. Poszczeg\u00f3lne fale ulegaj\u0105 superpozycji, oznacza to, \u017ce ich odchylenia od stanu normalnego dodaj\u0105 si\u0119 jak liczby zespolone, jako w bazie dyskretnej {{LinkWz\u00f3r|1.1}} i ci\u0105g\u0142ej {{LinkWz\u00f3r|1.1a}}:</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* </del>[<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">Combating_spam Dowiedz </del>si\u0119, <del class=\"diffchange diffchange-inline\">jak walczy\u0107 ze spamem </del>na <del class=\"diffchange diffchange-inline\">swojej wiki]</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{ElastycznyWiersz|{{CentrujWz\u00f3r|<MATH>\\psi(\\underline{x},t)=\\sum_i\\psi_i^'(\\underline{x},t)\\;</MATH>|1.1}}|{{CentrujWz\u00f3r|<MATH>\\psi(\\underline{x},t)=\\int_{\\underline k}\\psi^'(\\underline{x},t,\\underline{k})d^n\\underline k\\;</MATH>|1.1a}}}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Albo w bazie dyskretno-ci\u0105g\u0142ej {{LinkWz\u00f3r|1.1b}}:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\psi(\\underline{x},t)=\\sum_i\\psi_i^'(\\underline{x},t)+\\int_{\\underline k}\\psi^'(\\underline{x},t,\\underline{k})d^n\\underline k\\;</MATH>|1.1b}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Og\u00f3lnie superpozycj\u0119 dowolnej liczby fal zak\u0142adaj\u0105c, \u017ce {{Formu\u0142a|<MATH>\\psi_i^'\\;</MATH>}} okre\u015bla funkcj\u0119 falow\u0105 znalezienia cz\u0105stki w fali prawdopodobie\u0144stwa, wiedz\u0105c, \u017ce baz\u0119 funkcji wszystkich {{Formu\u0142a|<MATH>\\psi_i^'\\;</MATH>}}, kt\u00f3re s\u0105 wykorzystane w {{LinkWz\u00f3r|1.1}}, mo\u017cna znormalizowa\u0107, bo one tworz\u0105 przestrze\u0144 liniow\u0105 tych wersor\u00f3w bazowych, a wi\u0119c {{LinkWz\u00f3r|1.1}} mo\u017cna zapisa\u0107 w funkcjach bazy:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<math>\\psi(\\underline{x},t)=\\sum_{i}c_i\\psi_i(\\underline{x},t)\\;</MATH>|1.2a}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Lub w postaci ci\u0105g\u0142ej, tzn. gdy parametr charakteryzuj\u0105cy poszczeg\u00f3lne fale jest wielko\u015bci\u0105 ci\u0105g\u0142\u0105:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\psi(\\underline{x},t)=\\int c(\\underline {k})\\psi(\\underline{x},t,\\underline{k})d^n\\underline{k}\\;</MATH>|1.2b}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Lub m\u00f3wi\u0105c inaczej w {{LinkWz\u00f3r|1.1}} (suma) i {{LinkWz\u00f3r|1.1a}} (ca\u0142ka) normalizujemy funkcje falowe (tutaj nazwiemy je bazowymi) do siebie ortogonalne (w mechanice kwantowej s\u0105 tylko takie), wtedy przy funkcjach falowych pojawiaj\u0105 si\u0119 wsp\u00f3\u0142czynniki w sumie (baza dyskretna) i ca\u0142ce (baza ci\u0105g\u0142a), takie, by ko\u0144cowa uzyskana funkcja by\u0142a r\u00f3wnowa\u017cna z t\u0105 pocz\u0105tkow\u0105. St\u0105d zgodnie z zasad\u0105 Huygensa ko\u0144cowe r\u00f3wnanie {{LinkWz\u00f3r|1.2a}} w bazie dyskretnej i {{LinkWz\u00f3r|1.2b}} w bazie ci\u0105g\u0142ej, a tak\u017ce {{LinkWz\u00f3r|1.2d}} w bazie dyskretno-ci\u0105g\u0142ej, spe\u0142niaj\u0105 mechanik\u0119 kwantow\u0105, w kt\u00f3rej te wnioski s\u0105 przyj\u0119te jako postulat o funkcjach falowych.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Je\u015bli za\u0142o\u017cymy, i\u017c wielko\u015b\u0107 {{Formu\u0142a|<MaTH>\\underline{k}\\;</MaTH>}} jest skalarem, a {{Formu\u0142a|<MaTH>\\underline{x}\\;</maTh>}} jest wektorem wodz\u0105cym w przestrzeni tr\u00f3jwymiarowej, to {{Formu\u0142a|<MATH>\\vec{r}=\\underline{x}\\;</MATH>}}, w\u00f3wczas ostatnie r\u00f3wnanie b\u0119dzie mia\u0142o posta\u0107 nast\u0119puj\u0105c\u0105:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\psi(\\vec{r},t)=\\int c(k)\\psi(\\vec{r},t,k)dk\\;</MATH>|1.2c}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Gdy funkcja falowa jest sum\u0105 cz\u0119\u015bci dyskretnej opisanej wzorem {{LinkWz\u00f3r|1.2a}} i ci\u0105g\u0142ej {{LinkWz\u00f3r|1.2b}}, wtedy ko\u0144cowe r\u00f3wnanie na ca\u0142kowit\u0105 funkcje falow\u0105, wiedz\u0105c, \u017ce funkcja falowa dyskretna i ci\u0105g\u0142a s\u0105 do siebie ortogonalne:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{ElastycznyWiersz|1={{CentrujWz\u00f3r|<MATH>\\psi(\\underline{x},t)=\\psi_d(\\underline{x},t)+\\psi_c(\\underline{x},t)=\\underbrace{\\sum_{i}c_i\\psi_i(\\underline{x},t)}_{\\psi_d(\\underline{x},t)}+\\underbrace{\\int c(\\underline {k})\\psi(\\underline{x},t,\\underline{k})d^n\\underline{k}}_{\\psi_c(\\underline{x},t)}\\;</MATH>|1.2d}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|2={{CentrujWz\u00f3r|<MATH>\\int_{V}\\psi_d(\\underline{x},t)\\psi_c(\\underline{x},t)dV=0\\;</MATH>|1.2e}}|3={{CentrujWz\u00f3r|<MATH>1=\\int_V\\left|\\psi(\\underline{x},t)\\right|^2dV\\;</MATH>|1.2f}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Je\u015bli za\u0142o\u017cymy, i\u017c wielko\u015b\u0107 {{Formu\u0142a|<MaTH>\\underline{k}\\;</MaTH>}} jest skalarem, a {{Formu\u0142a|<MaTH>\\underline{x}\\;</maTh>}} jest wektorem wodz\u0105cym w przestrzeni tr\u00f3jwymiarowej, to {{Formu\u0142a|<MATH>\\vec{r}=\\underline{x}\\;</MATH>}}, w\u00f3wczas ostatnie r\u00f3wnanie b\u0119dzie mia\u0142o posta\u0107 nast\u0119puj\u0105c\u0105:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{ElastycznyWiersz|1={{CentrujWz\u00f3r|<MATH>\\psi(\\vec r,t)=\\psi_d(\\vec r,t)+\\psi_c(\\vec r,t)=\\underbrace{\\sum_{i}c_i\\psi_i(\\vec r,t)}_{\\psi_d(\\vec r,t)}+\\underbrace{\\int c(k)\\psi(\\vec r,t,k)dk}_{\\psi_c(\\vec r,t)}\\;</MATH>|1.2da}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|2={{CentrujWz\u00f3r|<MATH>\\int_{V}\\psi_d(\\vec r,t)\\psi_c(\\vec r,t)dV=0\\;</MATH>|1.2ea}}|3={{CentrujWz\u00f3r|<MATH>1=\\int_V\\left|\\psi(\\vec r,t)\\right|^2dV\\;</MATH>|1.2fa}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\u0141atwo zauwa\u017cy\u0107, \u017ce to\u017csamo\u015b\u0107 {{LinkWz\u00f3r|1.2da}} wynika bezpo\u015brednio z {{LinkWz\u00f3r|1.2d}}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Dualizm korpuskularno-falowy ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Dualizm korpuskularno-falowy jest cech\u0105 obiekt\u00f3w fizycznych, np. foton\u00f3w czy elektron\u00f3w, polegaj\u0105c\u0105 na tym, i\u017c w pewnych sytuacjach wykazuj\u0105 one cechy cz\u0105stek (korpusku\u0142), a w innych sytuacjach cechy fal. Mechanika kwantowa, przewiduje, i\u017c cz\u0105stka nie musi zachowywa\u0107 si\u0119 tylko i wy\u0142\u0105cznie jak fala czy cz\u0105stka, lecz mo\u017ce jednocze\u015bnie spe\u0142nia\u0107 cechy stanu po\u015bredniego. W\u00f3wczas nie nale\u017cy stosowa\u0107 ani teorii Huygensa (teoria fal), ani mechaniki klasycznej (teoria Newtona lub Einsteina w zale\u017cno\u015bci od pr\u0119dko\u015bci cz\u0105stki klasycznej), lecz w tym celu nale\u017cy pos\u0142u\u017cy\u0107 si\u0119 mechanik\u0105 kwantow\u0105 (klasyczn\u0105 lub relatywistyczn\u0105, w zale\u017cno\u015bci od warto\u015bci pr\u0119dko\u015bci, jak\u0105 taka cz\u0105stka posiada).