Question

For lecture #3, between the archived Lecture Notes #1 (PDF), Sections 1-3 and the Solutions (PDF) I managed to find the equation used to calculate the "Bohr radius" of an atom:
\[r_{n}=n^{2}h^{2} \epsilon _{o}/Z \pi m_{e}e^{2}\]

According to Wikipedia, the formula to calculate the Bohr radius is: \[r_{n}=n^{2}ℏ^{2}/Z k_{e} e^{2} m_{e}\] \[ℏ=h/2 \pi\] \[k_{e}=1/4 \pi \epsilon _{o}\]
Substitution does indeed yield the OCW equation. Could someone walk me through the units in these equations? Everything somehow works out to meters, but there is a lot going on and I'm not exactly sure how.

Answer 1

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Answer 2

Lets take the OCW equation first. \[r_n=\frac{n^2\hbar^2\varepsilon_0}{Z\pi e^2m_e}\]

first things to note are that the variables \(n\), \(\pi\), \(Z\) are all dimensionless meaning that they have no units, so we can ignore them leaving \[r_n=\frac{\hbar^2\varepsilon_0}{e^2m_e}\](\hbar\) has a factor of \(2\pi\) in it, but we wont worry about this. Lets identify what each of the variables are and what their units are, and we will convert the units into the base SI units. We will also put them into dimensional units for dimensional analysis, where the values M, T, L, and A for mass, time, length, and current respectively. Thus

Planks constant with units is \[\hbar=Js=\frac{Kgm^2}{s}=\frac{ML^2}{T}=ML^2T^{-1}\] Permittivity of free space \[\varepsilon_0=F/m=\frac{s^4A^2}{m^2Kg}=\frac{T^4A^2}{L^3M}=T^4A^2L^{-3}M^{-1}\]electronic charge \[e=C=As=AT\]and electron mass \[m_e=Kg=M\]

Having done this we can plug the dimensions into the Bohr radius equation to give

\[r=\frac{\hbar^2\varepsilon_0}{e^2m_e}=\frac{(ML^2T^{-1})^2(T^4A^2L^{-3}M^{-1})}{(AT)^2(M)}\]and hence
\[r=\frac{(M^2L^4T^{-2})(T^4A^2L^{-3}M^{-1})}{A^2T^2M}=\frac{MLT^2A^2}{A^2T^2M}=L\]Therefore the Bohr radius has units of length.

There is a lot of terms here in the equation, but it is just a matter of keeping your head and not loosing terms or powers (like I did on a few occasions when writing this answer). Dimensional analysis is a very powerful tool for checking to see if an equation is physically correct, since if the units on one side of teh equation do not balance the units (or dimensions) of the other side, then either you have made an error, or the equation is incorrect in the first place (the latter is coming about if you have derived the equation and made an error).