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Lect13_Current

# Lect13_Current - Quiz#5 1 Consider a particle of mass m...

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Quiz #5 1. Consider a particle of mass m , incident on a step potential of height V 0 , located at x =0 . The energy of the particle is above the step height, E > V 0 . Make the ansatz: Determine the wave vectors k 1 and k 2 in terms of E and V 0 . Solve for r and t using the appropriate boundary conditions ! " # > < + = \$ 0 0 ) ( 2 1 1 x te x re e x x ik x ik x ik % Quantum Reflections Consider scattering from a step potential, with an Energy above the barrier height: E x V 0 I II E x k i x k i I re e x 1 1 ) ( ! + = " x k i II te x 2 ) ( = ! h mE k 2 1 = ( ) h 0 2 2 V E m k ! = t r II I = + = 1 ) 0 ( ) 0 ( ! ! t ik r ik II I 2 1 ) 1 ( ) 0 ( ) 0 ( = ! " = " # # ) 1 ( 1 2 1 r k k r ! = + r k k r k k 1 1 2 2 ! = + 2 1 2 1 k k k k r + ! = 2 1 1 2 k k k t + = ( ) 2 2 1 2 2 2 1 2 1 2 2 | | k k k k k k r R + + ! = = Probabilities: ( ) 2 2 1 2 1 2 4 | | k k k t T + = = ( ) 1 2 5 2 2 1 2 2 2 1 2 1 ! + + " = + k k k k k k T R

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Probability Current Clearly we must have R + T = 1 So R = | r | 2 and T = | t | 2 must not be correct The problem is that in this case the velocity in region I is not the same as in II This suggests that we need to think in terms of a probability current
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Lect13_Current - Quiz#5 1 Consider a particle of mass m...

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