</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Energia kwantu energii w zale\u017cno\u015bci od cz\u0119stotliwo\u015bci ko\u0142owej lub cz\u0119sto\u015bci fali ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wiadomo, \u017ce fotony s\u0105 cz\u0105stkami o charakterze korpuskularnym. Wed\u0142ug teorii Plancka energi\u0119 takiego fotonu zapisujemy jako funkcj\u0119 jej cz\u0119sto\u015bci zdefiniowanej jako odwrotno\u015b\u0107 jej okresu drga\u0144. Je\u015bli fotony przyjmiemy jako fale, to energia cz\u0105stki wi\u0105\u017c\u0105ca jej charakter korpuskularny z jej charakterem falowym jest przedstawiana wed\u0142ug zale\u017cno\u015bci skwantowanej:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>E=h\\nu\\;</MATH>|1.4}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Je\u015bli zdefiniujemy cz\u0119stotliwo\u015b\u0107 ko\u0142ow\u0105 fali foton\u00f3w jako stosunek liczby 2&pi; przez okres drga\u0144 omawianej fali, to jego energi\u0119 w zale\u017cno\u015bci od jej cz\u0119stotliwo\u015bci ko\u0142owej drga\u0144 o sta\u0142ej proporcjonalno\u015bci r\u00f3wnej sta\u0142ej kre\u015blonej Plancka jest zdefiniowana jako {{Formu\u0142a|<MaTH>\\hbar={{h}/{2\\pi}}\\;</Math>}}, ta energia korpusku\u0142\u00f3w b\u0119d\u0105cych fotonami jest r\u00f3wna:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>E=h\\nu=h{{1}\\over{T}}={{h}\\over{2\\pi}}{{2\\pi}\\over{T}}=\\hbar\\omega\\;</MATH>|1.5}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Zatem na podstawie {{LinkWz\u00f3r|1.5}} energi\u0105 fotonu\u00a0 z jej cz\u0119stotliwo\u015bci\u0105 ko\u0142ow\u0105 jest przedstawiana wedle sposobu:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>E=\\hbar\\omega\\;</MATH>|1.6|Obramuj}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Nale\u017cy pami\u0119ta\u0107, \u017ce wzory {{LinkWz\u00f3r|1.4}} (wi\u0105\u017c\u0105cych energi\u0119 korpusku\u0142u z jej cz\u0119sto\u015bci\u0105 przy sta\u0142ej proporcjonalno\u015bci sta\u0142ej Plancka) i {{LinkWz\u00f3r|1.5}} (wi\u0105\u017c\u0105cych energi\u0119 korpusku\u0142y z jej cz\u0119stotliwo\u015bci\u0105 ko\u0142ow\u0105 przy sta\u0142ej proporcjonalno\u015bci sta\u0142ej kre\u015blonej Plancka) s\u0105 ze sob\u0105 r\u00f3wnowa\u017cne, tylko ten pierwszy wyra\u017ca si\u0119 poprzez cz\u0119sto\u015b\u0107 fali foton\u00f3w, a drugi przez cz\u0119stotliwo\u015b\u0107 ko\u0142ow\u0105 fali foton\u00f3w.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Efekt fotoelektryczny ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Rysunek|Photoelectric effect in a solid - diagram.svg|lk1|Zjawisko efektu fotoelektrycznego opisanego przez Alberta Einsteina}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">W efekcie fotoelektrycznym fotony o energii {{Formu\u0142a|<MATH>h\\nu\\;</MATH>}} (bo {{LinkWz\u00f3r|1.4}}) trafiaj\u0105 na ekran o pracy wyj\u015bcia W, i wybijaj\u0105 z niego elektrony o pr\u0119dko\u015bciach v. Cz\u0119\u015b\u0107 energii takiego fotonu o tej \u015bredniej energii jest marnowana na prac\u0119 wyj\u015bcia elektronu z metalu, a pozosta\u0142o\u015b\u0107 na energi\u0119 kinetyczn\u0105\u00a0 wybitego obiektu, zatem korzystaj\u0105c z zasady zachowania energii i z wyra\u017cenia na klasyczn\u0105 energi\u0119 kinetyczn\u0105 elektronu, to z zasady zachowania energii mamy wz\u00f3r:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>h\\nu={{mv^2}\\over{2}}+W\\;</MATH>|1.7}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Energi\u0119 kinetyczn\u0105 rozpatrujemy wed\u0142ug mechaniki klasycznej a nie relatywistycznej, bo energia \u015brednia takiego fotonu nie jest o wiele wi\u0119ksza od energii spoczynkowej rozwa\u017canego elektronu.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Fale de Broglie'a ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Energia fotonu w zale\u017cno\u015bci od jej cz\u0119sto\u015bci fali foton\u00f3w jest tak jak we wzorze {{LinkWz\u00f3r|1.4}}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wed\u0142ug wzoru {{LinkWz\u00f3r|1.4}} energia fotonu jest zale\u017cna liniowo od jego cz\u0119sto\u015bci fali foton\u00f3w, je\u015bli potraktowa\u0107 fotony jako fale maj\u0105cy pewn\u0105 d\u0142ugo\u015b\u0107 fali p\u0119dz\u0105cych z pr\u0119dko\u015bci\u0105 fazow\u0105 c, a wi\u0119c maj\u0105cych pewn\u0105 cz\u0119sto\u015b\u0107.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wed\u0142ug szczeg\u00f3lnej teorii wzgl\u0119dno\u015bci jego energia ca\u0142kowita wzgl\u0119dem jej masy relatywistycznej (foton nie ma masy spoczynkowej, jego masa spoczynkowa jest r\u00f3wna zero) mo\u017cna przedstawi\u0107 jako cz\u0105stki p\u0119dz\u0105ce z pr\u0119dko\u015bci\u0105 grupow\u0105 c napisan\u0105 wed\u0142ug:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>E=mc^2\\;</MATH>|1.8}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wzory {{LinkWz\u00f3r|1.4}} i {{LinkWz\u00f3r|1.8}} przedstawiaj\u0105 t\u0105 sam\u0105 energi\u0119 fotonu, raz jako fale p\u0119dz\u0105ce z pr\u0119dko\u015bci\u0105 fazow\u0105 r\u00f3wn\u0105 c, a za drugim razem jako cz\u0105stki p\u0119dz\u0105ce z pr\u0119dko\u015bci\u0105 grupow\u0105 c,\u00a0 wi\u0119c mo\u017cemy je przyr\u00f3wna\u0107 do siebie, dostajemy:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>h\\nu=mc^2\\;</MATH>|1.9}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wiemy, z definicji cz\u0119sto\u015bci dla foton\u00f3w p\u0119dz\u0105cych z pr\u0119dko\u015bci\u0105 \u015bwiat\u0142a (pr\u0119dko\u015b\u0107 fazowa), co mo\u017cna j\u0105 przedstawi\u0107 od d\u0142ugo\u015bci fali \u015bwiat\u0142a p\u0119dz\u0105cych z pr\u0119dko\u015bci\u0105 \u015bwiat\u0142a:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\nu={{1}\\over{T}}={{c}\\over{cT}}={{c}\\over{\\lambda}}\\Rightarrow \\nu={{c}\\over{\\lambda}}\\;</MATH>|1.10}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Podstawiaj\u0105c wz\u00f3r {{LinkWz\u00f3r|1.10}} przedstawiaj\u0105cy cz\u0119sto\u015b\u0107 fali foton\u00f3w w zale\u017cno\u015bci od d\u0142ugo\u015bci fali tego\u017c obiektu do {{LinkWz\u00f3r|1.9}}, to dostajemy r\u00f3wnowa\u017cne r\u00f3wnanie:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>{{hc}\\over{\\lambda}}=mc^2\\;</MATH>|1.11}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Skracaj\u0105c obustronnie r\u00f3wnanie {{LinkWz\u00f3r|1.11}} przez sta\u0142\u0105 pr\u0119dko\u015bci \u015bwiat\u0142a w pr\u00f3\u017cni c, to ono przyjmuje posta\u0107 wi\u0105\u017c\u0105c\u0105 d\u0142ugo\u015b\u0107 fali foton\u00f3w w zale\u017cno\u015bci o jej masy relatywistycznej:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>{{h}\\over{\\lambda}}=mc\\;</MATH>|1.12}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Poniewa\u017c p\u0119d fotonu jest wyra\u017cony wed\u0142ug wzoru {{Formu\u0142a|<MATH>p</ins>=<ins class=\"diffchange diffchange-inline\">mc\\;</MATH>}} wyst\u0119puj\u0105c\u0105 w szczeg\u00f3lnej teorii wzgl\u0119dno\u015bci dla cz\u0105stek bezmasowych, to wz\u00f3r na p\u0119d fotonu mo\u017cemy wykorzysta\u0107 do wzoru {{LinkWz\u00f3r|1.12}} podstawiaj\u0105c za t\u0105 wielko\u015b\u0107, by otrzyma\u0107 p\u0119d fotonu w zale\u017cno\u015bci od d\u0142ugo\u015bci