Electron beam in a Potential Barrier

[Domnu]

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

Solution
Answer: $$95.8219 \text{eV}$$

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

$$\sin^2 (2 k_2 a) = 0 \Leftrightarrow 2 k_2 a = n \pi$$​

Now, in our case, $$n = 3$$, because we have information regarding the third maximum. Substituting

$$k_2^2 = \frac{2m(E-V)}{\hbar^2}$$​

we have that $$V = 95.8219 \text{eV}$$. $$\blacksquare$$

Is my above work correct?

 

[Jhero]

Yes, your solution is correct. You have correctly used the equation for the transmission coefficient and solved for the potential barrier height using the given information. Well done!