Problem 64 - College Physics

Question

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.

Step-by-Step Solution

Answer

The uncertainty in mass as a fraction of the particle's mass is very small.

Step 1: Understand the Given Information

The mass of the \(\pi^0\) meson is about 264 times the mass of an electron (\(m_e = 9.11 \times 10^{-31} \text{ kg}\)). The average lifetime of the meson is \(8.4 \times 10^{-17} \text{ s}\).

Step 2: Apply the Heisenberg Uncertainty Principle

The uncertainty principle can be applied to energy and time as \(\Delta E \cdot \Delta t \geq \frac{\hbar}{2}\), where \(\Delta E\) is the uncertainty in energy, \(\Delta t = 8.4 \times 10^{-17} \text{ s}\), and \(\hbar = 1.055 \times 10^{-34} \text{ Js}\).

Step 3: Find the Uncertainty in Energy (\(\Delta E\))

Rearrange the uncertainty principle formula to find \(\Delta E\): \[\Delta E \geq \frac{\hbar}{2 \cdot \Delta t}\]Substitute the values into the formula: \[\Delta E \geq \frac{1.055 \times 10^{-34}}{2 \times 8.4 \times 10^{-17}}\]Calculate \(\Delta E\).

Step 4: Relate Energy Uncertainty to Mass Uncertainty

Using \(E = m c^2\), relate the uncertainties: \(\Delta E = \Delta m c^2\).So, \(\Delta m = \frac{\Delta E}{c^2}\).Substitute \(\Delta E\) and \(c = 3 \times 10^8 \text{ m/s}\) to find \(\Delta m\).

Step 5: Calculate \(\Delta m\) as a Fraction of the Meson's Mass

The meson mass \(m_{\pi^0} = 264 \times m_e\).Calculate \(\frac{\Delta m}{m_{\pi^0}}\) using the values of \(\Delta m\) and \(m_{\pi^0}\). This gives the uncertainty in mass as a fraction of the particle's mass.

Key Concepts

pi meson decay

The \(\pi^0\) meson, also known as a pi meson, is a neutral particle that plays a pivotal role in particle physics. Pi mesons are commonly found in high-energy particle collisions. They are quite unstable, with a very short average lifetime of \(8.4 \times 10^{-17} \, \text{s}\). This brief existence is due to their tendency to decay rapidly.
Once formed, the \(\pi^0\) meson decays into two gamma-ray photons. This decay process is significant in studies of subatomic particles because it offers insights into how energy and particles behave at fundamental levels.
Understanding pi meson decay is crucial for physicists, as it aids in exploring the forces and particles that constitute our universe. It also provides practical applications in various fields, including particle detectors used in experiments.

mass-energy equivalence

Mass-energy equivalence is a fundamental concept in physics introduced by Albert Einstein. It is expressed as the famous equation \(E=mc^2\), where \(E\) represents energy, \(m\) is mass, and \(c\) is the speed of light in a vacuum.
This equation tells us that mass can be converted into energy and vice versa. It shows that any change in the mass of a system will correspondingly change the energy of the system.

• When \(\pi^0\) mesons decay, the mass lost from the meson is transformed into energy, producing two gamma-ray photons.


• The concept of mass-energy equivalence is essential when calculating the energy produced from the decay of particles like the photonic emission of \(\pi^0\) mesons.

This principle is not only vital for understanding particle decay, but also for exploring realms such as nuclear reactions and the dynamics of the cosmos.

gamma-ray photons

Gamma-ray photons are highly energetic forms of electromagnetic radiation. They possess the largest amount of energy and the shortest wavelengths in the electromagnetic spectrum.
When a \(\pi^0\) meson decays, it releases energy in the form of two gamma-ray photons. These photons carry away the energy that was stored in the meson before it decayed. This type of radiation is a crucial tool in understanding subatomic processes.

• Gamma rays are invaluable in medical imaging and therapies, such as cancer treatment.


• They also help astronomers observe distant celestial phenomena, providing insights into energetic and often violent cosmic events.

The study and understanding of gamma-ray photons are indispensable in fields ranging from medical science to astrophysics due to their penetrative power and their origin from high-energy processes.

particle physics

Particle physics is the branch of physics that studies the fundamental constituents of matter and the forces that act between them. It explores the behavior of the tiniest known particles.The short-lived \(\pi^0\) meson forms part of our universe's building blocks, playing a role in understanding the strong force, which binds quarks together to form protons and neutrons.
Studies in particle physics have pioneered many technological advancements and deepened our comprehension of the universe. Research is conducted in particle accelerators, where particles like \(\pi^0\) mesons can be studied through high-energy collisions.

• Particle physics has led to the development of the Standard Model, which classifies all known subatomic particles.


• It has also been fundamental in medical technologies and improving data processing methods.

Understanding particle physics is vital for both theoretical investigations and practical applications, influencing technology and expanding our knowledge of the cosmos.