{"id":109,"date":"2024-10-13T11:32:19","date_gmt":"2024-10-13T08:32:19","guid":{"rendered":"https:\/\/blog.metu.edu.tr\/caglart\/?p=109"},"modified":"2024-10-13T11:51:06","modified_gmt":"2024-10-13T08:51:06","slug":"zenodan-einsteina-kuvantum","status":"publish","type":"post","link":"https:\/\/blog.metu.edu.tr\/caglart\/2024\/10\/13\/zenodan-einsteina-kuvantum\/","title":{"rendered":"ZENO’dan E\u0130NSTE\u0130N’a KUVANTUM – a –"},"content":{"rendered":"

F\u0130Z\u0130K -4 \/ a-<\/p>\n

\u0130\u00e7erik: Zeno, Maxwell, Boltzmann, Einstein, f\u0131r\u0131n, karacisim ve az\u0131c\u0131k matematik<\/p>\n

Zeno\u2019dan Einstein\u2019a gelen kuvantum yolu James Clerck Maxwell ile Ludwig Eduard Boltzmann\u2019a u\u011frar. Kapal\u0131, yani d\u0131\u015f\u0131 ile etkile\u015fimi s\u0131f\u0131r olan bir sistem i\u00e7indeki toplam olarak N_tane noktasal par\u00e7ac\u0131ktan, enerjisi Ei olan \u2018microstate\u2019 kapsam\u0131nda Ni tane noktasal par\u00e7ac\u0131k varsa, bu par\u00e7ac\u0131klar\u0131n da\u011f\u0131l\u0131m\u0131 alttaki Maxwell-Boltzman form\u00fcl\u00fc ile tan\u0131mlan\u0131r. (*) Bu form\u00fcl\u00fcn paydas\u0131ndaki J indisi par\u00e7ac\u0131klar\u0131n her birini sayar. Dikkate de\u011fer, N say\u0131s\u0131 bilinen herhangi bir do\u011fal say\u0131d\u0131r; a\u00e7\u0131k\u00e7as\u0131, sonsuz (adet tanecik olamayaca\u011f\u0131 olmad\u0131\u011f\u0131 i\u00e7in) de\u011fildir.<\/p>\n

\"\"<\/a> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 E\u015fitlik 1<\/p>\n

