(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The probability for a particle of energy E<<V_{0}to penetrate a potential barrier of height V_{0}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 V_{0}and 2v_{0}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 P_{right}>P_{left}. 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.

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# Homework Help: Probability of penetrating a potential barrier

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