ࡱ>  sQlZ5@,bjbj22 ӻXXr4444444H^^^8(_|`Hha fn|n|n|nXpXpXp`Rb4Xp6p"XpXpXp44|n|nPT(((Xp* 4|n4|n(Xp((6:,44|na Qu^} % T/RHH44444,XpXp(XpXpXpXpXpHH$Yl\HHl\ SADR}AJ 1 Uvod..............& .......................................................................................... 2 2.0 Kristalna ionska reaetka................................................................................. 3 2.1 Titranje kristalne reaetke................................................................................ 4 2.2 Akusti ko titranje........................................................................................... 7 2.3 Opti ko titranje..........................................................................& .................7 3 Harmoni ki oscilator u kvantnoj fizici...........................................................8 4 Fononi.............................................................................................................9 5 Varijacijska metoda......................................................................................10 6 Elektroni u polarnim kristalima ...................................................................14 7 Problem polarona u ionskom (polarnom) kristalu .......................................16 8 Zaklju ak.......................................................................................................230 Glava 1 Uvod Rjeaavanje nekih prirodnih problema vrlo je komplicirano ili nemogue rijeaiti egzaktno. Zato se u fizici rade razne aproksimacije kako bi se problem rijeaio barem pribli~no do neke zadovoljavajue to nosti. Takav je npr. problem gibanja elektrona kroz ionski kristal. Tu elektron meudjeluje s titranjima reaetke koja se opisuju kao fononska pobuenja. Rije  je o kvantnomehani kom problemu mnoatva estica koje meudjeluju i nije ga mogue rijeaiti egzaktno. U ovom radu emo taj problem rjeaavati aproksimativno koristei metodu srednjeg polja kombiniranu s varijacijskom metodom. Na taj na in emo pronai elektronsku valnu funkciju i pripadnu deformaciju reaetke koja nastaje kao rezultat elektron-fononskog meudjelovanja. Dakle postavlja se pitanje, koliko e ovo naae aproksimativno rjeaenje biti dobro i koliko e se ono razlikovati od ostalih rjeaenja istog problema? U radu emo se prvo upoznati (druga glava) s ionskom kristalnom reaetkom. Poato se gibanje iona u reaetci mo~e prikazati kao titranja harmoni kog oscilatora, u treoj emo se glavi ovog rada vrlo kratko osvrnuti na kvantnomehani ku energiju harmonijskog oscilatora. U etvrtoj glavi emo ukratko uvesti pojam fonona kao kvanta pubuenja titranja reaetke. Varijacijsku metodu u kvantnoj mehanici prikazat emo u petoj glavi. Kvantnomehani ki problem polarona u reaetki postavit emo u aestoj glavi. A naae varijacijsko rjeaenje bit e dano u sedmoj glavi. Odgovor na pitanje iz po etka uvoda morat emo potra~iti u zaklju ku tj. u osmoj glavi. Glava 22 2.1 Kristalna ionska reaetka Kristali se odlikuju visokim stupnjem regularnosti svoje strukture. Idealni kristal je prostorna tvorevina koja nastaje beskona nim ponavljanjem jednakih strukturnih jedinica. U svakoj elementarnoj strukturnoj jedinici kristala nalazi se jedan atom ili viae njih. Pri definiciji idealnog kristala pretpostavljamo da atomi miruju u svojim ravnote~nim polo~ajima. Naravno u prirodi zbog defekata (graaka u slaganju kristalne strukture), a i zbog samog titranja atoma nemamo idealne kristale. Defekti mijenjaju svojstva kristala ( vrstou, toplinsku i elektri nu vodljivost itd.). Nas e u daljnjem razmatranju zanimati ionski kristali. `to