fali foton\u00f3w p\u0119dz\u0105cych z pr\u0119dko\u015bci\u0105 \u015bwiat\u0142a, zatem:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>{{h}\\over{\\lambda}}</ins>=<ins class=\"diffchange diffchange-inline\">p\\;</MATH>|1.13}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Z r\u00f3wnania de Broglie'a {{LinkWz\u00f3r|1.13}} mo\u017cemy wyznaczy\u0107 d\u0142ugo\u015b\u0107 fali foton\u00f3w, kt\u00f3ra jest przedstawiana w zale\u017cno\u015bci od p\u0119du foton\u00f3w, wtedy:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\lambda={{h}\\over{p}}\\;</MATH>|1.14}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Powy\u017csze r\u00f3wnanie jest s\u0142uszne tylko dla fotonu (dla cz\u0105stek nie maj\u0105cej masy spoczynkowej i \u0142adunku), ale mo\u017cna je uog\u00f3lni\u0107 dla dowolnej cz\u0105stki o masie spoczynkowej r\u00f3\u017cnej od zera dla cz\u0105stek maj\u0105cych \u0142adunek {{Formu\u0142a|<MATH>q\\;</MATH>}}, czyli zachodzi: {{Formu\u0142a|<MATH>m_{0}\\neq 0\\;</MATH>}}, wtedy oznaczenia dla r\u00f3wnania {{LinkWz\u00f3r|1.14}} dla dowolnej cz\u0105stki:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{Formu\u0142a|<MATH>p\\;</MATH>}}- to jest p\u0119d uog\u00f3lniony relatywistyczny lub nierelatywistyczny cz\u0105stki,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{Formu\u0142a|<MATH>h\\;</MATH>}}- sta\u0142a Plancka,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>*<ins class=\"diffchange diffchange-inline\">{{Formu\u0142a|<MATH>\\lambda\\;</MATH>}}- d\u0142ugo\u015b\u0107 fal materii de Broglie dla dowolnej cz\u0105stki, je\u015bli potraktowa\u0107 cz\u0105stki masowe lub fotony jako fale o pewnej d\u0142ugo\u015bci fali i pr\u0119dko\u015bci fazowej.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Zapiszmy p\u0119d uog\u00f3lniony cz\u0105stki w zale\u017cno\u015bci od jej liczby falowej znaj\u0105c jej definicj\u0119 oraz sta\u0142\u0105 kre\u015blon\u0105 Plancka</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>p={{h}\\over{\\lambda}}=h{{1}\\over{\\lambda}}={{h}\\over{2\\pi}}{{2\\pi}\\over{\\lambda}}=\\hbar k\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.15}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Zatem p\u0119d uog\u00f3lniony cz\u0105stki jest napisany w zale\u017cno\u015bci od warto\u015bci liczby falowej:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>p=\\hbar k\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1</ins>.<ins class=\"diffchange diffchange-inline\">16|Obramuj}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Ale wiadomo, \u017ce wektory {{Formu\u0142a|<MATH>\\vec{p}\\;</MATH>}} (p\u0119d uog\u00f3lniony cz\u0105stki) i {{Formu\u0142a|<MATH>\\vec{k}\\;</MaTh>}} (wektor liczby falowej) s\u0105 wsp\u00f3\u0142liniowe i maj\u0105 te same zwroty), to r\u00f3wnanie {{LinkWz\u00f3r|1.16}} wektorowo mo\u017cemy zapisa\u0107 w postaci:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\vec {p}=\\hbar\\vec{k}\\;</MATH>|1.17}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Oczywiste jest, \u017ce wz\u00f3r skalarny {{LinkWz\u00f3r|1</ins>.<ins class=\"diffchange diffchange-inline\">16}} wynika ze wzoru wektorowego {{LinkWz\u00f3r|1.17}}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Napiszmy wz\u00f3r na energi\u0119 mechaniczn\u0105 cz\u0105stki znanego z mechaniki klasycznej przy pomocy wzoru {{LinkWz\u00f3r|1.17}}. Energia kinetyczna z definicji jest ona wyra\u017cona przy pomocy p\u0119du klasycznego danej cz\u0105stki . Je\u015bli wektor p\u0119du uog\u00f3lnionego wyrazimy przy pomocy wzoru wektorowego, czyli r\u00f3wnania {{LinkWz\u00f3r|1.17}}, to jego energia mechaniczna w zale\u017cno\u015bci od liczby falowej przy istnieniu potencja\u0142u magnetycznego, je\u015bli potraktowa\u0107 cz\u0105stki jako fale materii o pewnej d\u0142ugo\u015bci {{Formu\u0142a|<MATH>\\lambda\\;</MATH>}}, zapisujemy dla teorii Newtona w postaci:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\hbar\\omega=E=T+U=\\sum_i{{(\\vec p_i-q_i\\vec A_i)^2}\\over{2m_i}}+\\sum_{ij,i<j}q_i\\varphi_{ij}+\\sum_{i}q_i\\varphi_i+\\sum_{ij,i<j}U_{ij}+\\sum_iU_i=\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>{{Br}}<MATH>=\\sum_i\\underbrace{{{(\\hbar \\vec k_i-q_i\\vec A_i)^2}\\over{2m_i}}}_T+\\sum_{ij,i<j}q_i\\varphi_{ij}+\\sum_{i}q_i\\varphi_i+\\sum_{ij,i<j}U_{ij}+\\sum_iU_i\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.18}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A dla teorii Einsteina</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\hbar\\omega=E=E_r+E_p=\\sum_{i}\\sqrt{c^2(\\vec p_i-q_i\\vec A_i)^2+m_{0i}^2c^4}+\\sum_{ij,i<j}q_i\\varphi_{ij}+\\sum_iq_i\\varphi_i+\\sum_{ij,i<j}U_{ij}+\\sum_{i}U_i=\\;</MATH>{{Br}}<MATH>=\\sum_{i}\\sqrt{c^2(\\hbar \\vec k_i-q_i\\vec A_i)^2+m_{0i}^2c^4}+\\sum_{ij,i<j}q_i\\varphi_{ij}+\\sum_iq_i\\varphi_i+\\sum_{ij,i<j}U_{ij}+\\sum_{i}U_i</MATH>|1.18a}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Widzimy, \u017ce wz\u00f3r na energi\u0119 mechaniczn\u0105 jest zale\u017cny od masy cia\u0142a, liczby falowej\u00a0 je\u015bli traktowa\u0107 cz\u0105stki jako fale de Broglie'a, \u0142adunku cz\u0105stki oraz potencja\u0142u elektrycznego i magnetycznego.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Cia\u0142o doskonale czarne wed\u0142ug Plancka ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Foton jest bozonem wi\u0119c w dowolnym stanie mo\u017ce znajdowa\u0107 si\u0119 {{Formu\u0142a|<MATH>n=0,1,2,...,\\infty\\;</MATH>}}foton\u00f3w, a wi\u0119c energia tych foton\u00f3w w tym stanie jest skwantowana tzn. jest zale\u017cna od ca\u0142kowitego wsp\u00f3\u0142czynnika {{Formu\u0142a|<MATH>n\\;</MATH>}} i cz\u0119sto\u015bci foton\u00f3w, je\u015bli potraktowa\u0107 je jako fale, przedstawia si\u0119 jako {{Formu\u0142a|<MATH>E_n\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>}} powstaj\u0105ca po pomno\u017ceniu przez {{Formu\u0142a|<MATH>n\\;</MATH>}} energii pojedynczego fotonu {{LinkWz\u00f3r|1.4}}\u00a0 i oznaczamy j\u0105 przez {{Formu\u0142a|<MATH>E_n=nE\\;</MATH>}}:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Energia foton\u00f3w w dowolnym stanie, tzn. {{Formu\u0142a|<MATH>E_n\\;</MATH>}} {{LinkWz\u00f3r|1.4}} jest wielokrotno\u015bci\u0105 energii podstawowej {{Formu\u0142a|<MATH>E=h\\nu\\;</MATH>}} wed\u0142ug postulatu Plancka,\u00a0 ta energia fotonu jest zale\u017cna liniowo od cz\u0119sto\u015bci fali foton\u00f3w. Prawdopodobie\u0144stwo, \u017ce cz\u0105stki (fotony) maj\u0105 energi\u0119 {{Formu\u0142a|<MATH>E_n</MATH>}} jest okre\u015blona przez wz\u00f3r Boltzmanna w zale\u017cno\u015bci od temperatury w jakim uk\u0142ad si\u0119 on znajduje:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>P(E_n)=A e^{-{{E_n}\\over{k_BT}}}\\;</MATH>|1.19}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">We wzorze {{LinkWz\u00f3r|1.19}} energia danego poziomu E{{Sub|n}} jest napisana przez r\u00f3wnanie {{LinkWz\u00f3r|1.4}}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\u015arednia energia fotonu opisana jako \u015brednia energia fotonu w uk\u0142adzie wyra\u017cona