Why can a beam of electron and not scatter with a proton to form a H atom?

Question

*Consider aimimg a beam of free electrons towards an isolated free proton . When they scatter, an electron and a protons cannot combine to produce a H-atom because

A) Energy Conservation B)without simultaneously releasing energy in the form of radiation C) Momentum conservation D) Angular momentum conservation

NCERT Exemplar

What I believe is if a proton and electron should combine to form H, then the binding energy of products should be greater than the Binding energy of reactants. But this is just my presumption. Some of my peers argue it might be because of acceleration due to respective magnetic fields caused on each other...

Could someone give me a clear explanation on this?

Answer 1

Consider the center of momentum frame. In this frame, the electrons with mass $m_{e}$ have energy $E_{e}$ and momentum $\mathbf{k}$. The proton has mass $m_{p}$, energy $E_{p}$, and momentum $-\mathbf{k}$.

Afterwards we are supposed to be left with a hydrogen atom with energy $m_{H}$ and momentum $\mathbf{k'} = \mathbf{0}$. Writing out energy conservation in this frame gives

$E_{e} + E_{p} = \sqrt{m_{e}^{2} + \mathbf{k}^{2}} + \sqrt{m_{p}^{2} + \mathbf{k}^{2}} = m_{H}\, . $

We know that $m_{H} < m_{e} + m_{p}$ because of binding energy so this reaction isn't possible unless there is electromagnetic radiation emitted. So (a) & (b) are correct.