The neutral π^0 meson is an unstable particle produced in...

The neutral $\pi^{0}$ meson is an unstable particle produced in highenergy particle collisions. Its mass is about 264 times that of the electron, and it exists for an average lifetime of $8.4 \times 10^{-17} \mathrm{~s}$ before decaying into two gamma-ray photons. Assuming that the mass and energy of the particle are related by the Einstein equation $E=m c^{2},$ find the uncertainty in the mass of the particle and express it as a fraction of the particle's mass.

Key Concepts

Heisenberg Uncertainty Principle

This principle sets a fundamental limit to the accuracy with which pairs of physical properties, such as energy and time, can be known simultaneously. In the context of particle physics, a very short lifetime of an unstable particle implies a significant uncertainty in its energy. This energy uncertainty, derived using the time-energy uncertainty relation (?E?t ? ?), is crucial in quantifying the 'width' or instability of resonances in high-energy experiments.

Mass-Energy Equivalence

Mass-energy equivalence, encapsulated by Einstein’s famous equation E = mc², establishes a direct relation between the energy and the mass of a particle. By equating the uncertainty in energy (found via the uncertainty principle) to an uncertainty in mass (using ?m = ?E/c²), one can determine how the inherent energy spread of an unstable state translates into a spread or uncertainty in its mass.

Particle Decay and Resonance Width

Particles with short lifetimes, such as those encountered in high-energy collisions, decay rapidly into other particles. The brief existence of these unstable particles results in a significant energy uncertainty, often referred to as the resonance width in particle physics. This width is directly related to the particle’s lifetime and is indicative of the instability and the probabilistic nature of quantum decay processes.

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