su ionski kristali? Sama rije  ka~e da su to kristali u kojima su atomi povezani ionskom vezom. Ionska veza je karakteristi na za kristale sastavljene od pozitivnih i negativnih iona. U atomima koji se udru~uju u ionsku vezu, vanjske energijske podljuske su samo djelomi no popunjene. Tipi ni ionski kristali su alkalni halogenidi. U njima atom alkalijskog metala gubi, a atom halogenog elementa dobiva elektron. Energija kohezije za ionske kristale je 5-10 eV, zna  i imamo kristalni sustav u kojem ioni razli itog naboja meudjeluju kulonskim silama.  Slika 1. Skica kristalne reaetke ionskog kristala. 2.2 Titranje kristalne reaetke Opis titranja viaeatomnih reaetki je openito slo~eni problem koji zahtijeva rjeaavanje sustava vezanih diferencijalnih jednad~bi. Ovdje e se detaljnije razmotriti pojednostavljeni model beskona ne jednodimenzionalne reaetke s dva atoma u elementarnoj eliji. Pretpostavljamo da je linearna reaetka izgraena od dvije vrste atoma. Masu te~ih atoma ozna it emo sa M1, a lakaih sa M2: M1>M2 Konstanta reaetke je EMBED Equation.3, udaljenost izmeu susjednih razli itih atoma je EMBED Equation.3. Konstanta elasti nosti (druga derivacija potencijalne energije) jednaka je .  Slika 2 Titranje atoma u dvoatomnoj linearnoj reaetci Polo~aje atoma prikazujemo relativnim pomacima: EMBED Equation.3 n = indeks atoma EMBED Equation.3odstupanje od ravnote~nog polo~aja atoma Parni atom ima masu M1, a neparni M2. Ukupna energija je: EMBED Equation.3. Jednad~be gibanja glase: EMBED Equation.3 EMBED Equation.3. Njihova rjeaenja tra~imo u obliku ravnih valova: EMBED Equation.3 EMBED Equation.3. Amplitude ravnih valova razli ite su za razli ite vrste atoma (amplitude A i B). Ravni val uvratavamo u sustav jednad~bi gibanja, dobivamo homogeni sustav dvije jednad~be za nepoznate amplitude, A i B. EMBED Equation.3 EMBED Equation.3. Amplitude A i B su razli ite od nule samo ako je determinanta sustava jednaka nuli (homogeni sustav jednad~bi) EMBED Equation.3, iz ovog uvjeta dobivamo jednad~bu koja povezuje frekvenciju i valni broj: EMBED Equation.3. Dobivamo da jednad~ba ima dva rjeaenja : EMBED Equation.3, EMBED Equation.3 odgovara tzv. opti kim titranjima, a EMBED Equation.3 odgovara akusti kim titranjima. Ovisnost frekvencija o valnom broju za akusti ka i opti ka titranja prikazana je na slici 3. Translacijom valnog broja za EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 frekvencija titranja ostaje nepromijenjena: EMBED Equation.3 Frekvencija je periodi na funkcija valnog broja s periodom EMBED Equation.3 Sve mogue vrijednosti frekvencije dobivamo uzimajui valni broj iz intervala airine EMBED Equation.3Valni broj redovito ograni avamo na interval od EMBED Equation.3 do EMBED Equation.3: EMBED Equation.3 Podru je definirano gornjom relacijom odreuje prvu Brillouinovu zonu u jednoj dimenziji. Poveanjem valnog broja poveava se i akusti na frekvencija. Ona postaje maksimalna na rubu prve Brillouinove zone: EMBED Equation.3 Ovisnost opti ke grane frekvencije o valnom broju bitno je razli ita nego za akusti nu granu. U ishodiatu prve Brillouinove zone opti ka frekvencija odreena je relacijom EMBED Equation.3 Frekvencija EMBED Equation.3 je maksimalna opti ka frekvencija. `to je valni broj bli~i rubu zone, to je ni~a frekvencija opti kog titranja. Grani nim to kama prve Brillouinove zone pridru~ena je minimalna opti ka frekvencija EMBED Equation.3  Slika 3. Grafi ki prikaz disperzivne relacije 2.2.1 Akusti ko titranje EMBED Equation.3Slika 4. Akusti ko titranje atoma u dvoatomnoj linearnoj reaetci Pomaci atoma, A i B, su u fazi , a brzina airenja poremeaja: EMBED Equation.3 Brzina ovisi o ukupnoj masi atoma unutar jedne elije. 