Chapter6_33 - h r s . The transmission probability, P T ,...

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PHY3063 R. D. Field Department of Physics Chapter6_33.doc University of Florida Finite Step Potential (V 0 > E) x x 0 = 0 E Step Potential Classically Forbidden V 0 “Left Region” “Right Region” Boundary Conditions: The wavefunction ( i.e. probability amplitude) must be continuous and have a continuous 1 st derivative at the interface between the left and right region. ) 0 ( ) 0 ( = = = x x R L ψ implies R L L A B A = + 0 0 = = = x R x L dx d dx d implies R L L A ikB ikA κ = We can express B L and A R in terms of A L with L L A ir ir B + = 1 1 and L R A ir A + = 1 2 where 1 0 = = E V k r . Reflection and Transmission Probabilities: The reflection probability, P R , is defined to be the flux of particles traveling to the right in the right hand region ( incident flux ) divided by the flux of particles traveling to the left in the right hand region. Namely 1 1 1 1 1 | | | | 2 2 = + + = = = ir ir ir ir A B j j P L m k L m k L L R h
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Unformatted text preview: h r s . The transmission probability, P T , is defined by P T = 1- P R = 0 . Note that since no particles are created or destroyed P R + P T = 1 . Probability Density: The probability density is given by + + + = = ) 2 sin( 1 2 ) 2 cos( 1 ) 1 ( 2 2 | | | ) , ( | ) ( 2 2 2 2 2 kx r r kx r r A t x x L L L krx L R R e r A t x x 2 2 2 2 1 4 | | | ) , ( | ) ( + = = , where 2 2 h mE k = and 1 = E V r and = kr . Step Potential: V > E-8-6-4-2 2 x Probability Density Interference minimum! Non-Zero probability to be in the classically forbidden region!...
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