Problem 69 - College Physics

Question

\(\bullet\) 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.

Step-by-Step Solution

Answer

The uncertainty in mass is approximately 0.947% of the particle's mass.

Step 1: Understand the Problem

We need to use the average lifetime of the particle to determine the uncertainty in its energy using the energy-time uncertainty principle and then convert that energy uncertainty to mass uncertainty.

Step 2: Recall the Energy-Time Uncertainty Principle

According to the Heisenberg uncertainty principle, the uncertainty in energy \( \Delta E \) and the uncertainty in time \( \Delta t \) are related by: \\[ \Delta E \Delta t \geq \frac{\hbar}{2} \]where \( \hbar \) is the reduced Planck's constant, approximately \( 1.055 \times 10^{-34} \, \text{Js} \).

Step 3: Calculate Uncertainty in Energy

Given \( \Delta t = 8.4 \times 10^{-17} \, \text{s} \), we find \( \Delta E \) as follows:\[ \Delta E \approx \frac{\hbar}{2 \Delta t} = \frac{1.055 \times 10^{-34} \, \text{Js}}{2 \times 8.4 \times 10^{-17} \, \text{s}} \]Calculate \( \Delta E \) from this expression.

Step 4: Convert Energy Uncertainty to Mass Uncertainty

From the relationship \( E = mc^2 \), the uncertainty in the mass \( \Delta m \) can be expressed as:\[ \Delta m = \frac{\Delta E}{c^2} \] where \( c = 3 \times 10^8 \, \text{m/s} \) is the speed of light. Substitute \( \Delta E \) to find \( \Delta m \).

Step 5: Calculate the Fraction of the Particle's Mass

The mass of the particle is given as approximately 264 times the mass of the electron, where the mass of an electron \( m_e \) is \( 9.11 \times 10^{-31} \, \text{kg} \). Thus, the mass of the meson is:\[ m_{\pi} = 264 \times 9.11 \times 10^{-31} \, \text{kg} \]The fraction of the uncertainty to the mass is:\[ \frac{\Delta m}{m_{\pi}} \]Compute this final fraction.

Key Concepts

Neutral Pi Meson

The neutral pi meson, often denoted as the \(\pi^0\) meson, is a fascinating particle found in the realm of particle physics.It is part of the meson family, which are particles made up of quark-antiquark pairs.Uniquely, the neutral pi meson has no electric charge, distinguishing it from its charged counterparts.

This particle is quite unstable, with a very short average lifetime of\(8.4 \times 10^{-17}\) seconds.It plays a crucial role in high-energy particle physics due to its process of decay into two gamma-ray photons.Understanding the \(\pi^0\) meson involves examining its mass relative to more familiar subatomic particles.For instance, its mass is about 264 times that of an electron.This relative mass helps physicists study the behaviors and interactions of fundamental particles in atomic nuclei.

Mass-Energy Equivalence

Mass-energy equivalence is a vital concept in physics, brought into the realm of common knowledge by Albert Einstein.It is encapsulated in the famous equation \(E=mc^{2}\), which reveals that mass (\(m\)) and energy (\(E\)) are interchangeable.

• \(E\) - Energy of the particle
• \(m\) - Mass of the particle
• \(c\) - Speed of light in a vacuum, approximately \(3 \times 10^{8}\) m/s

Through this principle, even a small amount of mass can correspond to a substantial amount of energy.In our context, the energy of the neutral pi meson as it decays plays a significant role.By applying this principle, the energy uncertainty derived from the particle's limited lifetime allows us to calculate the mass uncertainty as well.This powerful relationship is at the heart of particle physics and cosmology.

Average Lifetime

The concept of average lifetime is crucial when dealing with unstable particles like the neutral pi meson.The average lifetime of a particle tells us how long it typically exists before decaying into other particles.
In our scenario, the \(\pi^0\) meson has an extremely short average lifetime of \(8.4 \times 10^{-17}\) seconds.This fleeting lifespan signifies that the meson rapidly transforms after being produced in a high-energy collision, resulting in its decay into gamma-ray photons.Measurement of such short lifetimes requires precision.This precision allows us to use the Heisenberg uncertainty principle to analyze particle behaviors and uncertainties in their properties like energy and mass.By knowing the average lifetime, we gain insight into the particle's stability and its subsequent interactions in the universe.

Gamma-Ray Photons

At the instance of decay, the neutral pi meson emits two gamma-ray photons.Gamma rays are a form of electromagnetic radiation, much like visible light but with much higher energy.These photons are incredibly energetic, making them essential in various fields of science and technology, from medical imaging to nuclear physics.
When the \(\pi^0\) meson decays, the transformation is an example of how energy is conserved through different forms.The mass of the meson is converted into the energy of the gamma-ray photons, showcasing mass-energy equivalence.Gamma-ray photons are uncharged and have high penetrating power.This makes them useful for studying fundamental physics processes where other means might fail due to interference.Using these photons, scientists explore the building blocks of the universe in both natural environments and specialized laboratory settings.