CHAPTER 4 SCRODIGER S EQUATIO A. Itroducto The old quatum theory have elaed successfully about le sectral hydroge atom. Ths theary also have show hyscal heomea atomc ad subatomc order fulflled rcle ad rule whch far dfferet from rcle ad rule whch fulflled macroscocs hyscal systems. evertheles the old quatum theory stll be ad hoc ad the fact that t ca ot be aled uerodcaly heomea atomc order. ow t s requred a ew quatum theory whch more geeral ad comrehesve. I 96 was foud a moto equato for hyscal system atomc order. The true or false of the quatum mechacs equato deed o the relevace betwee theorytcal result ad observatom result about the hyscal observables, Ths s based o the fact that hyscs s quattatve. Fgure 4.. Erw Schrodger Ths reseted several essetal thg whch must be cotaed ew mechacs theory. Most arts are fudametal thg old quatum theory as follow:. The classcal cocet about ath does ot have meag I quatum hyscs system, t oly ca be aled a statstcal meag about the osto robablty of artcle sace (Heseberg rcle).. Dualsm of wave ad artcle. 3. If artcle s reseted as a wave so for the wave fulflled suerosto rcle of wave. If ad s soluto of the wave equato so =c +c also the soluto of the wave equato (c ad c are costat). 4. I the equato s also ca be aled de Brogle ostulate ad Este s quatum h theory: ad E h. 8
5. The eergy coservato law o relatvstc case : E V, where lear m mometum, m mass of artcle, ad V otetal eergy. 6. For costat of V, E ad also must be costat ad artcle s reseted as a moochomats stadg wave (de Brogle). B. The Dfferece of Clasccal Physcs ad Quatum Physcs Klask o The future of artcle ca be determed from tal osto, tal mometum ad force whch eert o t. I macroscocs world all quattes ca be determed wth eough recsto so the classcal redcto s accordg to the observato result. o The aromato verso of quatum mechacs. o Based o the erceto of sese. o Descrbe a dvdual object sace ad ts chages travelg tme.. o Predct a evet. o Assume that objevtve realty was out of there. o We ca observe somethg wthout chage t. o It clam based o the absolut thruth that real uverse was the back of scree. Kuatum o The future of afrtcle does ot clear because ow era of artcle s ukow. Posto ad mometum of artcle ca ot be determed wth eough recso ( effect of Heseberg ucertaty) o It s more uversal. o Based o the behavor of subatomc artcle ad systems that ca ot be observed drecly. o Descrbe the behavor of system statstcally. o Predct the ossblty of evet. o It does ot assume that objectve realty s free from our eerece. o We ca ot observe somethg wthout chage t. o It clam that ca correlate eerece correctly. C. The Meag of Wave Fucto o Wave fucto usually has a form of comle umber: A B, whch cosst of real art ad mager art. o Comle cojugate: A B 9
o The desty of robablty: o ( A B)( A B) A B always be real. s roortoal wth robablty desty for obtag artcle rereseted by wave fucto. o Iterretato of Ma Bor 96 stated that f tme t s made measuremet obout osto of artcle related wth wave fucto of (,t) so the magtude of robablty P(,t)d of artcle wll be foud terval of ad +d s (, t) (, t) d. o Itegral of all sace must be fte. Fgure 4.. Ma Bor o dv 0, has meag s ot be foud a artcle. o The robablty for fdg artcle certa tme all sace must be equals to. So dv atau PdV. o The ormalzato of wave fucto s requred : dv. o The well behaved of wave fucto are: a. has a sge value a certa osto ad tme. b. must be kotu. o For artcle whch move as the robablty for fdg artcle s: P d D. Wave Equato The wave equato of y whch roagates as wth seed of v s: y y (4.) v t Soluto of equato (4.) s: 30
y F t (4.) v Where F a fucto whch ca be defereted. Based o equato (4.) t meas that there are solutos:. y F t roagates + as drecto v. y F t roagates as drecto v Geeral soluto for the moochromatc harmoc wave fucto wth a costat agular frequecy, ot be damed (costat amltude) ad roagates + as drecto s: y Ae t v (4.3) Fgure 4.3. Susodal Wave By rememberg : e cos s equato (4.3) ca be wrtte: y A cos t s t (4.4) v v E. Tme Deedet of Schrodger Equato: t v Based o equato of Ae By substtute : ad v equato (4.4) t s obtaed : Ae Ae t t 3
Because E h ad Et h t s obtaed: Ae (4.5) Equato (4.5) s reresetato of free artcle wth total eergy E ad mometum moves + as drecto. The equato (4.5) s deferetated two tmes, t s obtaed: Ae Ae ( Ae Et Et Et ) (4.6) Partal deferetal of equato (4.5) t: ( Ae t t E Ae Et Et ) E (4.7) t Total eergy E ada low seed fulflled : E V m Substtute E to t s obtaed: E V m (4.8) 3
33 From equato (4.7) t s obtaed: t E (4.9) From equato (4.6) t s obtaed: (4.0) By substtute equato (4.9) ad (4.0) (4.8) t s obtaed: V m t V m t (4.) Equato (4.) s a deedet tme of Schrodger equato oe dmesoal case. For three dmesoal case, the Schrodger equato s: V z y m t (4.) where oerator: z y Equato (4.) ca be wrtte smle term: V m t (4.3) F. Eectato Value Defto about robalty desty of P gve a way to redct the average value of the osto of artcle certa tme. The average value s called eectato value. The average osto of artcle fullflled:...... 3 3 3 (4.4) For artcle, the umber of must be relace wth P. At osto of artcle ca be foud terval of d.