2.2.2 Opti ko titranje  Slika 5. Opti ko titranje atoma u dvoatomnoj kristalnoj reaetci Frekvencija opti kog titranja slabo ovisi o valnom vektoru. Pomaci atoma, A i B, su u antifazi i obrnuto su proporcionalni masama atoma, tako da je te~iate jedini ne elije pribli~no nepomi no: EMBED Equation.3 Broj razli itih vrsta titranja (akusti ko, opti ka) jednak je broju atoma po jedini noj eliji. Samo jedno od njih (akusti ko) ima frekvenciju jednaku nuli kada je valni vektor jednak nuli. Ostala titranja su opti ka. U realnim kristalima u kojima su pomaci 3D vektori, postoje 3 akusti na titranja, te 3(n-1) opti ko, gdje je n broj atoma po jedini noj eliji. Glava 3 Harmoni ki oscilator u kvantnoj fizici U klasi noj fizici harmoni ki oscilator mo~e imati bilo koji iznos energije i ona je srazmjerna kvadratu amplitude titranja. U kvantnoj fizici energija estice koja se giba u ograni enom djelu prostora (kao HO ili gibanje elektrona oko atoma) mo~e imati samo to no zadane vrijednosti, kvantizirane vrijednosti. Kod harmonijskog oscilatora (HO) ona je jednaka: EMBED Equation.3,  je frekvencija titranja HO. Glava 4 Fononi Titranje kristalne reaetke mo~emo opisati mnoatvom harmoni kih oscilatora. Kvante pobuenja HO nazivamo fononima. Ako se HO nalazi u n-tom kvantnom stanju (n=0,1,2,...), ka~emo da postoji n fonona. Srednji iznos kvantnog broja EMBED Equation.3 HO, identificiramo sa srednjim brojem fonona: EMBED Equation.3. Srednji broj fonona nije konstantan, nego ovisi o temperaturi T. Energija pojedinog fonona jednaka je EMBED Equation.3. Energija HO: EMBED Equation.3. Fononi imaju impuls koji je jednak EMBED Equation.3, gdje je EMBED Equation.3valni broj ravnog vala kojeg opisujemo s harmoni kim oscilatorom. Vidjeli smo da imamo dvije vrste titranja u reaetci, akusti ko i opti ko, pa prema tome imamo akusti ke i opti ke fonone. Glava 5 Varijacijska metoda [1] Varijacijska metoda o kojoj emo diskutirati je korisna za procjenjivanje osnovnog energijskog stanja E0 kada to ne mo~emo izra unati to nom metodom. Pokuaat emo pogoditi osnovno energijsko stanje E0 razmatrajui probna kvantna stanja EMBED Equation.3, koja pokuaavaju da imitiraju osnovno kvantno stanje EMBED Equation.3. Definiramo EMBED Equation.3 da je EMBED Equation.3 , gdje smo se prilagodili mogunosti da EMBED Equation.3 nije normalizirano. Tada mo~emo dokazati sljedee: EMBED Equation.3 Ovo zna i da mo~emo dobiti najbolju procjenu za E0 razmatrajui razli ita EMBED Equation.3. Dokaz ovoga je izravan. ak kad i ne znamo vlastita stanja Hamiltonijana H, mo~emo zamisliti da se EMBED Equation.3 mo~e prikazati na sljedei na in EMBED Equation.3 , gdje suEMBED Equation.3 to ni vlastiti vektori od H: EMBED Equation.3. Tada imamo da je EMBED Equation.3 gdje smo koristili injenicu da EMBED Equation.3 u sumi je nu~no pozitivan. Isto tako iz ovog dokaza je o ito da jednakost postoji samo ako se EMBED Equation.3 podudara s EMBED Equation.3, jer tada svi koeficijenti EMBED Equation.3 nestanu za EMBED Equation.3 Dokazana nejednakost je jako va~na za procjenjivanje energije osnovnog stanja. ak i oskudna probna valna funkcija mo~e dati prili no dobru procjenu energije. Naravno, metoda niata ne govori o neslaganju izmeu EMBED Equation.3i EMBED Equation.3; sve ato znamo je da je EMBED Equation.3 vei (ili jednak) EMBED Equation.3. Varijacijska metoda sama po sebi nam niata ne govori koji se oblik probnog kvantnog stanja koristi za procjenu