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This 7 page Class Notes was uploaded by Citlalli Sauer on Tuesday October 20, 2015. The Class Notes belongs to PHYS410 at San Diego State University taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/225321/phys410-san-diego-state-university in Physics 2 at San Diego State University.
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Date Created: 10/20/15
Lecture 15 Outline nite square well 0 Wavepacket Clari cation Grif th s Figure 211 0 Finite D well even bound state solns 26 0 Finite D well scattering solns 26 Vx V0 Phase velocity vs Group velocity 0 Plane wave w Asincx wt w 27TT k 27TA QM dispersion relation was Eh 73622771 Phase velocity Uphase wk his2m Assume was wo w6c 160 x t eiw0k0w5t x 00615 O 0 ie wavepacket moves at vgmup wg dwdk hism All about dispersion relation Note that deep water w xgk gt Uphase 2119mm9 o In shallow water w xgd k gt Uphase vgmup httpgalileophysVirginiaeduclasses1OQNmorestuffAppletssinesGroupVelocityhtml FEW VO lt E lt 0 even funk Still considering the bound states V0 lt E lt 0 since Vx symmetry even or odd solns about a O Dcoscx V0 lt a lt a Fez m W x D F Via boundary cond xxa and d xdxxa EN Vxgta V93ltO Fez H D cosca and ltFe cD sinca Dividing one by the other H ktanUm FEW VO lt E lt 0 even solns Now that we have H ktanca Note KLE and Wh Introducing z 1m zo aimh We rewrite H2 k2 2mV07i2 and 1m z3 z2 and nally rearrange t0 tanz y 1 Z The graphical sOlns see gure 218 give us sOlns z which gives the allowed bound state energies En FEW V0 lt E lt 0 even soln En 2 o limits of tanz 2 0 1 Z a large wiolth depth ie large zO ax2mV0h intersects z zn occur just below zn n7T2 with n 13 2 En V a VWW mg h n27r2 2 En V Ea ovewe o om gt 0 2mlt2agt2 b 11b tt zn kna Sarne energies as ooDW even 1 solns ie for n 6 odd Odd FEW solns give the same En as ooDW odd 1 solns b Shallow narrow well ie srnall zO since z 1m gt 0 there is always one even bound state BUT the lowest state disappears below zO lt 7T 2 FEW VO lt O lt E scattering 0 Now looking at scattering states E gt O 0 so we have is 2mEh and H 2mEV0h A617 Bea m V a lt a T ECU N CsinOw D COSltLSL V a lt a lt a Fem V a gt a 0 Apply boundary cond xxia and d xdxxia flea m Be m Csin a D codw ikMe ik Baika MC COS a D sin a Fe m C Sil lltLa D COS a ikFeik MC COS a D sin a FEW Vo lt O lt E scatter solns 0 Four eqns four unknowns B C D F one xed Eliminating C and D sin2 a B 2 k2 F 2 21 H 6 2ikia F A oos2 a sin2 a De ning the transmission probability T F2 A2 rO2 4EE V0 Grif th s gure 219 sin2 2 a 2mE V0 T 11 h Re ection probability R B2A2 1 T