przy pomocy prawdopodobie\u0144stwa danego stanu o numerze n okre\u015blonej przez wz\u00f3r {{LinkWz\u00f3r|1.19}} jest pisana:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\overline{E}={{\\sum^{\\infty}_{n=0} E_n e^{-{{E_n}\\over{k_BT}}}}\\over{\\sum^{\\infty}_{n=0}e^{-{{E_n}\\over{k_BT}}}}}={{\\sum^{\\infty}_{n=0} n E e^{-{{n E}\\over{k_BT}}}}\\over{\\sum^{\\infty}_{n=0}e^{-{{n E}\\over{k_BT}}}}}=k_BT{{\\sum^{\\infty}_{n=0} n {{E}\\over{k_BT}} e^{-{{n E}\\over{k_BT}}}}\\over{\\sum^{\\infty}_{n=0}e^{-{{n E}\\over{k_BT}}}}}\\;</MATH>|1.20}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">gdzie</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* <ins class=\"diffchange diffchange-inline\">{{Formu\u0142a|<MATH>E=h\\nu\\;</MATH>}}, bo {{Formu\u0142a|<MATH>E_n=nE\\;</MATH>}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Obierzmy wielko\u015b\u0107, kt\u00f3ra zale\u017cy od temperatury uk\u0142adu i energii podstawowej stanu podstawowego (n=1) zale\u017c\u0105ca tylko od cz\u0119sto\u015bci fali foton\u00f3w w jakim mo\u017ce znajdowa\u0107 si\u0119 bezmasowy foton:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>x={{E}\\over{k_BT}}\\;</MATH>|1.21}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\u015arednia energia fononu w uk\u0142adzie napisana jest wed\u0142ug {{LinkWz\u00f3r|1.20}}, co po podstawieniu do niego {{LinkWz\u00f3r|1.21}}, kt\u00f3ry\u00a0 jest zale\u017cny od temperatury uk\u0142adu i jego stanu podstawowego E, a to z kolei jest zale\u017cny od cz\u0119sto\u015bci fotonu znajduj\u0105cych si\u0119 w naszym rozwa\u017canym uk\u0142adzie</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\overline{E}=k_BT{{\\sum^{\\infty}_{n=0} n x e^{-nx}}\\over{\\sum^{\\infty}_{n=0}e^{-nx}}}\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.22}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wiadomo z analizy matematycznej, \u017ce zachodzi to\u017csamo\u015b\u0107 wynikaj\u0105ca z w\u0142asno\u015bci szeregu pot\u0119gowego, bo eksponens e{{Sup|-nx}} tworzy pewnego rodzaju szereg pot\u0119gowy o ilorazie e{{Sup|-x{{Sub|0}}}}:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\sum^{\\infty}_{n=0}e^{-nx}=\\lim_{n\\rightarrow \\infty}{{1-e^{-nx}}\\over{1-e^{-x}}}={{1}\\over{1-e^{-x}}}\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.23}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">To\u017csamo\u015b\u0107 {{LinkWz\u00f3r|1.23}} mo\u017cemy wykorzysta\u0107 do policzenia mianownika wyra\u017cenia {{LinkWz\u00f3r|1</ins>.<ins class=\"diffchange diffchange-inline\">22}}</ins>. \u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Zr\u00f3\u017cniczkujemy obustronnie r\u00f3wnanie {{LinkWz\u00f3r|1.23}} wzgl\u0119dem x{{Sub|0}}, otrzymujemy:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\sum^{\\infty}_{n=0}-ne^{-nx}=-{{-(-e^{-x})}\\over{(1-e^{-x})^2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\Rightarrow\\sum^{\\infty}_{n=0}ne^{-nx}={{e^{-x}}\\over{(1-e^{-x})^2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\Rightarrow\\sum^{\\infty}_{n=0}nxe^{-nx}=x{{e^{-x}}\\over{(1-e^{-x})^2}}\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.24}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">To\u017csamo\u015b\u0107 {{LinkWz\u00f3r|1.24}} mo\u017cemy u\u017cy\u0107 do policzenia licznika r\u00f3wnania {{LinkWz\u00f3r|1.22}}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Po podstawieniu to\u017csamo\u015bci {{LinkWz\u00f3r|1.24}} i {{LinkWz\u00f3r|1.23}} do wzoru {{LinkWz\u00f3r|1.22}} oraz wykorzystuj\u0105c podstawienie {{LinkWz\u00f3r|1.21}} dostajemy wz\u00f3r na \u015bredni\u0105 energi\u0119 fotonu o danej cz\u0119sto\u015bci w uk\u0142adzie:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\overline{E}=k_BT{{x{{e^{-x}}\\over{(1-e^{-x})^2}}}\\over{{{1}\\over{1-e^{-x}}}}}=k_BTx{{e^{-x}}\\over{1-e^{-x}}}{{e^{x}}\\over{e^{x}}}=k_BT x{{1}\\over{e^{x}-1}}=k_BT{{h\\nu}\\over{k_BT}}{{1}\\over{e^{{{h\\nu}\\over{k_BT}}}-1}}=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">h\\nu{{1}\\over{e^{{{h\\nu}\\over{k_BT}}}-1}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.25}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A zatem \u015brednia energia ca\u0142kowita fotonu w uk\u0142adzie jest zale\u017cna od jej cz\u0119sto\u015bci i temperatury uk\u0142adu foton\u00f3w, jest wyra\u017cona wed\u0142ug wzoru {{LinkWz\u00f3r|1.25}}</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\overline{E}={{h\\nu}\\over{e^{{{h\\nu}\\over{k_BT}}}-1}}\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.26}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Zak\u0142adamy, \u017ce mamy sze\u015bcian o boku L, kt\u00f3rego warto\u015b\u0107 odchyle\u0144 amplitud fali foton\u00f3w na brzegach sze\u015bcianu jest jednakowa i r\u00f3wna zero</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\sum_i n_i{{\\lambda_i}\\over{2}}=L\\Rightarrow [n_1,n_2,n_3</ins>]<ins class=\"diffchange diffchange-inline\">[\\lambda_1,\\lambda_2,\\lambda_3]=2L\\;</MATH>|1.27}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Poniewa\u017c wektor liczb falowych nieujemnych {{Formu\u0142a|<MATH>\\vec{n}=[n_1,n_2,n_3]\\;</MATH>}} jest wektorem r\u00f3wnoleg\u0142ym do wektora\u00a0 {{Formu\u0142a|<MATH>\\vec{\\lambda}=</ins>[<ins class=\"diffchange diffchange-inline\">\\lambda_1,\\lambda_2,\\lambda_3]\\;</MATH>}}, tak\u017ce {{Formu\u0142a|<MATH>\\vec{\\lambda}^2=\\lambda^2\\;</MATH>}}\u00a0 jest to kwadrat d\u0142ugo\u015bci fali foton\u00f3w,\u00a0 te wspomniane wektory r\u00f3wnie\u017c maj\u0105 ten sam zwrot, zatem maj\u0105c wyra\u017cenie {{LinkWz\u00f3r|1.28}} podnosimy je do kwadratu, to z definicji iloczynu skalarnego dla wektor\u00f3w maj\u0105cych ten sam kierunek i zwrot</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>(n_1^2+n_2^2+n_3^2)\\lambda^2=\\left(2L\\right)^2\\Rightarrow n_1^2+n_2^2+n_3^2=\\left({{2L}\\over{\\lambda}}\\right)^2\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.28}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Promie\u0144 naszej kuli jest wyra\u017cony wed\u0142ug {{LinkWz\u00f3r|1.28}}, w kt\u00f3rej s\u0105 pewne warto\u015bci n{{Sub|i}}, kt\u00f3ra jest zale\u017cna od jakie\u015b d\u0142ugo\u015bci fali\u00a0 jak\u0105 foton mo\u017ce posiada\u0107:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>R={{2L}\\over{\\lambda}}={{2L}\\over{c}}\\nu\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1</ins>.<ins class=\"diffchange diffchange-inline\">29}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Ale musi zachodzi\u0107 n{{Sub|i}}&ge;0, aby ten warunek by\u0142 spe\u0142niony musimy rozwa\u017cy\u0107 {{Formu\u0142a|<MATH>{{1}\\over{8}}\\;</MATH>}} sfery o grubo\u015bci dR, jeszcze trzeba uwzgl\u0119dni\u0107 to, \u017ce foton ma spin 1, kt\u00f3ry wyst\u0119puje w dw\u00f3ch stanach, zatem liczba stan\u00f3w, kt\u00f3rych mo\u017ce znajdowa\u0107 si\u0119 foton o cz\u0119sto\u015bci podstawowej &nu; w danej obj\u0119to\u015bci V, jest zapisana wedle schematu:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>dn=2{{1}\\over{8}}4\\pi R^2 dR=\\pi R^2 dR=\\pi\\left({{2L}\\over{c}}\\right)^3\\nu^2 d\\nu=8\\pi V{{\\nu^2}\\over{c^3}} d\\nu\\;</MATH>|1</ins>.