Problems with Solution - Quantum Mechanics | PHYS 332, Study notes of Quantum Mechanics

Download Problems with Solution - Quantum Mechanics | PHYS 332 and more Study notes Quantum Mechanics in PDF only on Docsity! Leche 33 Prys 282 _ Sig of In #2 lost chs, we rewrdie Cp Homi lenian for a particle WO cenbead patenhal A 52 Re H = my —L + VéP) 2mr? We Waar that val Tang iy connie wih He, Ale wie Ui, Tas He serqy ciginstotes con always be cosbuded du cule a Wouy hak hay ore sinurltoneassly diggnstale of tf ark te, “Thus the cnegy eigenskeles are 1a, 40> hog z Commutes AaL™> = Bq la,®ad tol oway Lado = Nola, Xm Magnidide argulor rowmnbum quired Le fa £a0> = Aula, Zony B-Coapanent df angular monerten . We shall deal ust Le tag angular monertuon. equokars by an (almost) Paty opsalr Female. Te radial squahon wil requie a gifeel eguchne aporoads . Radial aguuahion Gosidar Le wnegy equahas . Hi Idm> = Enla,dm> We aim Yo recast 40s ia ams of wave tuinchms expressed in, sphoical Co-axdinales a, kmS aw Yon (",8,d) % do » we gowebiee he usual Conespeudace behueny states + waueFunchiaus , Recall tat in Gotesion 09 -ordinales : P(%4,2) = <xyzl YS Sioilarly ur sphoiedl 0 -orchinales Yir,0,8) = 10,d1¥S Most of the ules connecting Stakes 40 wauehunchans con he transcabed withadt olterohor. Ong. exception, whed con be vethed by te usucd wes for changa ag trom Gorlesion 4D sphaical co-ordurdlés is: AHP = aon 0,4) Olt,0,8) r2 ve a rote tis Example : Suppo Rat Y(r@,9)= ERMC Deming OPA and sip fy kis as much og possible * 2 pt w . <HY = i dr 2 [Roy| [soso (a jewe]' 5 0 Ne =| = (ve r IR Case Lena 00 : jar r Jace) ]- his is onalogus to te one ckmensoual ase whwe PIP= pl» Hae, Hue wavelynrehon vewon of tes cquohan ts: 4d wt 4 ak ax EK e PXMa [ ‘ take Ce we p p> Pp. {p> What is Ha equivalent sizdement oe padial mount? Yan may expect po a> A ele i Constant Exercise. « Deternune He radial prcbabilih, consiy for sude a wauehuchen . Use it to answer Bo quashon of wl Yee cost kely valee for Measuemant acteongs of t is, Telermine +e pobabilit, Hat re Ar VS pro tat lo € 0 6 fysAr Arswer Pf) gg VAe'pra TEAL Thay Ar Prob (céprP) \ Pie) dr = jaye en" Prob Cre ¢ crite (rns os tar [rosary ret) = IAT de Ar + Ar*] he tabi Oh, tuse ig. Pree (<br) fr épr) Prd (% £7 & wesAr) P+ 2c “PO as %® 20 We really require more tuen duistnbukon. “Thus bole allenph is A Orr Ag ipo Se Pry whoe A 1S constant. Then He tadedl Probalolihy donvihy Po) = [A\* 0 longer hes we ¢. This as Aralogaus to intensity of inegnehic waves. Recall Lot if Yo doserbes tee wave kan tha | on wie = Ee aot I= 11. But His woudl imply Hak fe total power pasiing Keough a sphoe at radius 6 tg {Ale r? Ore. Compensates Wa We 4 elt. Now consider a toe Ho radial Comp onant of Momenhon. Fist try hn AR Exacise, Does tein satish, Pe [Pro = pe Ip. Answer Pripp> ~p ik & [be e'Prve] = “US 42 7 Tale elf he = p(Aeiry) 4 +i _ Ae Prt Not quite | What modikeaton would we neod? Ke can se Rak frou Be previous eyuchian NG (Fe PM) =p Bera) yb a cies Tue wh ($+) B elprlk = Pr (Fe '\Prlc) “This suggests tat pr a> ak (2 te) On com veihy tis Fortally by taking Hie earlier ayprssian for 2a rodicl —compauant of mawentum ond le-expesting iin tems dy sploical co-avdundles.. Noie that the aboye LX pressian &s 2G dont to: leo. PelWs = -ik (3 A) ¥09), Feally ening fo te 00M equation lala 2 Bal a dim DH OFA Inde = By Ct,eb]a,2ad En Vo, (*,2,0) > peta de> - <,8,d1 Ula dm) 4 <t, ©] VIP) | ada) em 2mr? = En Wdia (,Q, 4)