Question

Q#5 An electron in a circular orbit about a proton can be described by classical mechanics if its angular momentum L is very much greater than h. Show that this condition is satisfied if the radius of the orbit r is very much greater than the Bohr radius a0 , i.e if

r>>a0 = 4πϵh2/e2me

Answer 1

According to Bohr, the quantization of the angular momentum of electrons is

$$L=mvr=n\hbar.$$

On the other hand, we know that the angular momentum, according to quantum mechanics, is

$$L=\sqrt{l(l+1)}\hbar=\sqrt{n^2-n}\cdot\hbar.$$

Of course, as we see, the two equation are different:

$$n>\sqrt{n^2-n}.$$

However, we can say that $n\approx\sqrt{n^2-n}$ for very large values of $n$. This criterion leads to $L>>\hbar$. Therefore, according to the first equation ($L=mvr=n\hbar$), we see that this is only possible for very large values of $r$.

Hence, to describe the motion of an electron in terms of classical mechanics, we need $r>>a_0$. As we saw above, this is only possible for very big $n$, which leads to $L>>\hbar$.