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# QUANTUMMECHANICS PHYS610A

SDSU

GPA 3.78

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This 13 page Class Notes was uploaded by Jamaal McGlynn on Tuesday October 20, 2015. The Class Notes belongs to PHYS610A at San Diego State University taught by M.Bromley in Fall. Since its upload, it has received 27 views. For similar materials see /class/225319/phys610a-san-diego-state-university in Physics 2 at San Diego State University.

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Date Created: 10/20/15

Lecture 18 Outline 3D radial bits 0 Spherical harmonics recount section 125 Radial Equation Section 126 In nite Spherical Well Section 126 in brief 3 D isotropic SHO Section 126 start on the H atorn Section 131 Angular Momentum eigenstates o Simultaneous eigenstates ie L2 L2 O L2 m Wag 1 m l 727 3 727quot o for O 1 andm 1 and LZ m hm 7m 1m 0 separation of variables gt spherical harmonics ngew gb A h2 a a 1 a L2YW 39 YW 2 1 YW sin6 66 8111636 sineagbl g h a g r YW 13 WW hm WW 2 e 3 e 6 e 0 ie good quantum numbers E mg En 7137715 0 when V0 then 11p O meanwhile 19 O L2 as Generators of Rotation c this bit is not assessable as seen in advanced course 0 Shankar 122126 rotation formalism eg a y plane N i 1 I LZ 7J OLzh RWOW 120 h N gt e 0 so working in position space LZ ih8 e Owp gt w cb 0 End of spherical harmonics Time indep SE V2 Vw Em using separable solns 03 6 gb RrY6 gb we found 2771742 aw 1 i sin g Ygt L O Y sin6 36 36 sin2 6 32gb constants F7 G396 gb EM 1 EM 1 O m 251 m n6 6 47v m Spherical Harmonics are all orthonormal eim 13lmoos 6 27r 7r WWK67 Yv tn 67 Sin 6 666 6mm 0 0 Radial Equation 0 Given separable solns 03 6 gb RU ng97 b 0 do a important Change of variable ur rRr 112dRgt M V7 E rr 1 Rdr T dr E2 RE dRi rduu i 7420B rdiu r 7 dr r2 dr 7 dr dr dr2 h2d2u may E 2m dr2 r2m 7 2 la u o the radial Schrodinger eqn has an effective potential E2 W 1 7 2 0 With normalisation fOOO R2r2dr fOOO u2dr 1 Ve 7 V0 In nite spherical well Outside well 03 6 gb O R0 ngw gb so want to nd bound state eigenfunctions inside well E2 d2u E2 E 1 V E 2m dr2 74 2m 7 2 la u d2u E 1 2 W Z l r2 k l u where k V ZmEh eigenenergies E unknown For E O we get a familiar 1 D eigenproblern d2u dr2 as 7 gt 0 then cosm 7 gt 00 thus B O k2u thus Mr Asinlcr BcosUm In nite sphere E O solns 0 At well surface un t a QC sincna 0 so kna mr o allowed energies for E O are 71 W12 EMZO 2m n22ma2 Vn123 o normalisation fOOO u2dr 1 means A xZa n007 9 sin Y0067 217m 8m 0 Energy Eng depends on two quantum numbers o Wavefunotion wngm depends on the three In nite sphere general soln o The general soln to 271 162 u o is given by the spherical Bessel and Neumann functions 71574 A 7 ngm B 7 TIM 7 Mas 92 11y SW W We 11y cosltxgt xdx a sinx sinx cosx joltxgt x j1ltxgt x2 x 0 Limit 9 gt 0 all jg oc mg ie j0x 1 and j1x 933 0 but as a gt 0 all ngx gt 00 so again impose B O cosx sinx 1 n0x 932 7 limit n0x gt O m In nite sphere Bessel bricks mu T2 The general soln to 277 M u W0 A 7 j 7 A 7 cr iiy Lina Finally impose boundary R0 a ugaa O allowed values of k are when aw O trivial for E 0 from before 1m mr for E gt O non trivial roots of Bessel ie computer n E2 2 knag SO Eng W ng remembering 23 1 degeneracy with m n m7 7 67 An J al 19 67 In nite sphere Bessel piccies Isotropic SHO o Hamiltonian H o i 135 132 5mm g2 22 0 assuming 03 6 gb WYXKQ gb gives radial SHO 2 2 d 2mElmw2r2h E1gtuO W 732 2 27717 2 0 as 7 gt 00 potential V0 gt mw2r2 solns gt eTy2 with y E r sotry fyeTy2 o with a power series expansion f 3 220 any 0 reqd to terminate at integer k O 1 2 as before 3 3 E2k hwn hw Coulomb potential Time indep SE V2 V E separable solns 03 6 gb ngw cb E2 d2u e2 ljL E2 1 uEu W 71mm r2 Coulomb potential here assumes in nitely heavy proton Continuum scattering states for E gt O and discrete bound states for E lt O which we want introduce H V ZmEh thus 1 d2u me2 1 1 1 u H2 dr2 27T 0h2li H7 m42 suggests introduce p H7 and p0 for constant u 0 radial equation 0 how the bloody hell do we go about solving this 0 with p H7 and p0 me227T60h243 we have 2 du1 1 132 p p2 0 rst consider the asympotic form as p gt 00 d2 N u hassolns u N A e B 6 dp2 N p N 0 second consider the asympotic form as p gt O d2u EM 1 m u dp2 p2 0 Limit p gt 00 has ep gt oo demands B O 1 hassolns up C pg1 D 7 p 0 Limit p gt O has p g gt oo demands D O

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