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 $15 \mathrm{~min}$, after which it decays into a proton, an electron, and an antineutrino. Use the energy-time uncertainty principle [Eq. $(28-3)]$ and the relationship between mass and rest energy to estimate the inherent uncertainty in the mass of a free neutron. Compare to the average neutron mass of $1.67 \times 10^{-27} \mathrm{~kg}$. (While 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

This principle, a cornerstone in quantum mechanics, states that the product of the uncertainty in energy and the uncertainty in time is bounded by a value on the order of the reduced Planck constant (?). It implies that for processes occurring in very short times, there is a correspondingly large uncertainty in the energy involved. This is particularly important for unstable particles, where the finite lifetime results in an intrinsic uncertainty in energy measurements.

Mass-Energy Equivalence

Mass-energy equivalence, encapsulated by Einstein’s equation E=mc², establishes that mass and energy are interchangeable. This principle allows us to relate the uncertainty in energy derived from the energy-time uncertainty principle directly to an uncertainty in mass. In other words, the shorter the lifetime of a particle, the greater the spread in its measured rest mass, as indicated by converting energy uncertainty into mass uncertainty via the c² factor.

Particle Lifetime and Inherent Mass Uncertainty

In the context of unstable particles, the finite lifetime (or decay time) introduces an inherent uncertainty in the particle’s energy and thus its mass. A longer lifetime results in a very small uncertainty, while particles with extremely short lifetimes show a significant spread in their mass measurements. This relationship is key to understanding how quantum mechanical effects manifest in the properties of unstable particles.

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