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soltn7 - Homework 7 Solutions Y4)=Acosls Bsinkr df =-tA rin...

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Homework 7 Solutions Y4:)=Acosls+Bsinkr df = -tA rin kr * kB "* tt dx )'Y = -k'Ar*kt- t2Esin tr dx' ( 4\ e -w" =( -2?E \,r cos u +asi,, r'r \ h ' ) \ h ' t rhe schrddinser equation is satisfied fi # =(#\E -urY * -k2 (4 cos tr +I sin O =(ff\,+ *s tr + I sin kt)' Therefore r=# 6.10 ,"=# (o.oe x ro-r Is)2 = 5.03 x 10-18 I=37.7 eY 8(e.rr x 10-31 tcg[to{o m)' \=37.2 ey Ez=!7.7 xzz =751 eV E3 = 37 .7 xgz = 339 eV Er=977 x42 =63 eV n1=)=n^,- r", " hc IZ0eV nm z=;-____;_=_ E",-E,,, En, -En, Fot ry= 4, ry =7, E,, - En, =903 eV -17'7 eY =ffi eY ' I = 2 19nn \ = 4, nt =2, 7=2.75 nsr \=4,nr=3, X= 4.N n!:r \=3, \ =1, 1=4.l2nllr ni=3, \=2,1=6.59 nm ni =2, nt =l ',1= 10.9 nm ( nx\ c;l6 (a) [email protected]= As{!: ), L = 3 A. Nomralization requires r =ltv4, a, =l fi "*.'( +\' = + 6 6 \ L . / r h2 8mU (a) *^=(?)" e='ipa'a'=(i)T,*'(TY=ii^'ao=|li-"Lu']=0.',uu
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(b) '=,,'.(Y), n=(?)u" v = ?ul *,1 rgon')d, = ?f -!-fY L'o \ L ) L\100tr) 'o I ll00E $n- dr=-l- 50trL 6 . I m0E\1 " \ 3 J J =; i#l'-(?)=]-s=0"" Yes: For large quantum nurnbers the pro,babitity approaches 1 n = 4 (c) GZI n = 4 Note that the n = 4 wavefunction has three nodes and is antisymmetric about the midpoint of the well. Gu After rearrangement, the Schrtidinger "q""do" is # = (p)U<rl -rtrt l -^ [email protected]=L^tzrz for the quantum oeciltator. Differealtiating y(x) = Qas-d'r 6v"" #=ao'vG)*c-"" # = -y# - * flr) - (2a xl&-o " = (za x)2
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