# soltn8 - Homework 8 Solutions 7-2(a To th leftofthe step...

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Homework 8 Solutions 7-2 (a) To th€ leftofthe step the partide is free with kinetic energy E and corresponding wavenurnber t1 =(T)u" , Y(x)= Aee" +gs-*'r r<0 To the right of the step the kinetic energy is reduced to E - U and the wavenumber is ' | 2n(E-U)1v2 now x2 =L D, I y?)=Cekz' +De-*r' .t>0 with D = 0 for waves incident on the step from the left. At x = 0 both rg and !! must be continuous: |z(0) = A + B = C d4 =*,ra-at=ik"c dx lo (b) Elininatingcgive s t+n=f<t-uor ^[*-t)=r(f *rJ.rnu,, . lBl2 Gtlkz-tl2 1k, -|kr12 ! =L-J =-:---:--:----- -' lAl (k1lkz+\z (\+k)2 T=1-R= 4krkz - (kt + k")' (c) As E -r U, k, -+ 0, and R -+ 1, T -r 0 (no transmission), in agreement with the result for any enerry E< U. For E-+-, t, -+ k, and R-+0, T-+l (perfect transmission) suggesting correcdy that very energetic partides do not see the step and so are unaffected by it. r-rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr r TL,c n-+1'drv', coekic;ul -is R=(*;)'"H* u,='La^t/r')L d^.d- L2= f2^(E-u)/h7v'. Lla" ca^ rrt.rr;\q l(a kr {c.*s o{ K, ral'o .- kv, r E:!rl (= '7kr= L El tl= (i+)" T= r-R E = l,?{t =) i- t,tr/-[11 r= L- F[-j o'loa Lc. .

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soltn8 - Homework 8 Solutions 7-2(a To th leftofthe step...

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