How Did They Derive Quantized Energy Levels in a 1D Atom?

[Pengwuino]

The book I am using did a simple little algebra and it is showing the quantized energy levels of an electron in a one-dimensional atom of width 0.1nm.

They did this:

$$\begin{array}{l}<br /> E_n = n^2 \frac{{h^2 }}{{8ml^2 }} \\ <br /> E_n = n^2 \frac{{h^2 c^2 }}{{8mc^2 l^2 }} \\ <br /> E_n = n^2 \frac{{(1239.8eV*nm)^2 }}{{(8)(0.511*10^6 eV)(0.1nm)^2 }} \\ <br /> \end{array}$$

But where did the c^2 go on top?

 

[Physics Monkey]

The $$c$$ has been put with the $$h$$ to give something with units of $$[E][T]\frac{[L]}{[T]} = [E][L]$$, and numerically $$h c = 1239.8$$ eV nm. This is one of those common estimation factors, although I remember it as $$\hbar c \sim 200$$ eV nm.

 

[Pengwuino]

Oh ok i get why they did this. They have a conversion at the front of the book for hc = (1239.8eV*nm) that i didn't notice or make the connection to.