<ins class=\"diffchange diffchange-inline\">30}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Energia foton\u00f3w wypromieniowana na jednostk\u0119 czasu, obj\u0119to\u015bci i jego cz\u0119sto\u015bci, o \u015bredniej energii {{Formu\u0142a|<MATH>\\overline{E}\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>}}, zale\u017cy od cz\u0119sto\u015bci, zatem moc wypromieniowania z uk\u0142adu fotonu o danej cz\u0119sto\u015bci foton\u00f3w jest r\u00f3wna:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>r_{\\nu}={{1}\\over{V}}{{dn}\\over{d\\nu}}\\overline{E}={{8\\pi\\nu^2}\\over{c^3}}\\overline{E}\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.31}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Je\u015bli przyjmowa\u0107 b\u0119dziemy fizyk\u0119 klasyczn\u0105, to \u015brednia energia fotonu dla jednego kierunku jest zapisana jako energia zale\u017cna tylko od temperatury uk\u0142adu kwantowego liczona z rozk\u0142adu Boltzmanna:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\overline E={{\\int_0^{\\infty} h\\nu e^{-\\beta h\\nu}d\\nu}\\over{\\int_{0}^{\\infty} e^{-\\beta h\\nu}d\\nu}}=-k_BT{{\\int_0^{\\infty}(-1) \\beta h\\nu e^{-\\beta h\\nu}d(-\\beta h\\nu)}\\over{\\int_0^{\\infty} e^{-\\beta h\\nu}d(-\\beta h\\nu)}}=-k_BT{{-\\beta h\\nu e^{-\\beta h\\nu}|_0^{\\infty}-\\int_0^{\\infty}e^{-\\beta h\\nu} d(-\\beta h\\nu)}\\over{-1}}=k_BT\\Rightarrow \\overline E=k_BT\\;</MATH>|1.32}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wz\u00f3r {{LinkWz\u00f3r|1.32}}, kt\u00f3ry jest wzorem na \u015bredni\u0105 energi\u0119 fotonu podstawiamy do {{LinkWz\u00f3r|1.31}}, co wtedy:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>r_{\\nu}={{8\\pi\\nu^2}\\over{c^3}}k_BT\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.33}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wz\u00f3r {{LinkWz\u00f3r|1.33}} nazywamy '''wzorem Rayleigha</ins>-<ins class=\"diffchange diffchange-inline\">Jeansa'''. Z tego wzoru wynika niesko\u0144czon\u0105 warto\u015b\u0107 r{{Sub|&nu;}} dla &nu; bardzo du\u017cego, co nazywamy '''katastrof\u0105 ultrafioletow\u0105'''</ins>. <ins class=\"diffchange diffchange-inline\">Gdy przyjmowa\u0107 b\u0119dziemy wed\u0142ug Plancka, tzn</ins>. <ins class=\"diffchange diffchange-inline\">podstawiaj\u0105c za \u015bredni\u0105 energi\u0119 wz\u00f3r {{LinkWz\u00f3r|1</ins>.<ins class=\"diffchange diffchange-inline\">26}} zale\u017cny nie tylko od temperatury uk\u0142adu, ale te\u017c od cz\u0119sto\u015bci foton\u00f3w wypromieniowan\u0105 z uk\u0142adu, to moc wypromieniowana z uk\u0142adu jest wyra\u017cona:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>r_{\\nu}={{8\\pi\\nu^2}\\over{c^3}}{{h\\nu}\\over{e^{{{h\\nu}\\over{k_BT}}}-1}}\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.34}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Rysunek|Wiens_law.svg|lk2|Rozk\u0142ad Plancka dla r\u00f3\u017cnych temperatur. Moc (kJ/s) promieniowania cia\u0142a </ins>o <ins class=\"diffchange diffchange-inline\">powierzchni 1m{{Sup|2}} do k\u0105ta bry\u0142owego pe\u0142nego w zakresie d\u0142ugo\u015bci fal 1 nm.}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Gdy uwzgl\u0119dnimy \"h\" bardzo ma\u0142e, czyli matematycznie m\u00f3wi\u0105c h\u21920, to wz\u00f3r {{LinkWz\u00f3r|1.34}} przechodzi w r\u00f3wnanie:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>r_{\\nu}={{8\\pi\\nu^2}\\over{c^3}}{{h\\nu}\\over{e^{{{h\\nu}\\over{k_BT}}}-1}}\\rightarrow {{8\\pi\\nu^2}\\over{c^3}}{{h\\nu}\\over{{{h\\nu}\\over{k_BT}}}}={{8\\pi\\nu^2}\\over{c^3}}k_BT\\;</MATH>|1.35}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Czyli po tych zabiegach dochodzimy zn\u00f3w do wzoru '''Rayleigha-Jeansa''' wyprowadzonej w pozycji {{LinkWz\u00f3r|1.33}}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Dla du\u017cych cz\u0119sto\u015bci, tzn.:&nu;<big>&#187;</BIG>0, to wz\u00f3r {{LinkWz\u00f3r|1.34}}, w kt\u00f3rym eksponens wyst\u0119puj\u0105cy w mianowniku staje si\u0119 bardzo du\u017cy w stosunku do jedynki, zatem t\u0105 jedynk\u0119 wyst\u0119puj\u0105c\u0105 w mianowniku mo\u017cemy pomin\u0105\u0107, co staje si\u0119 jasne:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>r_{\\nu}={{8\\pi\\nu^2}\\over{c^3}}{{h\\nu}\\over{e^{{{h\\nu}\\over{k_BT}}}-1}}\\rightarrow{{8\\pi\\nu^2}\\over{c^3}}{{h\\nu}\\over{e^{{{h\\nu}\\over{k_BT}}}}}={{8\\pi h\\nu^3}\\over{c^3}}e^{-{{h\\nu}\\over{k_BT}}}={{8\\pi\\nu^2}\\over{c^3}}h\\nu e^{-{{h\\nu}\\over{k_BT}}}=\\;</MATH>{{Br}}<MATH>={{8\\pi\\nu^2}\\over{c^3}}\\underbrace{h\\nu n</ins>(<ins class=\"diffchange diffchange-inline\">h\\nu</ins>)<ins class=\"diffchange diffchange-inline\">}_{\\overline{E}}\\;</MATH>|1.36}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wz\u00f3r Wiena {{LinkWz\u00f3r|1.36}}\u00a0 mo\u017cemy r\u00f3wnie\u017c otrzyma\u0107 z {{LinkWz\u00f3r|1.31}}, podstawiaj\u0105c do niego za \u015bredni\u0105 energi\u0119 fotonu iloczyn energii fotonu {{LinkWz\u00f3r|1.4}} i ilo\u015bci foton\u00f3w wynikaj\u0105cych z rozk\u0142adu Boltzmanna, wiedz\u0105c, \u017ce potencja\u0142 chemiczny w nim jest r\u00f3wny zero {{Formu\u0142a|<MATH>\\mu=0\\;</MATH>}}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Prawo Stefana-Boltzmanna ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Ca\u0142kowita energia wypromieniowana przez cia\u0142o doskonale czarne o wszystkich mo\u017cliwych cz\u0119sto\u015bci liczona jest przy pomocy wzoru wyra\u017cenia dla jednej cz\u0119sto\u015bci, kt\u00f3ra jest przedstawiona przez wz\u00f3r {{LinkWz\u00f3r|1.34}},\u00a0 jest to ca\u0142k\u0105 mocy promieniowania dla wszystkich cz\u0119sto\u015bci fotonu w jakim wyst\u0119puje foton na jednostk\u0119 obj\u0119to\u015bci</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>R_{\\nu}=\\int^{\\infty}_{\\nu=0}r_{\\nu}d\\nu=\\int^{\\infty}_{\\nu=0}{{8\\pi\\nu^2}\\over{c^3}}{{h\\nu}\\over{e^{{{h\\nu}\\over{k_BT}}}-1}}d\\nu={{8\\pi h}\\over{c^3}}\\int^{\\infty}_{\\nu=0}{{\\nu^3}\\over{e^{{{h\\nu}\\over{k_BT}}}-1}}d\\nu=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\begin{Bmatrix}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">x={{h\\nu}\\over{k_BT}}\\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\nu={{xk_BT}\\over{h}}\\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">d\\nu={{k_BT}\\over{h}}dx</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\end{Bmatrix}={{8\\pi h}\\over{c^3}}\\int^{\\infty}_{0}{{\\left({{k_BT}\\over{h}}\\right)^4x^3 }\\over{e^x-1}}dx=\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>{{Br}}<MATH>=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{8\\pi k_B^4}\\over{c^3h^3}}T^4\\int^{\\infty}_{0}{{x^3}\\over{e^x-1}}dx={{8\\pi k_B^4}\\over{c^3h^3}}T^4{{\\pi^4}\\over{15}}={{8\\pi^5k_B^4}\\over{15c^3h^3}}T^4=\\sigma T^4</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1</ins>.<ins class=\"diffchange diffchange-inline\">37}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">W r\u00f3wnaniu {{LinkWz\u00f3r|1</ins>.