Gelgelelim, pek \u00e7ok olguyu a\u00e7\u0131klamakta kullan\u0131lan bu temel form\u00fcl\u00fcn \u00e7ok s\u0131radan, her g\u00fcn herkesin rastlad\u0131\u011f\u0131 rastlayabilece\u011fi bir olguyu a\u00e7\u0131klamakta yetersiz kald\u0131\u011f\u0131 \u00e7ok ge\u00e7meden anla\u015f\u0131ld\u0131.
\nHemen her evde bir elektrikli f\u0131r\u0131n vard\u0131r. Hemen her evin yak\u0131nlar\u0131nda da bir simit veya ekmek f\u0131r\u0131n\u0131 vard\u0131r. Bu f\u0131r\u0131nlar\u0131n ortak \u00f6zelliklerinden biri \u015fudur: Her f\u0131r\u0131n\u0131n i\u00e7indeki s\u0131cakl\u0131k d\u00f6rt y\u00fcz Celsius derece kadar olmas\u0131na kar\u015f\u0131n, f\u0131r\u0131n\u0131n g\u00f6zleme penceresi veya kapa\u011f\u0131 a\u00e7\u0131ld\u0131\u011f\u0131nda bakan\u0131 hemen hi\u00e7 rahats\u0131z etmez. \u00c7\u00fcnk\u00fc o mertebede \u0131s\u0131 \u00e7\u0131kmaz o aral\u0131ktan. Bu t\u00fcr, yani sahip oldu\u011fu enerjiyi yay\u0131mlamay\u0131p yahut en fazlas\u0131 \u00e7ok k\u00fc\u00e7\u00fck bir orandakini yay\u0131mlayan nesnelere KARAC\u0130S\u0130M denir.
\nBu karacisimlerin bir ortak \u00f6zelli\u011fi de \u015fudur: yayd\u0131klar\u0131 \u0131\u015f\u0131man\u0131n i\u00e7inde her frekanstan (her dalga boyundan dense de do\u011frudur) \u0131\u015f\u0131n vard\u0131r.
\n\u015e\u00f6yle d\u00fc\u015f\u00fcnelim. Diyelim ki, bir mikrodalga f\u0131r\u0131n\u0131 sabit bir frekanstan \u0131s\u0131tma yap\u0131yor. Yani, f\u0131r\u0131na gelen elektrik, \u0131s\u0131t\u0131c\u0131y\u0131 sabit bir frekansta titre\u015ftirip bu frekansta \u0131\u015f\u0131ma yay\u0131lmas\u0131na sebep oluyor. Bu mikrodalga f\u0131r\u0131n\u0131n kapa\u011f\u0131 a\u00e7\u0131l\u0131p \u00e7\u0131kan (s\u0131zan dense de olur) \u0131\u015f\u0131man\u0131n frekans da\u011f\u0131l\u0131m\u0131 (izgesi, \u2018spectrum\u2019) incelendi\u011finde \u015fu g\u00f6r\u00fcl\u00fcr: F\u0131r\u0131n \u0131s\u0131t\u0131c\u0131s\u0131n\u0131n titre\u015fim frekans\u0131n\u0131 da i\u00e7eren vel\u00e2kin t\u00fcm frekanslarda ama ye\u011finli\u011fi (\u2018amplitude\u2019) frekansa g\u00f6re de\u011fi\u015fim g\u00f6steren bir da\u011f\u0131l\u0131m g\u00f6zlenir.
\nOysa, J. C. Maxwell\u2019in b\u00fcy\u00fck katk\u0131lar\u0131yla geli\u015ftirilen (Klasik) Elektrom\u0131knat\u0131sl\u0131k Kuram\u0131na g\u00f6re tek frekansta (\u0131s\u0131t\u0131c\u0131 frekans\u0131) ve bu frekans de\u011fi\u015ftirilse bile bu frekans\u0131n karesi ile orant\u0131l\u0131 enerji yay\u0131l\u0131m\u0131 g\u00f6zlenmesi gerekiyordu. Dahas\u0131, bu enerji yay\u0131l\u0131m\u0131, mor \u00f6tesi frekanslar\u0131 gibi, X-\u0131\u015f\u0131n\u0131 ve hele gamma-\u0131\u015f\u0131n\u0131 frekanslar\u0131 yan\u0131nda \u00f6nemsenmeyecek denli k\u00fc\u00e7\u00fck olan frekans de\u011ferlerinde bile, D\u00fcnya\u2019da \u00fcretilemeyecek b\u00fcy\u00fck miktarlarda enerjiye denk gelmekteydi. \u0130\u015fte bu \u00e7eli\u015fkiye de mor \u00f6tesi y\u0131k\u0131m (\u2018ultraviolet catastrophe\u2019) denmi\u015fti. Y\u0131k\u0131m (\u2018catastrophe\u2019) s\u00f6zc\u00fc\u011f\u00fc ile \u015fu ima ediliyordu sanki: Maxwell kuram\u0131 yanl\u0131\u015f m\u0131 veya hi\u00e7 de\u011filse eksik miydi acaba?
\nBu soru uzun y\u0131llar belini b\u00fckt\u00fc fizik\u00e7ilerin.
\n\u0130lkin, t\u00fcm karacisimlerin enerji-frekans izgelerinin benze\u015f oldu\u011fu farkedildi. \u00c7ok d\u00fc\u015f\u00fck frekanslarda, frekans\u0131n karesi ile (parabolik) b\u00fcy\u00fcyen enerji b\u00f6yle devam etmiyor ama bir zirve yap\u0131p b\u00fcy\u00fck frekanslarda giderek azal\u0131yordu.
\nGarip! B\u00fcy\u00fck frekanslarda \u0131\u015f\u0131ma enerjisi frekans ile artm\u0131yor, tersine azal\u0131yordu.
\n\u0130kincileyin, bu g\u00f6zlemsel karacisim \u0131\u015f\u0131ma izgesinin matematiksel form\u00fcl\u00fc benze\u015ftirme (\u2018fitting\u2019) y\u00f6ntemiyle elde edildi. Frekansa ba\u011fl\u0131 \u0131\u015f\u0131ma ye\u011finli\u011fini veren form\u00fcl \u015fu \u015fekildeydi:<\/p>\n

