Question

A free neutron (that is, a neutron on its own rather than in a nucleus) is not a stable particle. Its average lifetime is 15 min, after which it decays into a proton, an electron, and an antineutrino. Use the energy-time uncertainty principle \([\mathrm{Eq} .(28-3)]\) and the relationship between mass and rest energy to estimate the inherent uncertainty in the mass of a free neutron. Compare with the average neutron mass of \(1.67 \times 10^{-27} \mathrm{kg} .\) (Although the uncertainty in the neutron's mass is far too small to be measured, unstable particles with extremely short lifetimes have marked variation in their measured masses.)

Short answer

Answer: No, the inherent uncertainty in the mass of a free neutron is much smaller (about 3.9 x 10^{-8} times smaller) than the average neutron mass. Therefore, for neutrons, the uncertainty in mass is far too small to be measured.

Step-by-step solution

Calculate the uncertainty in energy

Using the energy-time uncertainty principle, we can calculate the uncertainty in energy as follows: \((\Delta E)(\Delta t) \geq \hbar/2\) We are given the average lifetime (uncertainty in time) of a neutron as 15 minutes, so let's convert it to seconds: \(\Delta t = 15 \times 60 = 900 \, s\) Now, we can find the uncertainty in energy: \((\Delta E) = \frac{\hbar/2}{\Delta t} = \frac{1.054571800\times10^{-34} \, J \cdot s/2}{900 \, s} = 5.86 \times 10^{-38} \, J\)

Find the uncertainty in mass

We can use the relationship between mass and rest energy to find the uncertainty in mass: \(E = mc^2\) or \(\Delta E = (\Delta m)c^2\) where \(\Delta E\) is the uncertainty in energy and \(\Delta m\) is the uncertainty in mass. We've found \(\Delta E\) in step 1, so now we can solve for the uncertainty in mass: \(\Delta m = \frac{\Delta E}{c^2} = \frac{5.86 \times 10^{-38} \, J}{(3\times10^8\, m/s)^2} = 6.51 \times 10^{-35} \, kg\)

Compare the uncertainty in mass to the average neutron mass

The inherent uncertainty in the mass of a free neutron was found to be \(6.51 \times 10^{-35} \, kg\). Compare this with the average neutron mass of \(1.67 \times 10^{-27} \, kg\): \(\frac{\Delta m}{\text{average neutron mass}} = \frac{6.51 \times 10^{-35} \, kg}{1.67 \times 10^{-27} \, kg} \approx 3.9 \times 10^{-8}\) Thus, the inherent uncertainty in the mass of a free neutron is much smaller (about 3.9 x 10^{-8} times smaller) than the average neutron mass. This means that for neutrons, the uncertainty in mass is far too small to be measured. However, as mentioned in the exercise, unstable particles with extremely short lifetimes may have marked variation in their measured masses, even if the uncertainty in the mass is small.