P d Eectato value for fdg artcle s: d d (4.5) If s a ormalzed wave fucto so d that t s obataed a eectato value: d (4.6) The same rocedure s aled for fdg eectato value of <G()> as otesal eergy of V() as follow: G( ) G( ) d (4.7) Eectato value of mometum <> ca ot obtaed by ths way, because accordg to the Heseberg ucertaty, there s o fucto of (). If we determe of so that =0terme, because: Smlar thg s occurred eectato value of ergy <E>. G. Tme Ideedet of Schrodger Equato Based o equato: Ae Et E t Ae E t e where s a osto deedet of wave. t e (4.8) 34
Substtute equato (4.8) tme deedet of Schrodger equato (4.): t Is obtaed: V m E E E t t e e m Ve E t m E V 0 (4.9) Equato (4.9) s steady state Schrodger equato oe dmeso. For three dmesoal case: m y z E V 0 (4.0) A steady state Schrodger equato oly ca be solved for certa eergy E (quatfcato eergy). Kuatfcato of eergy s characterstc of all hyscal system whch s stable. A ear aalogy about quatfcato of eergy solve Schrodger equato s case of roe wth legth of whch s stregted wth bded of both edge. H. Partcle Oe dmesoal Bo The aalogy of artcle oe dmesoal bo s a stadg stadg wave of roe wth both edge s bded. The wave fucto wall equals to zero. ddg 0 The relato betwee wde of bo ad wave legth of s show Fgure 4.4. 35
3 =(/3) = = Fgure 4.4. The relato betwee wde of bo ad wave legth of The de-brogle wave legth of afrtcle geeral ca be formulated :, =,, 3, (4.) Ketc eergy of artcle: mv K mv m m (4.) h h Because mv mv Substtute equato (4.) equato (4.), t s obtaed ketc eergy of artcle K: h K (4.3) m I ths model, otetal eergy of artle V=0, that the eergy whch beloged to the artcle: E E h K m h 8m h m (4.5) For quatum umber =, the eergy of artcle s: 36
E h 8m The eergy of afrtcle wth quatum umber of ca be wrtte E as follow: E E (4.6) Based o (4.6) t ca be cocluded that: o Eergy of artcle s quatfed. o Quatum umber s charactersato of eergy state o Mmum eergy of artcle 0. The smlest roblem quatum mechacs s roblem about artcle oe dmesoal bo whch has ufte hard wall. The moto of artcle fted as betwee =0 ad = caused by the ufte hard wall. A artcle does ot loose ts eergy whe t colldes the wall. It meas that the total eergy of artcle s costat. Accordg to formal vew quatum mechacs, otetal eergy of artcle V to be fte both edge of the wall, mea whle ts otetal eergy the bo s costat equals to zero. V= 0 Fgure 4.5. Potetal Well Because artcle ca ot has fte eergy, so artcle does ot robable out of bo, so the wave fucto equals to zero for 0 ad. Our task s determg the wave fucto the bo. Schrodger equato the bo: d m E 0 d (4.7) 37
Equato (4.7) has soluto: me me As Bcos (4.8) Whch ca be roved aga equato (.7) wth A ad B are ostats that must be deremed. Ths solvg must be lmted wth boudary requremet 0 for = 0 ad =. Because cos 0 = the secod term does ot equal to zero for = 0, so B must be zero (B=0). The because s 0 =0 so the term whch cosst of sus always roduce 0 for =0 as whch requred, but t wll be zero at = f oly: me for =,, 3, (4.9) It caused zero value of sus occurred at the agle of π, π, 3π, From equato (4.9) t s clear that eergy that ca belog to the artcle has a certa value that called ege value. The ege value of eergy states fulflled: E where =,, 3, (4.30) m The wave fucto of artcle oe dmesoal bo whch has eergy of E s: me As (4.3) By substtute equato (4.30) (3.3) t s obtaed: As (4.3) Whch state ege fucto accordg to value of E. The ormalzato of the wave fucto s: d A s d A 0 0 38
A A So the ormalzed wave fucto of artcle oe dmesoal bo s: s ; =,, 3, (4.33) Probablty desty of,, da s show Fgure 4.6 as follow: 3 E 3 = 3 (/6) (3/6) (5/6) E = (/4) (3/4) E = 0 ½ Fgure 4.6. Probablty Desty for Fdg Partcle Oe Dmesoal Bo 39
Based o 4.6 t ca be descrbed that: o, value of, da 3 always ostve ad because ts wave fucto s ormalzed so the for certa equals to the robablty P for fdg artcle that ot. o The bggest robablty for fdg artcle whch has eergy of E s laced at osto of ½. o The bggest robablty for fdg artcle whch has eergy of E s laced at osto of ¼ ad ¾. o The bggest robablty for fdg artcle whch has eergy of E 3 s laced at osto of /6, 3/6, ad 5/6. o The bggest robablty for fdg artcle s deffretet deed o the wave fucto of artcle, osto ad ts eergy state. o I clasccal vew s stated that t has the same robablty for fdg artcle at each ot oe dmesoal bo. I. Referece Almuf Aref, dkk. (00). Buku Mater Pokok Fska Atom. Jakarta: Pusat Peerbta Uverstas Terbuka. Haryad S. (983). Persamaa Gelombag Schrodger. Badug: ITB. Yusma Wyatmo. (003). Fska Moder. Yogyakarta: Pustaka Pelajar.. Yusma Wyatmo. (008). Fska Atom dalam Persektf, Klask, Semklask, da Kuatum. Yogyakarta: Pustaka Pelajar 40