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Probability of penetrating a potential barrier

  1. Dec 5, 2017 #1
    1. The problem statement, all variables and given/known data
    The probability for a particle of energy E<<V0 to penetrate a potential barrier of height V0 and width d is approximately [tex]\frac{16E}{V_0}exp\left[\frac{-2d\sqrt{2m(V_0-E)}}{\hbar}\right][/tex].
    An electron moves between two potential barriers of height V0 and 2v0 that are of widths 2a and a respectively . For what range of energies is it more likely to exit through the right barrier than through the left?

    2. Relevant equations
    Given in question.

    3. The attempt at a solution
    [tex]P_\text{left}=\frac{16E}{V_0}exp\left[\frac{-4a\sqrt{2m(V_0-E)}}{\hbar}\right][/tex]

    [tex]P_\text{left}=\frac{16E}{2V_0}exp\left[\frac{-2a\sqrt{2m(2V_0-E)}}{\hbar}\right][/tex]

    I have attempted to solve the inequality Pright>Pleft. Due to the two square roots I had to square the entire expression twice to obtain an expression without surds, leaving me to solve:
    [tex]9E^2+\left(\frac{10\hbar^2}{4a^2}(\ln{2})^2-12V_0\right)E+4V_0^2-\frac{3\hbar^2}{a^2}(\ln{2})^2V_0>0[/tex]

    I know I can solve this to find a range of energies, however I think there should be an easier way to solve it, since it is an exam question with limited time.
     
  2. jcsd
  3. Dec 5, 2017 #2

    PeroK

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    Just looking at the maths - why not just cancel the ##\frac{16E}{V_0}## before you start?
     
  4. Dec 5, 2017 #3
    I did that as part of the maths which led to the final inequality, I just didnt show the steps as its just rearranging the original inequality.
     
  5. Dec 5, 2017 #4

    PeroK

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    Yes, of course! The clue is ##E << V_0## I think. That suggests a binomial expansion to me.
     
  6. Dec 5, 2017 #5
    I hadn't thought of that at all. I'll give it a try, thanks!
     
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