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**Problem**

An electron beam is sent through a potential barrier [tex] 4.5 \AA[/tex] long. The transmission coefficient exhibits a third maximum at [tex]E = 100 \text{eV}[/tex]. What is the height of the barrier?

**Solution**

Answer: [tex]95.8219 \text{eV}[/tex]

We know that the transmission coefficient reaches a maximum of [tex]1[/tex] only when the energy of the electron beam, [tex]E = 100 \text{eV}[/tex], exceeds the potential barrier [tex]V[/tex], so [tex]E > V[/tex]. Now, we know that in this case, the transmission coefficient attains a maximum where

[tex]\sin^2 (2 k_2 a) = 0 \Leftrightarrow 2 k_2 a = n \pi [/tex]

Now, in our case, [tex]n = 3[/tex], because we have information regarding the third maximum. Substituting

[tex]k_2^2 = \frac{2m(E-V)}{\hbar^2}[/tex]

we have that [tex]V = 95.8219 \text{eV}[/tex]. [tex]\blacksquare[/tex]

Is my above work correct?