<ins class=\"diffchange diffchange-inline\">37}} dokonali\u015bmy podstawienia w ko\u0144cowym wyniku:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\sigma={{8\\pi^5k_B^4}\\over{15c^3h^3}}\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.37a}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Jest ona zale\u017cna od sta\u0142ych fizycznych, tzn. od sta\u0142ej Boltzmanna ({{Formu\u0142a|<MATH>k_B\\;</MATH>}}), sta\u0142ej pr\u0119dko\u015bci \u015bwiat\u0142a w pr\u00f3\u017cni ({{Formu\u0142a|<MATH>c\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>}}), sta\u0142ej Plancka ({{Formu\u0142a|<MATH>h\\;</MaTH>}}) i jednej sta\u0142ej matematycznej {{Formu\u0142a|<MATH>\\pi\\;</MaTH>}}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Ca\u0142kowita energia na jednostk\u0119 obj\u0119to\u015bci jest zapisana wed\u0142ug wzoru</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>R_{\\nu}=\\sigma T^4\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.37b}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">We wzorze {{LinkWz\u00f3r|1.37b}} sta\u0142a &sigma; zosta\u0142a wyja\u015bniona w punkcie {{LinkWz\u00f3r|1.37a}}.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Powy\u017csze wyra\u017cenie jest zale\u017cne od czwartej pot\u0119gi temperatury T (wyra\u017conej w kelwinach) przy sta\u0142ej proporcjonalno\u015bci\u00a0 {{LinkWz\u00f3r|1.37a}}, jest to moc wypromieniowana przy cia\u0142o doskonale czarne przy wszystkich mo\u017cliwych cz\u0119sto\u015bciach na jednostk\u0119 obj\u0119to\u015bci.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=== Prawo przesuni\u0119\u0107 Wiena ===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wyznaczmy dla jakich &nu; funkcja r{{Sub|&nu;}} wz\u00f3r {{LinkWz\u00f3r|1.34}} (rozk\u0142ad Placka cia\u0142a doskonale czarnego) przyjmuje maksimum dla danej temperatury T uk\u0142adu, czyli dla jakich &nu;, nat\u0119\u017cenie promieniowania jest najwi\u0119ksze, czyli matematycznie m\u00f3wi\u0105c maksimum wyst\u0119puje, gdy pochodna nat\u0119\u017cenia promieniowania </ins>na <ins class=\"diffchange diffchange-inline\">jednostk\u0119 obj\u0119to\u015bci w rozk\u0142adzie Plancka jest r\u00f3wna zero, czyli musimy policzy\u0107 pochodn\u0105 wyra\u017cenia {{LinkWz\u00f3r|1.34}} wzgl\u0119dem cz\u0119sto\u015bci foton\u00f3w z jakich\u00a0 mo\u017ce drga\u0107 fala, je\u015bli przyjmowa\u0107, \u017ce fotonowi odpowiada pewna fala wed\u0142ug teorii korpuskularno-cz\u0105steczkowej:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>0={{dr_{\\nu}}\\over{d\\nu}}={{8\\pi h}\\over{c^3}}{{d}\\over{d\\nu}}{{\\nu^3}\\over{e^{{{h\\nu}\\over{k_BT}}-1}}}=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{8\\pi h}\\over{c^3}}{{\\left(3\\nu^2\\left(e^{{{h\\nu}\\over{k_BT}}}-1\\right)-\\nu^3{{h}\\over{k_BT}}e^{{{h\\nu}\\over{k_BT}}}\\right)}\\over{\\left(e^{{{h\\nu}\\over{k_BT}}}-1\\right)^2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\;</MATH>|1.38}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Warto\u015b\u0107 zerow\u0105 przyjmuje licznik wyra\u017cenia {{LinkWz\u00f3r|1.38}} a mianownik jest nier\u00f3wny zero dla niezerowych cz\u0119sto\u015bci promieniowania wypromieniowanego z uk\u0142adu dla dowolnej sko\u0144czonej temperatury wi\u0119kszej od zera, otrzymujemy:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>0=3\\nu^2\\left(e^{{{h\\nu}\\over{k_BT}}}-1\\right)-\\nu^3{{h}\\over{k_BT}}e^{{{h\\nu}\\over{k_BT}}}\\;</MATH>|1.39}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Dzielimy r\u00f3wnanie {{LinkWz\u00f3r|1.39}} obustronnie przez kwadrat cz\u0119sto\u015bci foton\u00f3w &nu;{{Sup|2}}, tak\u017ce wiemy, \u017ce w og\u00f3lno\u015bci cz\u0119sto\u015b\u0107 fali foton\u00f3w jest r\u00f3\u017cna od zera, zatem dochodzimy do wniosku, \u017ce:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>0=3\\left(e^{{{h\\nu}\\over{k_BT}}}-1\\right)-\\nu{{h}\\over{k_BT}}e^{{{h\\nu}\\over{k_BT}}}\\;</MATH>|1.40}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Obierzmy podstawienie we wzorze {{LinkWz\u00f3r|1.40}}, tzn. za wielko\u015b\u0107 zale\u017cn\u0105 od temperatury uk\u0142adu i cz\u0119sto\u015bci fali foton\u00f3w wypromieniowan\u0105 z uk\u0142adu o maksymalnym nat\u0119\u017ceniu, kt\u00f3ra jest wielko\u015bci\u0105 bezwymiarow\u0105, czyli dokonajmy podstawienia:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>x={{h\\nu}\\over{k_BT}}\\;</MATH>|1.41}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">R\u00f3wnanie {{LinkWz\u00f3r|1.40}} po podstawieniu do niego wielko\u015bci bezwymiarowej {{LinkWz\u00f3r|1.41}} przedstawia si\u0119:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>3(e^x-1)-xe^x=0\\Rightarrow 3e^x-xe^x-3=0\\;</MATH>|1.42}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Rozwi\u0105zuj\u0105c r\u00f3wnanie {{LinkWz\u00f3r|1.42}} numerycznie dla x&ne;0, w kt\u00f3rych jest \u015bci\u015ble okre\u015blone x, a zatem je\u015bli mamy x, to ze wzoru {{LinkWz\u00f3r|1.41}} dostajemy r\u00f3wnanie po wyznaczeniu cz\u0119sto\u015bci zale\u017c\u0105cej od x i od temperatury uk\u0142adu:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\nu=x{{k_BT}\\over{h}}\\;</MATH>|1.43}} </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wiemy jednak, \u017ce cz\u0119sto\u015b\u0107 fali jest zale\u017cna od odwrotno\u015bci d\u0142ugo\u015bci fali foton\u00f3w w spos\u00f3b: {{Formu\u0142a|<MATH>\\nu={{c}\\over{\\lambda}}\\;</MATH>}} oraz przyjmujemy, \u017ce fotony b\u0119dziemy przyjmowa\u0107 jako fale o d\u0142ugo\u015bci &lambda; rozchodz\u0105cych si\u0119 z pr\u0119dko\u015bci\u0105 fazow\u0105 c, wtedy:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>{{c}\\over{\\lambda}}=x{{k_BT}\\over{h}}\\Rightarrow\\lambda={{ch}\\over{xk_B}}{{1}\\over{T}}\\Rightarrow\\lambda={{C}\\over{T}}\\;</MATH>|1.44}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">W r\u00f3wnaniu {{LinkWz\u00f3r|1.44}} widzimy, \u017ce czym wi\u0119ksza temperatura uk\u0142adu, to jest mniejsza d\u0142ugo\u015b\u0107 fali promieniowania o najwi\u0119kszej mocy wypromieniowana z uk\u0142adu.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A zatem z r\u00f3wnania {{LinkWz\u00f3r|1.44}}, dostajemy nast\u0119pne r\u00f3wnowa\u017cne r\u00f3wnanie:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\lambda T=C\\;</MATH>{{Tekst|, gdzie:}} <MATH>C=2.9\\cdot 10^{-3} m\\cdot K</MATH>|1.45}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Iloczyn d\u0142ugo\u015bci fali promieniowania o najwi\u0119kszej mocy z uk\u0142adu przez temperatur\u0119 uk\u0142adu jest wielko\u015bci\u0105 sta\u0142\u0105 i niezale\u017cn\u0105 od innych parametr\u00f3w charakteryzuj\u0105cych uk\u0142ad.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Paczki falowe w nowej teorii kwant\u00f3w ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">''' Paczka falowa ''' inaczej zwany pakiet falowy, jest to fala skupiona w ograniczonym obszarze przestrzeni. Swobodn\u0105 paczk\u0119 falow\u0105 mo\u017cna traktowa\u0107 jako superpozycj\u0119 (z\u0142o\u017cenie) harmonicznych fal p\u0142askich o r\u00f3\u017cnych cz\u0119stotliwo\u015bciach ko\u0142owych wed\u0142ug ''' zasada Huygensa '''.