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Schrodinger equation for potential drop

  1. Oct 18, 2011 #1

    cep

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    1. The problem statement, all variables and given/known data

    Hello! I'm looking at a situation where there is a finite potential Vo for x<0, but zero potential for x>0. For a particle moving from left to right, I'm wondering what coefficients for the solution to the Schrodinger equation are equal to zero, and also how to prove that there is reflection even for a potential drop. hbar is h/(2π).

    2. Relevant equations

    Time-independent Schrodinger equation

    3. The attempt at a solution

    Here's what I'm thinking:

    For x<0, ψ(x) = 1/√(k0)(Arighteik0x+Alefte-ik0x)
    where k0 = √[2m(E+Vo)/hbar2]

    For x>0, ψ(x) = 1/√(k1)(Brighteik1x+Blefte-ik1x)
    where k1 = √[2m(E)/hbar2]

    I think that Bleft is zero, as there is nothing to cause reflection past the potential drop. How can I prove this, and that Aleft is non-zero (ie, potential drop produces reflection)? I know that the wave function and its derivative must be continuous at x=0-- is that sufficient?

    Thank you!
     
  2. jcsd
  3. Oct 18, 2011 #2

    cep

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    Sorry, it lost the sub/superscripts. Here are the rewritten equations

    For x<0, ψ(x) = 1/√(k0)*(Arighteik0x+Alefte-ik0x)
    where k0 = √[2m(E+Vo)/hbar2]

    For x>0, ψ(x) = 1/√(k1)*(Brighteik1x+Blefte-ik1x)
    where k1 = √[2m(E)/hbar2]
     
  4. Oct 18, 2011 #3

    vela

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    There's nothing to prove. Just state the reason you gave for why Bleft=0. That's enough.
     
  5. Oct 18, 2011 #4

    cep

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    ... really? I'm skeptical-- the problem asks for a proof.
     
  6. Oct 18, 2011 #5

    cep

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    *proof that a potential drop causes reflection, not proof of which coefficient is zero.
     
  7. Oct 19, 2011 #6
    If you know that B-left is 0 then you know what the form of the equation is on the right side (x>0)

    Now its just a boundary condition problem. you need to find A-left and A-right so that the two equations match up at x = 0. Set the two equations equal to each other at x = 0 and also set their derivates to be equal at x = 0. Then its just 2 equations 2 unknowns.

    You should find that for the two equations to match up A-left cannot be 0 and there's your proof of reflection
     
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