Many particles are far too short-lived for their lifetimes...

Many particles are far too short-lived for their lifetimes to be measured directly. Instead, tables of particle properties often list "width," measured in energy units and indicating the width of the distribution of measured rest energies. For example, the $Z^{0}$ has mass $91.18 \mathrm{GeV}$ and width $2.5 \mathrm{GeV} .$ Use the energy-time uncertainty relation to estimate its corresponding lifetime.

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Key Concepts

Energy-Time Uncertainty Principle

This principle in quantum mechanics states that there is a fundamental limit to the precision with which certain pairs of physical properties, such as energy and time, can be simultaneously known. Specifically, the product of the uncertainty in energy and the uncertainty in time is bounded from below by a constant (typically ?/2). This principle is essential when dealing with processes that occur on very short time scales, such as the decay of unstable particles, where the energy spread (or width) of a particle state is related to its lifetime.

Decay Width and Lifetime Relationship

In the context of particle physics, the decay width of a particle is a measure of the spread in its energy distribution and is inversely related to its lifetime. By applying the energy-time uncertainty relation, one can estimate the lifetime of a particle by calculating the ratio of the reduced Planck constant to the decay width. This relationship implies that a larger width, indicating a broader energy distribution, corresponds to a shorter lifetime, and vice versa.

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