\"\"\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/a> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 E\u015fitlik 2<\/p>\n

E\u015fitlik 2\u2019de \uf06e simgesi frekans\u0131, Tsimgesi s\u0131cakl\u0131\u011f\u0131, h simgesi Planck sabitini, k simgesi ise, Boltzmann sabitini g\u00f6stermektedir. \u0130lgili grafik de \u015fu ba\u011flant\u0131da incelenebilir: https:\/\/www.google.com\/search?q=karacisim<\/p>\n

B\u00f6ylelikle \u2018mor\u00f6tesi y\u0131k\u0131m\u2019 sorunsal\u0131 E\u015fitlik 2\u2019deki form\u00fcl\u00fcn nas\u0131l elde edilece\u011fi sorunsal\u0131na indirgenivermi\u015fti. Pek \u00e7ok fizik\u00e7i ve pek \u00e7ok matematik\u00e7i pek \u00e7ok \u00e7e\u015fitli simgelerin pek \u00e7ok kombinasyonunu denedi o form\u00fcl\u00fc elde edivermek i\u00e7in.
\n\u00d6yle ya, ucuz malzemelerden alt\u0131n elde etmeye \u00e7al\u0131\u015fm\u0131\u015f olanlar da ayn\u0131 y\u00f6ntemi izlememi\u015f miydi? Isaac Newton da o \u00fcnl\u00fc k\u00fctle\u00e7ekim form\u00fcl\u00fcn\u00fc (Mm\/r2) elde etmek i\u00e7in \u00f6nce bu \u00e7ekimin sadece ve sadece cisimlerin k\u00fctlelerine ve aradaki uzakl\u0131\u011fa ba\u011fl\u0131 olaca\u011f\u0131n\u0131 varsay\u0131p bu \u00fc\u00e7 simgenin yerleriyle oynayarak Kepler Yasalar\u0131\u2019n\u0131n \u00fc\u00e7\u00fcn\u00fc birden veren o form\u00fcl\u00fc deneyerek bulmam\u0131\u015f m\u0131yd\u0131? Bkz., Isaac Newton: The Principia, Mathematical principles of natural philosophy, a new translation” by I Bernard Cohen and Anne Whitman, preceded by “A Guide to Newton’s Principia” by I Bernard Cohen, University of California Press, 1999.<\/p>\n

