∙The neutral π^∘ meson is an unstable particle produced in...

$\bullet$ The neutral $\pi^{\circ}$ meson is an unstable particle produced in high-energy particle collisions. Its mass is about 264 times that of the electron, and it exists for an average lifetime of $8.4 \times 10^{-17}$ s before decaying into two gamma-ray photons. Assuming that the mass and energy of the particle are related by the Einstein relation $E=m c^{2},$ find the uncertainty in the mass of the particle and express it as a fraction of the particle's mass.

Key Concepts

Mass–Energy Equivalence

A cornerstone of special relativity, mass–energy equivalence is encapsulated in Einstein’s equation E = m c², which shows that mass can be converted into energy and vice versa. This principle allows one to relate the energy uncertainties of a particle, as determined by its lifetime, to uncertainties in its mass, providing a framework for understanding phenomena in high-energy physics.

Unstable Particles and Decay Width

In particle physics, unstable particles decay after a finite lifetime, and this lifetime is inversely related to the uncertainty in their energy states (or decay width). The shorter the lifetime of an unstable particle, the greater the inherent uncertainty in its energy (and consequentially its mass), which is a direct application of the uncertainty principle. This concept is crucial in analyzing the properties and behaviors of particles produced in high-energy collisions.

Energy-Time Uncertainty Principle

This quantum mechanical principle states that the uncertainty in energy (?E) multiplied by the uncertainty in time (?t) is on the order of the reduced Planck constant (?). It highlights that there is a fundamental limit to the precision with which energy and time can be simultaneously known, and is particularly significant in describing the energy spread (or width) of unstable particles that exist only for short periods of time.

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