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This 12 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 39 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 13 Outline Bound and Scattered 0 out of the in nite box Section 52 a into a nite well problem 526 Completeness and matrix elements 0 well eigenstates E wnw then Was Z o with orthonormal states E f mugs wnada 6mm WK wx wx n1 n2 3 a laxl l matrix elements E a wnada lt93gt WIIEIW Z Cincdmlxlm mn1 Ut Z Me mnth thus 11125 Z one iEnthm easy time dependent soln propagator and completeness 0 Since Hamiltonian is Hermitian H 132 2m expand at I Z Cnngt and Ut Z ltnle z Enth 0 example of expanding wall little theorems 1D bound states a Two little theorems that we ve just whipped past a There is no degeneracy in 1 D bound states b The eigenfunctions of H can always be Chosen to be real for real H while we are working in the sir basis eg if 412 d2 Va in mm then w M w Em a so um and u have same eigenvalue 0 but in W Mi 2 also has same eigenvalue 0 Finally 1 um rum2239 377 due to and w I W Wz 1iCWr 14 Bound and Scattering States a Depends on Whether particle can escape to a too a Two forms of our linear combination of stationary states 1 00 hk2 x11 932 k elm m dk lt gt m m qgtlt gt 1115 25 Z cnwnaeiEquotth o if E lt V oo AND E lt Voo bound state a if E gt V oo AND E gt Voo scattering state Finite Square Well boundl Firstly considering the bound states V0 lt E lt 0 so need to look at SE outside vs inside the well 712 dZib 712 dZib E 39 39 E 2m das2 w md 2m das2 Vow w d2 dZib d ajg H b and W k2w Where 714 V 2mE and hk 2mE V0 General soln w a lt a lt a Csinka D coska my lt a BeXplta and my gt a FeXp lta Finite Square Well boundZ QM is wavefunotion matching boundary conditions since Va symmetry even or odd solns about a 0 D COSUCCE Fe W x D7 F Via boundary cond ibaxa and dwadaxa V0ltazlta EN Vasgta Vaslt0 Fe Dooska and ltFe w kD sinka Dividing one by the other H ktanka We rewrite H2 k2 2mV07i2 and Ho ka tanka Finite Square Well b0und3 o Graphical solns give us allowed k H Ha ka tanka and H2a2 k2a2 2ma2V0h2 a s0 hkn 2mEn V0 gives bound state energies En Finite Square Well bound4 a large widthdepth ie large radius R ax2mV0h intersects occur just below kna mr 2 with n 13 ax2mEn V0 kna n7T2 h 2 2 2 n 7T 71 En lO 2m2a2 Eabove well bottom Same energies as ooElW even on solns ie for n E odd Odd FEW solns give the same En as ooDW odd on solns b Shallow narrow well ie small radius since 16ch gt 0 there is always one even bound state BUT the lowest odd state disappears below ka lt 7T 2 Finite Square Well scatterl o NOW looking at scattering states E gt 0 a so we have k x2mEh and H 2mE V0h Ae m Bequot V a lt a wECE N Csinlta DCOSlta V a lt a lt a Fail V a gt a 0 Apply boundary cond waxia and dwadaxia 148 116 B8116 Csinlta D COSlta ikAeik Balm MC COSlta D sinlta Fe m C sinlta D COSlta ikFeik MC COSlta D sinlta Finite Square Well scatter Four eqns four unknowns B C D7 F one xed Eliminating C and D sin2lta 2 2 k F 2643 H B Zz ka F A cos2lta kg2 sin2lta De ning the transmission probability T F2A2 V2 2a T 11 0392 2 E V 4EEVo 8m h m 0 Re ection probability R B2 A2 1 T Finite Square Well scatter3 a special energies E V0 Where the well is transparent a which coincides with in nite square well energies
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