osnovnog energijskog stanja.. esto se moramo pouzdati u intuiciju. esto je mogue iskoristiti i poznavanje asimptotskog ponaaanja valne funkcije na velikim udaljenostima. U praksi, esto okarakteriziramo probno kvantno stanje s jednim ili viae parametara EMBED Equation.3i ra unamo EMBED Equation.3 kao funkciju od EMBED Equation.3. Tada tra~imo minimalnu vrijednost od EMBED Equation.3 mijenjajui vrijednosti parametara. Ekstremalna (minimalna) vrijednost od EMBED Equation.3 dana je s nizom jednad~bi: EMBED Equation.3 koje odreuju optimalne vrijednosti parametaraEMBED Equation.3 Ako valna funkcija probnog kvantnog stanja ve ima formu funkcije egzaktnog osnovnog stanja, mi ovom metodom dobivamo pravu valnu funkciju osnovnog stanja. Na primjer, zamislimo da netko ima mo da pogodi valnu funkciju osnovnog stanja vodikovog atoma EMBED Equation.3 gdje je EMBED Equation.3 nepoznati parametar koji treba odrediti. Tada nalazimo da EMBED Equation.3 ima minimum jednak pravoj vrijednosti energije osnovnog stanja vodikovog atoma,EMBED Equation.3 kada je a jednak Bohr-ovom radijusu EMBED Equation.3. U drugom primjeru pokuaat emo procijeniti osnovno energijsko stanje beskona no (jednodimenziona kutija) duboke potencijalne jame. Potencijalna energija je EMBED Equation.3 Egzaktno rjeaenje je, kao ato znamo: EMBED Equation.3 Ali pretpostavimo da smo ovo znali. O ito je da valna funkcija mora ia ezavati za EMBED Equation.3 te u osnovnom stanju valna funkcija ne smije imati oscilacije. Najjednostavnija analiti ka funkcija koja zadovoljava oba uvjeta je parabola koja ide kroz EMBED Equation.3 EMBED Equation.3 pri tome se ne mu imo s normalizacijom EMBED Equation.3 Ovdje nema varijacijskih parametara. Mo~emo izra unati EMBED Equation.3sljedei : EMBED Equation.3. Izvanredno je da pomou jednostavne probne funkcije mo~emo doi do energije osnovnog stanja unutar greake od 1.3%. Mnogo bolji rezultati mogu se postii ako koristimo slo~enije probne funkcije. Pokuaajmo s EMBED Equation.3 gdje sada EMBED Equation.3 ozna uje varijacijski parametar. Izravni ra un daje rezultat EMBED Equation.3 koji ima minimum za EMBED Equation.3 koji nije jako razli it od ranije razmatranog EMBED Equation.3 (parabola). 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Varijacijska metoda daje ispravnu energiju osnovnog stanja unutar greake od 0.3%. Ovo je jako dobar rezultat uzimajui u obzir jednostavnost upotrebljene funkcije. Kako dobro ova probna funkcija imitira stvarnu valnu funkciju osnovnog stanja? Za pretpostaviti je da mo~emo odgovoriti na pitanje bez eksplicitnog ra unanja preko integrala EMBED Equation.3 Pretpostavimo da je EMBED Equation.3 normalizirana, onda imamo EMBED Equation.3 EMBED Equation.3 , gdje je EMBED Equation.3 energija drugog pobuenog stanja; prvo pobueno stanje (k=1) ne doprinosi nejednakosti zbog svoje neparnosti. Iz ovog izraza izlazi da je EMBED Equation.3. Glava 6 Elektroni u polarnim kristalima Zamislimo da se elektron giba u polarnom kristalu, kao ato je natrijev klorid. Elektron meudjeluje s ionima, koji nisu kruto fiksirani. Elektron u svom susjedstvu stvara izobli enje kristalne reaetke, i ako se elektron giba, podru je izobli enja se giba s njime. Ovaj elektron, zajedno s izobli enom okolinom, zove se polaron. Jedna posljedica izobli enja reaetke je ta da je energija elektrona sni~ena. Nadalje, dok se elektron giba ioni se moraju gibati tako da se prilagode poremeaju, efektivna inercija elektrona (ili da upotrebimo prihvaeni izraz, masa polarona) je vea nego jednostavna masa elektrona koju bi imao da je