Gelgelelim ne denense olmuyordu taa ki, rivayete g\u00f6re, Boltzmann\u2019\u0131n son derece \u00f6nemli bir fikri Planck\u2019\u0131n kula\u011f\u0131na f\u0131s\u0131ldayana dek. O m\u00fcthi\u015f b\u00fcy\u00fck, \u00e7a\u011f de\u011fi\u015ftirici fikir \u015fuydu: Enerji s\u00fcrekli olmay\u0131p paket\u00e7ikler halindedir.
\nBa\u015fka bir deyi\u015fle, s\u0131cak cisimlerin yayd\u0131\u011f\u0131 \u0131\u015f\u0131madaki fotonlar\u0131n enerjisi s\u00fcrekili olmay\u0131p basamakl\u0131 yani paket\u00e7ikli (\u2018quantised\u2019) haldedir.
\nBu fikrin ortaya \u00e7\u0131k\u0131\u015f\u0131ndan sonras\u0131 kolayd\u0131.
\n\u0130lkin, serbest bir fotonun enerjisi E=Planck sabiti \u00e7arp\u0131 frekans olarak tan\u0131mland\u0131. Hemencecik ard\u0131ndan da, T s\u0131cakl\u0131\u011f\u0131ndaki kapal\u0131 bir (yani, yal\u0131t\u0131lm\u0131\u015f ve d\u0131\u015far\u0131 enerji ve par\u00e7ac\u0131k s\u0131zd\u0131rmayan veya bu s\u0131z\u0131nt\u0131n\u0131n ihmal edilebilecek denli k\u00fc\u00e7\u00fck oldu\u011fu) sistemdeki fotonlar\u0131n paketlenece\u011fi ve her birinin enerjisinin, n bir do\u011fal say\u0131y\u0131 g\u00f6stermek \u00fczere En=nxhxfrekans bi\u00e7iminde yaz\u0131labilece\u011fi varsay\u0131ld\u0131.
\nAcaba, bir f\u0131r\u0131n\u0131n i\u00e7indeki fotonlar\u0131n adedi say\u0131labilir mi? Say\u0131lamaz elbette, hani \u2018sonsuz\u2019 denir ya o kabilden. (**)
\nPeki ya \u2018sonsuz\u2019 foton i\u00e7eren f\u0131r\u0131nda yeni fotonlar \u00fcretilebiliyorsa? \u0130\u015fte bu soruyu sormakta iseniz, Hilbert\u2019in Sonsuz Oteli\u2019ndesiniz demektir. \u201c\u2014Ho\u015f geldiniz!\u201d Bir koltu\u011fa veya kanepeye kurulup veya yata\u011f\u0131n\u0131za uzan\u0131p 1, 2, 3, \u2026 SONSUZ ba\u015fl\u0131kl\u0131 yaz\u0131y\u0131 okuyabilirsiniz. Bkz., https:\/\/blog.metu.edu.tr\/caglart\/?s=Hilbert
\nHerhangi bir enerji de\u011ferine sahip foton say\u0131s\u0131n\u0131n belirsizli\u011fi nedeniyle bu yaz\u0131n\u0131n tepesindeki E\u015fitlik 1\u2019in fotonlar i\u00e7in revize edilmesi gerekmekteydi. Bunu da, devam\u0131 olan Zeno-Planck-Einstein ili\u015fkisine de bir sonraki Fizik – 4 \/ b- Zeno\u2019dan Einstein\u2019a Kuvantum yaz\u0131m\u0131zda de\u011finelim.<\/p>\n

(*) https:\/\/en.m.wikipedia.org\/wiki\/Maxwell%E2%80%93Boltzmann_distribution
\nhttps:\/\/www.gtu.edu.tr\/Files\/UserFiles\/90\/MF1-KARA_CSM_IIMASI.pdf
\n(**) \u2018Say\u0131labilir sonsuzluk\u2019 (\u2018countable infinity\u2019) ile \u2018say\u0131lamaz sonsuzluk\u2019 (\u2018uncountable infinity\u2019) konusunu tart\u0131\u015fmay\u0131 bir ba\u015fka yaz\u0131ya b\u0131rak\u0131yoruz.<\/p>\n","protected":false},"excerpt":{"rendered":"

F\u0130Z\u0130K -4 \/ a- \u0130\u00e7erik: Zeno, Maxwell, Boltzmann, Einstein, f\u0131r\u0131n, karacisim ve az\u0131c\u0131k matematik Zeno\u2019dan Einstein\u2019a gelen kuvantum yolu James Clerck Maxwell ile Ludwig Eduard Boltzmann\u2019a u\u011frar. Kapal\u0131, yani d\u0131\u015f\u0131 ile etkile\u015fimi s\u0131f\u0131r olan bir sistem i\u00e7indeki toplam olarak N_tane noktasal par\u00e7ac\u0131ktan, enerjisi Ei olan \u2018microstate\u2019 kapsam\u0131nda Ni tane noktasal par\u00e7ac\u0131k varsa, bu par\u00e7ac\u0131klar\u0131n da\u011f\u0131l\u0131m\u0131 […]<\/p>\n","protected":false},"author":1425,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[1],"tags":[],"class_list":["post-109","post","type-post","status-publish","format-standard","hentry","category-genel"],"_links":{"self":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts\/109","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/users\/1425"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/comments?post=109"}],"version-history":[{"count":0,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/posts\/109\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/media?parent=109"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/categories?post=109"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/caglart\/wp-json\/wp\/v2\/tags?post=109"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}