A free neutron (that is, a neutron on its own rather than in...

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.)

Key Concepts

Energy-Time Uncertainty Principle

The energy-time uncertainty principle is a fundamental concept in quantum mechanics which states that the uncertainty in energy (?E) multiplied by the uncertainty in time (?t) is on the order of or greater than the reduced Planck constant (?). This principle implies that if a system exists for only a short time, there is an inherent spread in its energy, meaning that unstable particles with brief lifetimes exhibit an uncertainty in their measured energy, and consequently, their mass.

Mass-Energy Equivalence

Mass-energy equivalence, expressed by Einstein's famous equation E = mc², establishes that mass and energy are interchangeable. This concept is crucial in converting an uncertainty in a particle's energy to an uncertainty in its mass by dividing the energy uncertainty by the speed of light squared.

Unstable Particle Decay

Unstable particle decay refers to the process by which particles that are not in a stable state transform into other particles over time. Free neutrons, for example, decay into protons, electrons, and antineutrinos. The lifetime of these unstable particles is directly related to the uncertainty in their energy and mass, making the decay process an important consideration in high-energy physics experiments.

Particle Lifetime and Mass Uncertainty

There is a direct connection between the short lifetime of an unstable particle and the magnitude of uncertainty in its mass. Short-lived particles, such as free neutrons with lifetimes on the order of minutes or less, inherently have a wider spread in their energy and mass measurements. This is a consequence of the energy-time uncertainty principle, which quantitatively relates the brief existence of the particle to the variation in its mass.

Transcript

Please give Ace some feedback