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Schrodinger equation for step potential

  1. Nov 29, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider the one-dimensional Schrodinger equation for the step potential, that is for U(x) = 0 for x<0, and gif.gif for gif.gif , gif.gif . Consider a particle with mass m and energy E < U. Assume the particle is initially at x<0.

    a) Calculate the penetration depth Δx at which the probability density of the transmitted wave "decayed" to half the value that it has at x=0.

    b) Apply your result of part a) to an electron inside a metal block for which the work function is 4 eV.

    c) Apply your resilt of part a) to a "macroscopic" particle with mass m=10^-15 kg and velocity v=10^-2 m/sec. Assume a barrier height that is 1.5 as high as the kinetic energy of the particle.

    2. Relevant equations


    3. The attempt at a solution
    I have no idea where start with this one, I have a clue on how to find the transmission and reflection probabilities but I am not sure this is relevant for finding Δx. In this question are we suppose to use the equation Δp Δx >= ħ/2π ?
     
  2. jcsd
  3. Nov 29, 2015 #2
    The question does not ask you to calculate transmission probabilities. Just write down the form of the wave function for x > 0.
     
  4. Nov 29, 2015 #3
    For x>0 I believe the wave function would be u(x) = Ae^(ikx)
     
  5. Nov 29, 2015 #4
    So imaginary k will give you the decay. Adjust with a factor ln(2).
     
  6. Nov 29, 2015 #5
    I don't quite understand what you mean by adjust by a factor of ln(2). Also still not seeing how k will help find penetration depth Δx
     
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