Chapter6_6 - 2 1 2 2 2 t x x V x t x m i t t x h h...

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PHY3063 R. D. Field Department of Physics Chapter6_6.doc University of Florida Probability Flux Probability Flux: Look at the time dependence of the probability that the particle lies in the region x 1 x x 2 Ψ Ψ = 2 1 ) , ( ) , ( ) , , ( * 2 1 x x dx t x t x t x x P . We see that ) , ( ) , ( 2 2 2 2 1 ) , ( ) , ( ) , ( ) , ( ) , , ( 2 1 * * 2 * 2 2 2 * * 2 * 2 2 2 2 2 * * * 2 1 2 1 2 1 2 1 2 1 t x j t x j dx x x x im dx x x im dx V x m V x m i dx t x t t x t t x t x dt t x x dP x x x x x x x x = Ψ Ψ Ψ Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ + Ψ Ψ + Ψ Ψ = Ψ Ψ + Ψ Ψ = h h h h h where Ψ Ψ Ψ Ψ x t x t x x t x t x im t x j ) , ( ) , ( ) , ( ) , ( 2 ) , ( * * h and were I used Schrödinger’s equation Ψ + Ψ = Ψ ) , ( ) ( ) , ( 2 1 ) , ( 2 2 2 t x x V x t x m i t t x h h Ψ + Ψ = Ψ ) ,
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Unformatted text preview: ( ) ( ) , ( 2 1 ) , ( * 2 * 2 2 * t x x V x t x m i t t x h h Continuity Equation: Similarly it is easy to show that ) , ( ) , ( = ∂ ∂ + ∂ ∂ x t x j t t x ρ where ρ (x,t) is the probability density 2 ) , ( ) , ( t x t x Ψ = Traveling Plane Wave: For a traveling plane wave h / ) ( ) ( ) , ( Et x p i t kx i x Ae Ae t x − − = = Ψ ω and 2 ) , ( A t x = and ) , ( | | | | ) , ( 2 2 t x v A m p A m k t x j x x = = = h (flux) = (velocity) x (probability) x x 2 x 1 j(x 1 ,t) j(x 2 ,t) P(x 1 ,x 2 ,t) Probability Flux!...
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