reaetka vrsto fiksirana. Kvantnomehani ka analiza gibanja polarona je krajnje slo~ena. Mi emo zato napraviti brojne aproksimacije ije opravdanje u realnim slu ajevima mo~e biti upitno. Tako emo doi do idealiziranog problema, koji je bio prou avan od brojnih istra~iva a. Ovaj je problem bio istra~ivan ne samo zbog svoje mogue povezanosti s realnim ponaaanjem elektrona u kristalu, ve i zbog toga ato on predstavlja jednostavni primjer meudjelovanja estice i polja. Varijacijska metoda je bila vrlo uspjeana u rjeaavanju ovog idealiziranog problema. Kao prvo, ak ako su ioni vrsto vezani u kristalu, elektron e se gibati u vrlo kompliciranom potencijalu. Rjeaenje Schrdingerove jednad~be za elektron karakterizirano je valnim brojem k. Energetski nivoi ovog rjeaenja su openito slo~ene funkcije od valnog broja. Ipak, pretpostavljamo da relacija izmeu energije E i valnog broja k je i dalje kvadrati nog oblika, kao ato je EMBED Equation.3, gdje je m konstanta (ne nu~no masa elektrona u vakuumu). Sljedee, sila kojom elektron djeluje na kristalnu reaetku je takva da gura od sebe negativne ione i privla i pozitivne ione. Logi no je zato pretpostaviti da elektron najja e djeluje na opti ka fononska titranja u kojima se pozitivni i negativni ioni gibaju u suprotnim smjerovima. Opti ku deformaciju reaetke tretirat emo kao skup harmoni kih oscilatora. Slabu ovisnost frekvencije titranja opti kih fonona, EMBED Equation.3, o valnom broju uobi ajeno je zanemariti pa emo predpostaviti da je  konstantna. Naa zadatak je nai elektri nu silu izmeu opisane opti ke deformacije reaetke valnog broja k i elektrona. Zanemarujemo atomsku strukturu i tretiramo kristal kao jednostavan kontinuirani dielektrik koji prijenosi valove polarizacije stvorene opti kim titranjima. Ako je EMBED Equation.3 polarizacija, piaemo u formi longitudinalnih valova EMBED Equation.3, gdje je c konstanta proporcionalnosti. Gustoa naboja zbog deformacije reaetke je EMBED Equation.3. Pa je U potencijalna energija elektrona (opisanog ravnim valom) zadana jednad~bom EMBED Equation.3, ije je rijeaenje  EMBED Equation.3 . Poato su energija i moment elektrona povezani s izrazom EMBED Equation.3EMBED Equation.3mo~emo pisati Hamiltonijan itavog sistema kao EMBED Equation.3EMBED Equation.3, gdje smo produkt konstanti e i c zamijenili konstantom g. Prvi lan izraza je energija elektrona u krutoj reaetci, gdje je r polo~aj elektrona. Drugi lan predstavlja energiju titranja opti kih fonona (polarizacije), gdje smo pretpostavili da svi valovi polarizacije imaju istu frekvenciju i gdje je EMBED Equation.3 koordinata titranja k-tog ravnog vala. Zadnji lan je energija meudjelovanja elektrona i vibracija reaetke. Ispred energije meudjelovanja imamo predfaktor g ija je to na vrijednost: EMBED Equation.3  je bezdimenzionalna konstanta, a u slu aju kao ato je kristal natrijevog klorida, tipi na vrijednost ove konstante [21] EMBED Equation.3 je oko 5, gdje su EMBED Equation.3 redom statisti ka i visoko-frekventna dielektri na konstanta. Glava 7 Problem polarona u ionskom (polarnom) kristalu Hamiltonijan (energija) Energija koja opisuje meudjelovanje elektrona i elektri ki polarizirane reaetke u ionskom kristalu je: EMBED Equation.3 gdje je M elektronska masa, EMBED Equation.3 frekvencija opti kih fononskih titranja, EMBED Equation.3 je amplituda opti kog titranja valnog broja EMBED Equation.3, EMBED Equation.3 je impuls opti kog titranja, te g konstanta vezanja izmeu elektrona i fononske deformacije (fononskih titranja). EMBED Equation.3 i EMBED Equation.3 su kompleksni