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Probability for Quantum tunnelling

  1. Dec 19, 2012 #1
    1. The problem statement, all variables and given/known data
    A particle with the energy E < V[itex]_{0}[/itex] (V[itex]_{0}[/itex] > 0) moves in the potential V(x) = 0, x<0 ; V(x)= V[itex]_{0}[/itex], 0<x<d and V(x)= 0, x>d. Measure the probability that the particle will tunnel through the barrier by calculating the absolute value of the ratio squared, |[itex]\Psi[/itex](d)/[itex]\Psi[/itex](0)|[itex]^{2}[/itex] between the values of the wave function at x=d and x = 0

    Calculate the probability for an electron, when V[itex]_{0}[/itex]- E=1 eV and d = 1 Å.


    2. Relevant equations
    [itex]\Psi[/itex](x) = ae[itex]^{\kappa*x}[/itex]+be[itex]^{-\kappa*x}[/itex], [itex]\kappa[/itex] = [itex]\sqrt{2m( V_{0}-E)/\hbar^{2}}[/itex] for E<V[itex]_{0}[/itex]


    3. The attempt at a solution

    Firstly I get:

    [itex]\kappa[/itex] = [itex]\sqrt{2m(1)/\hbar^{2}}[/itex] for E<V[itex]_{0}[/itex]

    However, the problem is with this wave function:

    [itex]\Psi[/itex](x) = ae[itex]^{\kappa*x}[/itex]+be[itex]^{-\kappa*x}[/itex]

    In order to calculate the ratio, |[itex]\Psi[/itex](d)/[itex]\Psi[/itex](0)|[itex]^{2}[/itex], I think I have to define a and b somehow, but I don't know where to start.

    Thanks!
     
  2. jcsd
  3. Dec 20, 2012 #2
    You have to solve Shroedinger's equation in all three regions. Then you need to apply the appropriate boundary conditions.
     
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