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Rysunek|Wave packet (no dispersion).gif|i3|Paczka falowa jako superpozycja fal harmonicznych z pewnego przedzia\u0142u dla liczb falowych}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Aby w tym celu usun\u0105\u0107 ca\u0142kowit\u0105 lokalizacj\u0119 cz\u0105stki a jej delokalizacj\u0105 wprowadza si\u0119 funkcj\u0119 falow\u0105 opisuj\u0105c\u0105 fal\u0119 p\u0142ask\u0105 o d\u0142ugo\u015bci fali zale\u017cnej od jej liczby falowej, kt\u00f3ra te\u017c charakteryzuje fal\u0119 w spos\u00f3b: {{Formu\u0142a|<MATH>\\lambda={{2\\pi}\\over{k}}\\;</Math>}} propaguj\u0105c\u0105 si\u0119 w kierunku osi x, kt\u00f3r\u0105 mo\u017cna przedstawi\u0107 w postaci:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<math>\\psi(x,t)=Ce^{i(\\omega t-kx)}\\;</MATH>|1.46}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wykorzystajmy wz\u00f3r {{LinkWz\u00f3r|1.6}} oraz {{LinkWz\u00f3r|1.17}}, to wyra\u017cenie (funkcj\u0119 falow\u0105) {{LinkWz\u00f3r|1.46}} zapisujemy jako funkcj\u0119 energii cz\u0105stki o \u015bci\u015ble okre\u015blonym p\u0119dzie, gdy ona znajduje si\u0119 w po\u0142o\u017ceniu x i w czasie t, kt\u00f3ra przedstawia si\u0119:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\psi(x,t)=Ce^{{{i}\\over{\\hbar}}(Et-px)}\\;</MATH>|1.47}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wprowad\u017amy superpozycj\u0119 fal o liczbach falowych z przedzia\u0142u (k{{Sub|0}}-&Delta;k,k{{Sub|0}}+&Delta;k) o r\u00f3\u017cnych amplitudach przy pomocy liczb funkcji falowych z definiowanych wedle schematu {{LinkWz\u00f3r|1.46}} i obierzmy jego ca\u0142k\u0119 po omawianych zakresie zmienno\u015bci liczby falowej k.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\psi(x,t)=\\int^{k_0+\\Delta k}_{k_0-\\Delta k}C(k)e^{i(\\omega t-kx)}dk\\;</MATH>|1.48}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Roz\u0142\u00f3\u017cmy w szereg Taylora cz\u0119stotliwo\u015b\u0107 ko\u0142ow\u0105 drga\u0144 cz\u0105stki wzgl\u0119dem funkcji falowej fali k naszej rozwa\u017canej fali p\u0142askiej i napiszmy ten nasz szereg Taylora do drugiego rz\u0119du wyrazy w\u0142\u0105cznie, a dalsze wyrazy oznaczamy wielokropkami:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<math>\\omega(k)=\\omega_0+\\left({{d\\omega}\\over{dk}}\\right)_{k=k_0}(k-k_0)+{{1}\\over{2}}\\left({{d^2\\omega}\\over{dk^2}}\\right)_{k=k_0}(k-k_0)^2+...\\;</MATH>|1.49}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Zak\u0142adamy, \u017ce jest ma\u0142e odchylenie zmiennej k od punktu k{{Sub|0}}, to w wyra\u017ceniu {{LinkWz\u00f3r|1.49}} wyrazy kwadratowe i wy\u017csze pomijamy, wtedy w wyra\u017ceniu {{LinkWz\u00f3r|1.48}}, w kt\u00f3rym b\u0119dziemy zak\u0142ada\u0107, \u017ce C(k) s\u0142abo zale\u017cy od k w tym\u017ce rozwa\u017canym przedziale zmienno\u015bci k, zatem mo\u017cemy przej\u0105\u0107 w przybli\u017ceniu, \u017ce zachodzi C(k)&asymp;C(k{{Sub|0}}), wtedy je piszemy:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\psi(x,t)=e^{i(\\omega_0 t-k_0x)}\\int^{k_0+\\Delta k}_{k_0-\\Delta k}C(k)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">e^{i\\left[\\left({{d\\omega}\\over{dk}}\\right)_{k=k_0}(k-k_0)t-(k-k_0) x\\right]}dk\\simeq C(k_0)e^{i(\\omega_0t-k_0x)}\\int^{\\Delta k}_{-\\Delta k}e^{i\\left[\\left({{d\\omega}\\over{dk}}\\right)_{k=k_0}t-x\\right]\\xi}d\\xi=\\;</MATH>{{Br}}<MATH></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">=2C(k_0)\\Delta ke^{i(\\omega_0t-k_0x)}{{e^{i\\left[\\left({{d\\omega}\\over{dk}}\\right)_{k=k_0}t-x\\right]\\Delta k}-e^{-i\\left[\\left({{d\\omega}\\over{dk}}\\right)_{k=k_0}t-x\\right]\\Delta k}}\\over{2i\\left[\\Delta k\\left({{d\\omega}\\over{dk}}\\right)_{k=k_0}t-x\\right</ins>]<ins class=\"diffchange diffchange-inline\">}}=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">2C(k_0)\\Delta ke^{i(\\omega_0t-k_0x)}{{\\sin\\left[\\left({{d\\omega}\\over{dk}}\\right)_{k=k_0}t-x\\right]\\Delta k}\\over{\\left[\\left({{d\\omega}\\over{dk}}\\right)_{k=k_0}t-x\\right]\\Delta k}}\\;</math>}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Na podstawie oblicze\u0144 wyst\u0119puj\u0105cych w ostatnich dokonanych operacjach na formu\u0142ach, otrzymujemy ostateczny wz\u00f3r dla paczki falowej, kt\u00f3ra jest superpozycj\u0105 fal prostych o bardzo ma\u0142ym zakresie zmienno\u015bci warto\u015bci sta\u0142ej\u00a0 falowej k wok\u00f3\u0142 liczby falowej k{{Sub|0}}, wtedy:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\psi(x,t)=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">2C(k_0)\\Delta ke^{i(\\omega_0t-k_0x)}\\left\\{{{\\sin\\left[\\left({{d\\omega}\\over{dk}}\\right)_{k=k_0}t-x\\right]\\Delta k}\\over{\\left</ins>[<ins class=\"diffchange diffchange-inline\">\\left({{d\\omega}\\over{dk}}\\right)_{k=k_0}t-x\\right]\\Delta k}}\\right\\}\\;</MATH>|1.50|Obramuj}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wyra\u017cenie {{LinkWz\u00f3r|1.50}} przedstawia pewn\u0105 paczk\u0119, kt\u00f3ra jest superpozycj\u0105 r\u00f3\u017cnych fal o ma\u0142ym zakresie zmienno\u015bci liczby falowej wok\u00f3\u0142 punktu k{{Sub|0}}, kt\u00f3rego wykres w czasie jest przedstawiony obok.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Pr\u0119dko\u015b\u0107 grupowa paczki falowej i pr\u0119dko\u015b\u0107 cz\u0105stki oraz dow\u00f3d wzoru na fale de Broglie'a i energi\u0119 kwantu Plancka ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Policzmy pochodn\u0105 energii cz\u0105stki wzgl\u0119dem jej p\u0119du uog\u00f3lnionego w mechanice klasycznej Newtona:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MaTH>{{\\partial E}\\over{\\partial\\vec p_i}}={{\\partial}\\over{\\partial\\vec p_i}}\\left(\\sum_i{{(\\vec p_i-q_i\\vec A_i)^2}\\over{2m_i}}+\\sum_{ij,i<j}q_i\\varphi_{ij}+\\sum_iq_i\\varphi_i+\\sum_{ij,i<j}U_{ij}+\\sum_{i}U_i\\right)={{(\\vec p_i-q_i\\vec A_i)}\\over{m_i}}={{\\vec p_{kli}}\\over{m_i}}={{m_i\\vec v_i}\\over{m_i}}=\\vec v_i\\;</MaTH>|1.51}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Widzimy, \u017ce wed\u0142ug mechaniki klasycznej pochodna energii cz\u0105stki wzgl\u0119dem p\u0119du uog\u00f3lnionego jest r\u00f3wna pr\u0119dko\u015bci cz\u0105stki klasycznej, co nie powinno by\u0107 zaskoczeniem.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Nast\u0119pnie rozpatrzmy cz\u0105stk\u0119 relatywistyczn\u0105 o energii relatywistycznej wyra\u017conej przy pomocy p\u0119du uog\u00f3lnionego cz\u0105stki i jej masy spoczynkowej m{{Sub|0}}, kt\u00f3ra mo\u017ce by\u0107 r\u00f3wna zero, znaj\u0105c definicj\u0119 p\u0119du klasycznego:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<mATH>{{\\partial E}\\over{\\partial\\vec p_i}}={{\\partial}\\over{\\partial\\vec p_i}}\\left(\\sum_{i}\\sqrt{c^2(\\vec p_i-q_i\\vec A_i)^2+m_{0i}^2c^4}+\\sum_{ij,i<j}q_i\\varphi_{ij}+\\sum_iq_i\\varphi_i+\\sum_{ij,i<j}U_{ij}+\\sum_{i}U_i\\right)={{2c^2(\\vec p_i-q_i\\vec A_i)}\\over{2\\sqrt{c^2(\\vec p_i-q_i\\vec A_i)^2+m_{0i}^2c^4}}}=\\;</MATH>{{Br}}<MATH>=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{c^2\\vec p_{kli}}\\over{\\sqrt{c^2p^2_{kli}+m_{0i}^2c^4}}}={{c^2\\vec p_{kli}}\\over{E_i}}={{c^2\\vec p_{kli}}\\over{m_ic^2}}={{\\vec p_{kli}}\\over{m_i}}={{m_i\\vec v_i}\\over{m_i}}=\\vec v_i\\;</MATH>|1.52}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Widzimy, \u017ce wed\u0142ug szczeg\u00f3lnej teorii wzgl\u0119dno\u015bci, \u017ce pochodna energii cz\u0105stki wzgl\u0119dem p\u0119du uog\u00f3lnionego jest r\u00f3wna pr\u0119dko\u015bci cz\u0105stki klasycznej.