brojevi (u kvantnoj mehanici to su hermitski operatori-matrice). Radi se o problemu mnoatva estica koje meudjeluju i nije ga mogue rijeaiti egzaktno. Stoga emo se poslu~iti aproksimativnim metodama, to nije koristit emo metodu srednjeg polja kombiniranu s varijacijskom metodom za iznala~enje elektronske valne funkcije i deformacije reaetke. Energiju meudjelovanja aproksimirat emo s efektivnim potencijalom koji treba samosuglasno odrediti: EMBED Equation.3 =EMBED Equation.3 Sada je mogue napisati energiju za elektron koji se giba u jednom efektivnom potencijalu stvorenom fononskom deformacijom, te energiju fononskih titranja koji su pod djelovanjem vanjskog polja od elektrona: EMBED Equation.3 EMBED Equation.3. Pri emu je ukupna energija: EMBED Equation.3 Prvo ato uo avamo je da su fononska titranja opisana harmonijskim oscilatorima koji su izlo~eni djelovanju vanjske sile. Efekt djelovanja te sile je da harmonijski oscilatori titraju oko novog polo~aja iju vrijednost mo~emo odrediti deriviranjem potencijalne energije harmonijskih oscilatora po polo~aju: EMBED Equation.3 Dakle novi polo~aj oko kojeg harmonijski oscilatori titraju je: EMBED Equation.3 a energija harmonijskih oscilatora e biti EMBED Equation.3 Sada je mogue napisati izraz za potencijalnu energiju elektrona: EMBED Equation.3 =EMBED Equation.3. Pretpostavit emo da frekvencije harmonijskih oscilatora (opti kih titranja) ne ovise o valnom broju . Dakle: EMBED Equation.3 za svaki EMBED Equation.3. Za varijacijsku probnu valnu funkciju elektrona biramo: EMBED Equation.3, funkciju sli nu valnoj funkciji osnovnog stanja elektrona u vodikovom atomu. Nepoznatu konstantu A mo~emo odrediti iz uvjeta normalizacije valne funkcije: EMBED Equation.3 dok e se vrijednost konstante ( odrediti iz uvjeta da je ukupna energija minimalna: EMBED Equation.3 Naimo konstantu A: EMBED Equation.3 =EMBED Equation.3. Normalizacijska konstanta je: EMBED Equation.3 Srednja vrijednost ravnog vala je: EMBED Equation.3 = =EMBED Equation.3 =EMBED Equation.3. Tada je elektronski potencijal: EMBED Equation.3 =EMBED Equation.3 =EMBED Equation.3 =EMBED Equation.3 =EMBED Equation.3 =EMBED Equation.3. Pod uvjetom da je mogue raditi integraciju do beskona nosti. A ato nije!!! Postoji kona ni valni broj EMBED Equation.3 do kojeg se treba integrirati. Ipak rezultat je dobar, ako je EMBED Equation.3 gdje je EMBED Equation.3 konstanta reaetke. EMBED Equation.3 je u stvari podru je lokalizacije elektrona , i za to podru je pretpostavljamo da je vee od konstante reaetke, tj. EMBED Equation.3, ato zna i da je taj uvjet zadovoljen. Naa rezultat je dobar pod uvjetom da je podru je lokalizacije puno vee od jedini ne elije. Ponovo se dobiva rezultat da potencijalna energija na velikim udaljenostima opada kao kulonski potencijal , ali na malim udaljenostima potencijal je kona an, tj. nema divergencije uobi ajene za kulonski potencijal. Na malim udaljenostima potencijal je jama kona ne dubine.  Slika 6. Graf funkcije  EMBED Equation.3  Koristei probnu valnu funkciju ra unamo srednju kineti ku energiju EMBED Equation.3 =EMBED Equation.3 =EMBED Equation.3 jer je EMBED Equation.3 Potencijalna energija elektrona je: EMBED Equation.3 =EMBED Equation.3 =EMBED Equation.3 =EMBED Equation.3 Kona na elektronska energija : EMBED Equation.3 Ukupna energija sustava je: EMBED Equation.3 EMBED Equation.3 =EMBED Equation.3 Ukupna energija sustava EMBED Equation.3 =EMBED Equation.3 Deriviranjem ukupne energije po EMBED Equation.3 nalazimo da minimum odgovara: EMBED Equation.3 Da bi vrijediloEMBED