Before the neutrino hypothesis, the beta decay process was thought to be the transition,

np+e¯

If this was 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.

Hint: Use the law of conservation of momentum.
Step 1: Conserve the momentum.
Before β-decay, the neutron is at rest. Hence, En=mnc2,pn=0
pn=pp+pe
orpp+pe=0pp=pe=p
Also,Ep=(mp2c4+pp2c2)12,Ee=(me2c4+pe2c2)12
Step 2: Conserve the energy.
From the conservation of energy,
(mp2c4+p2c2)12+(me2c4+p2c2)12=mnc2
mpc2936MeV,mnc2938MeV,mec2=0.51MeV
Step 3: Find the energy of the electron and the proton.
Since the energy difference between n and p is small, pc will be small, pc<<<mpc2, while pc may be greater than mec2.
mpc2+p2c22mp2c4mnc2-pc
Tofirstorderpcmnc2-mpc2=938MeV-936MeV=2MeV
This gives the momentum of proton or neutron. Then,
Ep=(mp2c4+p2c2)12=9362+22
936MeV
Ee=(me2c4+p2c2)12=(0.51)2+22
2.06MeV