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Z szczeg\u00f3lnej teorii wzgl\u0119dno\u015bci i teorii klasycznej Newtona wiadomo, \u017ce pr\u0119dko\u015b\u0107 cz\u0105stki jest r\u00f3wna pochodnej energii cz\u0105stki wzgl\u0119dem p\u0119du uog\u00f3lnionego, bo ({{LinkWz\u00f3r|1.51}} i {{LinkWz\u00f3r|1.52}}), a je\u017celi potraktowa\u0107 cz\u0105stki jako fal\u0119, to wtedy ta pr\u0119dko\u015b\u0107 powinna by\u0107 r\u00f3wna pr\u0119dko\u015bci grupowej fali, kt\u00f3ra jest r\u00f3wna pochodnej cz\u0119stotliwo\u015bci k\u0105towej wzgl\u0119dem liczby falowej, a je\u017celi przyr\u00f3wna\u0107 j\u0105 do pr\u0119dko\u015bci cz\u0105stki, to dochodzimy do wniosku, \u017ce spe\u0142niony jest wz\u00f3r na energi\u0119 kwantu {{LinkWz\u00f3r|1.6}} i wz\u00f3r na fale de Broglie'a {{LinkWz\u00f3r|1.17}}, bo</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MaTh>\\left(\\vec v_{gi}={{\\partial\\omega}\\over{\\partial\\vec k_i}}={{\\partial \\hbar\\omega}\\over{\\partial \\hbar \\vec k_i}}\\wedge\\vec v_i={{\\partial E}\\over{\\partial\\vec p_i}}\\wedge v_{gi}=v_i\\right)\\Rightarrow E=\\hbar\\omega\\wedge \\vec p_i=\\hbar \\vec k_i\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.53}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Pr\u0119dko\u015b\u0107 grupowa fali o wektorze falowym {{Formu\u0142a|<MATH>\\vec k_i\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>}} przedstawia pr\u0119dko\u015b\u0107 energii fali, a w przypadku korpuskularnym cz\u0105stka porusza si\u0119 z pewn\u0105 pr\u0119dko\u015bci\u0105, kt\u00f3ra okre\u015bla pr\u0119dko\u015b\u0107 poruszania si\u0119 energii (korpusku\u0142u), a wi\u0119c obie te pr\u0119dko\u015bci s\u0105 sobie r\u00f3wne</ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Warunek Bragg\u00f3w, a do\u015bwiadczenie, fale materii ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Rysunek|Bragg.svg|i5|Rysunek pozwalaj\u0105cy wyprowadzenie warunku Bragg\u00f3w</ins>.<ins class=\"diffchange diffchange-inline\">}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">R\u00f3\u017cnica dr\u00f3g optycznych mi\u0119dzy g\u00f3rnym a dolnym promieniem jest zale\u017cna od odleg\u0142o\u015bci pomi\u0119dzy warstwami mi\u0119dzy dwoma p\u0142aszczyznami atom\u00f3w i od k\u0105ta {{Formu\u0142a|<MATH>\\theta\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>}} do p\u0142aszczyzny z atomami, pod kt\u00f3r\u0105 pada fala elektromagnetyczna X:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<math>{{\\delta}\\over{d}}=\\sin\\theta\\Rightarrow \\Delta s=2\\delta=2d\\sin\\theta\\;<</ins>/<ins class=\"diffchange diffchange-inline\">MATH>|1.54}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">R\u00f3\u017cnica faz mi\u0119dzy promieniem drugim a pierwszym, bo promie\u0144 drugi przebywa d\u0142u\u017csz\u0105 drog\u0119 optyczn\u0105 ni\u017c pierwszy, jest r\u00f3wna</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>{{\\Delta \\phi}\\over{\\Delta s}}={{2\\pi}\\over{\\lambda}}\\Rightarrow\\Delta \\phi={{2\\pi}\\over{\\lambda}}\\Delta s={{2\\pi}\\over{\\lambda}}2d\\sin\\theta={{4d\\pi\\sin\\theta}\\over{\\lambda}}\\;<</ins>/<ins class=\"diffchange diffchange-inline\">math>|1.55}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">R\u00f3wnanie fal materii pierwszej i drugiej uwzgl\u0119dniaj\u0105 przesuni\u0119cie fazowe tych\u017ce fal przed ugi\u0119ciem fal, a tak\u017ce przesuni\u0119cie drugiej fali wzgl\u0119dem pierwszej po ugi\u0119ciu tego\u017c obiektu,\u00a0 zapisujemy je\u00a0 wedle sposobu</ins>:</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{ElastycznyWiersz|1={{CentrujWz\u00f3r|<MATH>\\psi_1=Ae^{i(\\omega t-ks+\\delta_0)}\\;</MAth>|1.56}}|2={{CentrujWz\u00f3r|<MATH>\\psi_2=Ae^{i(\\omega t-ks+\\delta_0-\\Delta\\phi)}\\;</MaTH>|1.57}}}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Wed\u0142ug zasady Huygensa musimy doda\u0107 fale {{LinkWz\u00f3r|1.56}} do {{LinkWz\u00f3r|1.57}}, kt\u00f3re ulegaj\u0105 superpozycji w bardzo du\u017cej odleg\u0142o\u015bci od kryszta\u0142u, co mo\u017cna zapisa\u0107 w przybli\u017ceniu, \u017ce te dwie fale poruszaj\u0105 </ins>si\u0119 <ins class=\"diffchange diffchange-inline\">po liniach prostych i r\u00f3wnoleg\u0142ych do siebie przed doj\u015bciem do kryszta\u0142u i po jego wyj\u015bciu:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>\\psi=\\psi_1+\\psi_2=A\\left(e^{i\\left(\\omega t-ks+\\delta_0\\right)}+e^{i\\left(\\omega t-ks+\\delta_0-\\Delta\\phi\\right)}\\right)=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Ae^{i\\left(\\omega t-ks+\\delta_0-{{\\Delta\\phi}\\over{2}}\\right)}\\left(e^{i\\left({{\\Delta\\phi}\\over{2}}\\right)}+e^{i\\left(-{{\\Delta\\phi}\\over{2}}\\right)}\\right)=\\;</MATH>{{Br}}<MATH>=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">2Ae^{i\\left(\\omega t-ks+\\delta_0-{{\\Delta\\phi}\\over{2}}\\right)}\\cos\\left({{\\Delta\\phi}\\over{2}}\\right)=</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">2Ae^{i\\left(\\omega t-ks+\\delta_0-{{\\Delta\\phi}\\over{2}}\\right)}\\cos {{2d\\pi\\sin\\theta}\\over{\\lambda}}\\;</MATH>|1.58}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">W wyra\u017ceniu {{LinkWz\u00f3r|1.58}} kosinus przyjmuje warto\u015b\u0107 jeden lub minus jeden, gdy w wyra\u017ceniu pod kosinusem jest n&pi;, to modu\u0142 wspomnianego wyra\u017cenia przyjmuje warto\u015b\u0107 maksymaln\u0105, gdy zachodzi zale\u017cno\u015b\u0107:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MATH>n\\pi={{2d\\pi\\sin\\theta}\\over{\\lambda}}\\Rightarrow n\\lambda=2d\\sin\\theta\\;</MaTH>|1.59}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Na podstawie ostatniego wyra\u017cenia w {{LinkWz\u00f3r|1.59}} dochodzimy do wniosku, \u017ce r\u00f3wnanie Bragg\u00f3w jest zapisywane w postaci:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CentrujWz\u00f3r|<MaTH>2d\\sin\\theta=n\\lambda\\;</MATH>|1.60|Obramuj}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Jest ona zale\u017cna od odleg\u0142o\u015bci mi\u0119dzy dwoma p\u0142aszczyznami d, od d\u0142ugo\u015bci fali &lambda; i od k\u0105ta &theta; padania promieniowania elektromagnetycznego X, dzi\u0119ki kt\u00f3remu fala elektromagnetyczne ulegnie wzmocnieniu na ekranie</ins>, na <ins class=\"diffchange diffchange-inline\">kt\u00f3rej badamy wzmocnienia fal elektromagnetycznych dla jakiego\u015b parametru n, je\u015bli w og\u00f3le istnieje.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><noinclude>{{SkomplikowanaStronaKoniec}}</noinclude></ins></div></td></tr>\n"
}
}