Equation.3, parametri moraju zadovoljavati uvjet: EMBED Equation.3 Sada u izraz za ukupnu energiju sustava uvratavamo EMBED Equation.3 pa za energiju dobivamo: EMBED Equation.3. Da bi provjerili koliko je naae aproksimacijsko rjeaenje to no usporedit emo ga s drugim aproksimacijskim rjeaenjima koja su imala puno slo~enije probne funkcije. Npr. Feynmanovo rjeaenje iz njegove knjige Quantum Mechanics and Path Integrals, za male vrijednosti EMBED Equation.3, nalazi da je ukupna energija polarona [32] EMBED Equation.3, nasuprot za velike vrijednosti EMBED Equation.3, energija je [3] EMBED Equation.3. Glava 8 Zakju ak Prou avali smo gibanje polarona kroz kristalnu reaetku., Rradi se o sistemu s velikim brojem estica  EMBED Equation.3 . Da olakaamo slo~enu kvantno-mehani ku analizu gibanja koristili smo se aproksimacijama, to nije koristili smo aproksimaciju srednjeg polja kombiniranu s varijacijskom metodom. Aproksimacijom srednjeg polja energiju meudjelovanja izmeu elektrona i kristalne reaetke paproksimirali rikazali smo s efektivnim potencijalom  EMBED Equation.3 . Tada smo mogli napisati energiju za elektron koji se giba u efektivnom potencijalu. Fononska titranja opisali smo s harmonijskim oscilatorima (HO) koji su izlo~eni djelovanju vanjske sile. Rezultat djelovanja vanjske sile je da HO titraju oko novog polo~aja ravnote~e s energijom  EMBED Equation.3 . Nakon izra una energije fononskih titranja mogli smo napisati potencijalnu energiju elektrona  EMBED Equation.3 . U daljnjem ra unanju pretpostavili smo da frekvencije titranja HO ne ovise o valnom broju k. Sada primjenjujemo varijacijsku metodu, kojom se nastojimo ato viae pribli~iti pravome rijeaenju. Biramo probnu funkciju (eksponencijalna funkcija) elektrona, koja ovisi o varijacijskom parametru . Vrijednost varijacijskog parametra dobijemo iz uvjeta da energija bude minimalna. Iza unavamo normalizacijsku konstantu A. Pomou probne funkcije izra unavamo efektivnu potencijalnu energiju elektrona  EMBED Equation.3 . dobijemo srednju vrijednost ravnog vala, koju uvratavamo u elektronski potencijal. Iz kompjuterski nacrtanog elektronskog potencijala vidimo da se na velikim udaljenostima ponaaa kao kulonski potencijal, a na malim udaljenostima kao potencijal kona ne dubine. Dalje pomou probne funkcije ra unali smo kineti ku i potencijalnu energiju elektrona, iz kojih smo dobili ukupnu energiju elektrona. I na kraju izra unali smo ukupnu energiju sustava kao funkciju varijacijskog parametra (. Vrijednost varijacijskog parametra dobijemo iz uvjeta da energija bude minimalna. Dobiveno rjeaenje usporedili smo s Faynmanovim rijeaenjem iz njegove knjige Quantum Mechanics and Path Integrals. Primjeujemo da se naae rijeaenje poklapa s njegovim u slu aju velikih  EMBED Equation.3 . S obzirom na jednostavnost naae probne funkcije, relativno kratak i jednostavan ra un u usporedbi s Faynmanovim, zakju ujemo da je naae rijeaenje jako dobro za velike  EMBED Equation.3 . EMBED Equation.3 Bibliografija [1] J.J. Sakurai , Modern Quantum Mechanics (Addison-Wesley Publishing Company, Inc.,1985.). [1] J.H. Wilkinson, and C. Reinsch, Handbook for Automatic Computation, (Vol. II, Linear Algebra) Springer-Verlag (1971) [2] M. Abramovitz and I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New York (1972) [3] H. Roeder, J. Comp. Phys. 124, 115 (1996) [214] R.P. Feynman, Statistical Mechanics (W.A. Benjamin, Inc. 1972). [325] R.P. Feynman, Quantum Mechanics and Path Integrals (Mc Graw-Hill, Inc. 1965.) [36] R.P. 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