Answer to Question #123229 in Quantum Mechanics for asghar ali

Question

Q#1 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

Let us consider the circular motion of electron, so the acceleration is

$a =\dfrac{v^2}{r}.$ The force due to which the motion exists is $F = \dfrac{1}{4\pi\varepsilon_0}\cdot\dfrac{e^2}{r^2}.$ Therefore, $v^2 = \dfrac{e^2}{4\pi\varepsilon_0m_er}.$

The angular momentum is $L=m_evr .$

If $L \gg h,$ or $L^2\gg h^2, \;\;\; m_e^2v^2r^2 \gg h^2, \;\;\; m_e^2\cdot \dfrac{e^2}{4\pi\varepsilon_0m_er}\cdot r^2 \gg h^2, \;\;\; r \gg \dfrac{h^24\pi\varepsilon_0}{m_ee^2}$ .