Lect16_Scattering1D - Lecture 16: Probability Current II...

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Lecture 16: Probability Current II and 1D Scattering Phy851 Fall 2009
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Continuity Equation This is the standard continuity equation, valid for any kind of fluid For energy eigenstates (stationary states), we need: This gives: Must have spatially uniform current in steady state (of course j can be zero) dt t x dP x j x j ) , ( ) ( ) ( = + ε dt t x d x j x j ρ 2 ) , ( ) ( ) ( = + dt t x d x j x j ) , ( 2 ) ( ) ( = + ) , ( ) , ( t x dt d t x j dx d = ) 0 , ( ) , ( 0 ) , ( x t x t x dt d = = d dx j ( x , t ) = 0 j ( x , t ) = j 0 d dt j ( x , t ) = 0 j ( x , t ) = j ( x ,0)
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Derivation of the probability current:
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Current of a plane wave For a plane wave we have: The corresponding probability current is: So for a plane wave, we find: ikx ae x = ) ( ψ j = i h 2 m −ψ ' ( ) density velocity 0 0 ) , ( v t x j r r ρ = ( ) m k a ik ik m a i h h 2 2 ) ( 2 = = This result is fairly intuitive
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Quantum Interference terms Consider a superposition of plane waves: The probability density is: The probability current density is: Note that the interference term in j ( x,t ) vanishes for k 2 = - k 1 This is always the case for Energy Eigenstates Currents are then purely additive There is still interference in the probability density due to the presence of left and right currents, just not in the probability current . x ik x ik e a e a x 2 1 2 1 ) ( + = ψ ( ) . . ) ( ) ( 2 1 2 2 2 1 1 2 c c e a a a a x x k k i + + + = ρ Interference Term Interference Term j ( x ) = a 1 2 h k 1 m + a 2 2 h k 2 m + a 1 a 2 e i ( k 2 k 1 ) x + c . c ( ) h ( k 1 + k 2 ) 2 m
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Lect16_Scattering1D - Lecture 16: Probability Current II...

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