lecture_12_07c

lecture_12_07c - Exam 1 grades Exam 1(ave=182.8-13.0 14 12...

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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 1 Exam 1 - grades Exam 1  (ave=182.8+/-13.0) 0 2 4 6 8 10 12 14 170 175 180 185 190 195 200 More grade Each bin represents the number of students with grades less than or equal to the indicated number but greater than the next bin grade, e.g. 5 had grades >190 but < or = to 195.
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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 2 Steps and barriers Consider stationary plane waves ~exp(ikx) Step potential: U(x)=0 for x<0 =U 0 for x>0 impulsive force -U 0 /∆x 0 0 U 0 x - η 2 2 m d 2 dx 2 y ( x ) + U ( x ) y ( x ) = Ey ( x ) Time independent 1-dim Schrödinger equation
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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 3 Barrier penetration For x < 0 h 2 k 1 2 2 m = E or k 1 = 2 mE h and y e ik 1 x But for x > 0 h 2 k 2 2 2 m = E - U 0 2 cases k 2 = 2 m ( E - U 0 ) h for E > U 0 and y e ik 2 x or k 2 = i 2 m ( U 0 - E ) h for E < U 0 and y e - | k 2 | x Note the exponential falloff for 2nd case Analog: light wave from air to glass or other medium Need reflected waves to match at x=0
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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 4 Barrier wave functions So for E > U 0 at x < 0 can write y ( x ) = Ae ik 1 x + Be - ik 1 x at x > 0 have y ( x ) = Ce ik 2 x forward moving Matching ψ (0) and d ψ /dx| 0 relates A, B to C B/A = (k 1 -k 2 )/(k 1 +k 2 ) and A=C (k 1 +k 2 )/2k 1 Reflection probability is (B/A) 2 =R and Transmission coefficient is T=1-R For E < U 0 at x < 0 can write y ( x ) = Ae ik 1 x + Be - ik 1 x but at x > 0 have y ( x ) = Ce - | k 2 | x forward decaying
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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 5 Barrier waves in time Multiply ψ by e -i ϖ t which turns each term into traveling plane wave , A and C parts to the right, B to the left - reflected For decaying case partial penetration into barrier as allowed by uncertainty in x p. Quantum Scattering demo of traveling waves & barriers or wells
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