Before the neutrino hypothesis,the beta decay process was thought to be the transition $n \to p + e^-$. If this were true,show that if the neutron was at rest,the proton and electron would emerge with fixed energies and calculate them. Experimentally,the electron energy was found to have a large range.

(N/A) According to the law of conservation of energy,the total energy of the neutron at rest must equal the sum of the energies of the proton and the electron:
$E_n = E_p + E_e$
$m_n c^2 = \sqrt{p^2 c^2 + m_p^2 c^4} + \sqrt{p^2 c^2 + m_e^2 c^4}$
Rearranging the terms:
$m_n c^2 - \sqrt{p^2 c^2 + m_p^2 c^4} = \sqrt{p^2 c^2 + m_e^2 c^4}$
Squaring both sides:
$(m_n c^2)^2 + (p^2 c^2 + m_p^2 c^4) - 2 m_n c^2 \sqrt{p^2 c^2 + m_p^2 c^4} = p^2 c^2 + m_e^2 c^4$
$m_n^2 c^4 + m_p^2 c^4 - m_e^2 c^4 = 2 m_n c^2 \sqrt{p^2 c^2 + m_p^2 c^4}$
Squaring again and solving for $p^2 c^2$,we find that $p$ is uniquely determined by the masses of the particles $(m_n, m_p, m_e)$. Since $p$ is fixed,the energies $E_p = \sqrt{p^2 c^2 + m_p^2 c^4}$ and $E_e = \sqrt{p^2 c^2 + m_e^2 c^4}$ must also be fixed values. This contradicts the experimental observation of a continuous energy spectrum for beta particles.