Atomic Physics PHYS 4261
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This 0 page Class Notes was uploaded by Kylie Bartoletti DVM on Monday November 2, 2015. The Class Notes belongs to PHYS 4261 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/234287/phys-4261-georgia-institute-of-technology-main-campus in Physics 2 at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
Physics 4261 Homework Set 2 Central Force Problem and Hydrogen Atom Friday January 25 2008 Please review Chapter 21 of Foot s book and the lecture notes 5 1 Prove that the kinetic energy of a 2 particle system is equal to the sum of kinetic energy associated with COM and relative motion ie 132 E Z m f A 2 where M is the total mass and u is the reduced mass N In the textbook the angular equation is solved using an algebraic method In this problem you will apply a direct approach to the differential equation below d z dF 1M 1 a m2 17 x2 Derive Equation 1 from the eigenvalue equation for the angular momentum operator LZY8 7121109 p Apply the parity operator to the equation What does this tell you about the solutions For the special case m0 this reduces to Legendre s equation Substitute FAF0 1 7 0 into this a power series solution F x 2L1ka and nd the appropriate k recursion relation between aHZ and ak aka P Show that gt k k 2 as k gt 00 What does this mean for the solution k at the poles 8 07 HINT Look at similar series Use this to nd the eigenvalue 7 The resulting solutions are the Legendre polynomials Px de ned for lxlg 1 D The Legendre polynomials may be written in the form P x Ld lkxz 71y 2 1 dx and the associated Legendre polynomials are P x 1 ixz39 Z Xm Px which satisfy Equation 1 above for m is Tabulate the Legendre polynomials 1 for 0I2 and 3 Verify that they are normalized t2 111310012 2m quot L n L 3X n ml ax a4 5amp1 L L m L 39 2quot La ling 3quot hurlquot 3 U L 1 SvHrcm k SL4 M x SMQ 74 WW5 ML MAAUL K 3 7539 U39 39 32 quotug Tu 5 139 2 h 7 1n gnu 2 M1 mm 3X x 3 MWLDX a 39L 1 Li 39 5 1 vm Ham 3X 3quot MN 23 x 7 1 Fl L a 2M 3 3 ht Quill 3X 7 MHzL W 2 4L 1 quotMM 31 f3939 i 5 Z 1 a A mmL 0X 39 y 3quot 2 A 2 2 gt a 142740124 70245 EQMWM ewk r 1392 2 Hbmf 422 BJ H 2 k A quotLucio A 1 2 J 3 l J 3 L 5 M w L 1 48 9 Ska 3839 933 941 9Jp 90 y Y I39m K 445 39 Shag 3 de a g 3 T W 0 4343 t sk 9 M sh 16 E9 T gm Mu W 9 my 4gt an m L a NM 3 E 1W N etim M mW m a w w m w A99 S g 3 SM3 33 39 6 65 swL kkk x use Tm 39 g 4mg so 5979 019 99 ale 9 4590 ampla Mar HAj Maw PM em m Tic AF Dlx li Fo B O Pad oftta u Them x L c an Po 046313 A Fx M w 3 2 if l x 35 3 PCquot 9 FOX is asL i m M wt SMQ H1 LAWN Marl l d fw 1 51kg P1 F m LL and eJuw 1ch Flag Pb Dr i4 Fx PM quot0 0453 Aon we 2ngFAFo F Equx 3 f EMU 0 c x Pea 45 2 3 K W XkZ QK hk lXHZ XRE 2 gathh h E uk all1 be M h AEQVx 0 L30 ILo n L1 so 2 akh Vt 0 X 2 maxVt AZP hut0 a 22 5 gmka gakxkbzh m i0 ya hUzh A kiln L1 kw Gui am chuuun 5 2 n knD 2k ward2 an a Qt qu gt an HBQQ thnfokhajr zzk LL an 931 MOO1 5 km SW 1044 Wt lowme Seam H x X1 H gt dud AW X211 LA emuv45 MAL To us unguk 3th at fie fbh n xt 9 off uc m wn ruva Kl lungm 39gt 0 pm 5M 11LAM TM A Um Mm Lang 1 l FECX E 145w ai akxk
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