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Inconsistent Reflection and Transmission values-step potential & E < V

  1. Apr 8, 2014 #1
    I can show that for a step potential and E < V, that the wave function is fully reflected and has no transmission into the potential interval (interval 2), x =0 at interval boundary, by

    Wave equation for interval 1: Ψ1 = A1eik1x + B1e-ik1x

    Wave equation for interval 2: Ψ2 = A2eκ2x

    where B1 = A1 (k1 - iκ2)/(k1 + iκ2)

    and

    A2 = A1 (2k1)/(k1 + ik2)

    Further B1/A1 = (k1 - iκ2)/(k1 + iκ2)

    |B1|/|A1| = (k12 - iκ22)1/2/(k21 + iκ22)1/2 = 1

    and |B1| = |A1| by which we can conclude full reflection.

    However, if I calculate the reflection and transmission value by the probability current density, I find,

    For interval 1

    j1x = ħk1/m (|A1|2 -|A1|2|ρ(E)|2)

    where

    |ρ(E)| = (k1 - iκ2)/(k1 + iκ2)

    But with this value for |ρ(E)| the probability current density in interval 1 does not match the probability current density of the incident wave, i.e.

    jincidentx = ħk1/m (|A1|2)

    But these must match for full reflection.

    Not sure how I am getting this result.
     
  2. jcsd
  3. Apr 13, 2014 #2
    I'm not entirely sure I got your question, but if E<V then you should be getting full reflection, however as best as I can imagine it, not all of them reflect at the barrier (step), some of them go a little further before coming back. So you will find some probability density out past the step and correspondingly you wont find all of your particles on the same side as your source. But, you will also find that they all reflect (eventually).
     
  4. Apr 13, 2014 #3
    Thank you. I believe my confusion relates to the interpretation of

    j1x = ħk1/m (|A1|2 -|A1|2|ρ(E)|2)


    I see when |ρ(E)|2 = 1

    then

    j1x = ħk1/m (|A1|2 -|A1|2|)

    reflects the identical probability current density flux proceeding to the right and then to the left, i.e. full reflection.
     
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