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Chapter 39: Problem 53

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 GeV. Use the energy-time uncertainty relation to estimate its corresponding lifetime.

Short Answer

Expert verified
The calculated lifetime (approximately) of the \(Z^{0}\) particle will be the solution provided by Step 2. Be careful to include correct and consistent units in your answer.

Step by step solution

01

Understanding the energy-time uncertainty principle

The energy-time (also known as time-energy) uncertainty principle is an intrinsic property of quantum systems, according to which there is a limit on the precision with which the time at which an event occurs and the energy of the event can be jointly defined. This can be mathematically expressed as: \( \Delta E \, \Delta t \geq \frac{h}{4\pi} \), where \( \Delta E \) is the uncertainty in energy (in our case, it is given as 2.5 GeV), \( \Delta t \) is the uncertainty in time (which we will be calculating), and \( h \) is the reduced Planck's constant (\( h \approx 6.582 \times 10^{-25} \, \mathrm{GeV.s} \)). Remember to keep the units consistent.
02

Calculating the lifetime

Since we want an estimate of the lifetime of the \(Z^{0}\) particle, we can set the inequality to be an equality. By rearranging the energy-time uncertainty principle for \( \Delta t \) will yield:\( \Delta t = \frac{h} {\Delta E \times 4\pi} = \frac{6.582 \times 10^{-25} \, \mathrm{GeV.s}} {2.5 \, \mathrm{GeV} \times 4\pi} \). Solving this equation will fetch the value of \( \Delta t \) which is the estimated lifetime of \(Z^{0}\) particle. Remember that the units should accord to the problem statement (which might require some conversion).

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