Question

The neutral pion (\(\pi^0\)) 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. Using the relationship \(E = mc^2\) between rest mass and energy, find the uncertainty in the mass of the particle and express it as a fraction of the mass.

Short answer

The uncertainty in the pion's mass as a fraction is about \(2.4 \times 10^{-14}\).

Step-by-step solution

Identify Given Values

We are given the average lifetime of the neutral pion as \(\Delta t = 8.4 \times 10^{-17}\, \text{s}\) and its mass relative to the electron \(m_{\pi^0} = 264 m_e\). We need to find the uncertainty in the mass as a fraction of the mass.

Understand Energy-Mass Uncertainty Relationship

According to the energy-time uncertainty principle, \(\Delta E \cdot \Delta t \geq \frac{\hbar}{2}\), where \(\Delta E\) is the energy uncertainty and \(\hbar\) is the reduced Planck's constant \(\left(\hbar = 1.0545718 \times 10^{-34}\, \text{Js}\right)\).

Relate Energy Uncertainty to Mass Uncertainty

Since energy is related to mass through \(E = mc^2\), the uncertainty in mass \(\Delta m\) can be expressed as \(\Delta E = \Delta m \cdot c^2\). So, \(\Delta m \cdot c^2 \cdot \Delta t \geq \frac{\hbar}{2}\).

Solve for Mass Uncertainty

Rearranging the energy-mass uncertainty relationship gives \(\Delta m \geq \frac{\hbar}{2c^2 \Delta t}\). Substitute \(\hbar = 1.0545718 \times 10^{-34}\, \text{Js}\), \(c = 3 \times 10^8\, \text{m/s}\), and \(\Delta t = 8.4 \times 10^{-17}\, \text{s}\).

Calculation Details

\[\Delta m \geq \frac{1.0545718 \times 10^{-34}}{2 \cdot (3 \times 10^8)^2 \cdot 8.4 \times 10^{-17}}\] Compute this value to find \(\Delta m\).

Express as Fraction of Mass

Calculate the fraction \(\frac{\Delta m}{m_{\pi^0}}\) where \(m_{\pi^0} = 264 m_e\). Since the electron mass \(m_e = 9.10938356 \times 10^{-31}\, \text{kg}\), find \(m_{\pi^0} = 264 \times 9.10938356 \times 10^{-31}\, \text{kg}\). Use this to find the fraction.

Key concepts

Neutral Pion

The neutral pion, denoted as \(\pi^0\), is a type of meson, which is a particle made up of a quark and an antiquark. It is one of several types of pions, which are important players in particle physics, responsible for mediating the strong force that holds atomic nuclei together. Neutral pions have no electric charge, which distinguishes them from their charged counterparts. Despite their crucial role in physics, neutral pions are incredibly unstable. They are typically produced in high-energy particle collisions, such as those occurring in particle accelerators.

The mass of a \(\pi^0\) is about 264 times that of an electron. Electrons have a relatively small mass, known to be approximately \(9.10938356 \times 10^{-31}\, \text{kg}\). At this mass, a \(\pi^0\) remains stable only for an exceedingly short amount of time, specifically about \(8.4 \times 10^{-17}\, \text{s}\) before it decays into lighter particles like gamma-ray photons. Learning about neutral pions helps us understand fundamental forces and interactions that occur in the subatomic world.

Mass-Energy Equivalence

Mass-energy equivalence is a profound principle in physics, famously encapsulated by Einstein's equation, \(E = mc^2\). This equation reveals that mass and energy are two forms of the same thing and can be converted into each other. \(E\) denotes energy, \(m\) represents mass, and \(c\) is the speed of light, which is approximately \(3 \times 10^8\, \text{m/s}\). This principle is fundamental in understanding many phenomena in the universe.

In the context of the neutral pion, mass-energy equivalence allows us to connect an uncertainty in the particle's mass to its energy. When a particle like the \(\pi^0\) decays, its mass is converted into energy, emitting gamma-ray photons. The precise energy of these photons can vary, leading to an uncertainty in energy. Using the mass-energy equivalence concept, one can determine the uncertainty in mass from the uncertainty in energy.

Lifetime of Particles

The lifetime of particles like the neutral pion is an essential concept in particle physics. This lifetime, also known as the mean lifetime, is a measure of how long a particle exists before it decays into other particles. For a neutral pion, this lifetime is remarkably short, about \(8.4 \times 10^{-17}\, \text{s}\). Such short lifetimes characterize many fundamental particles, especially those that are highly unstable.

The uncertainty principle, related to Heisenberg's Uncertainty Principle, tells us that the product of the uncertainty in energy \(\Delta E\) and the uncertainty in time \(\Delta t\) is roughly equal to or greater than a constant value, \(\frac{\hbar}{2}\), where \(\hbar\) is the reduced Planck's constant. This principle means that the shorter the lifetime of the particle (\(\Delta t\)), the greater the uncertainty in its energy \(\Delta E\).

This lifetime-uncertainty relationship is vital for understanding how experiments in particle physics measure energies and lifetimes of short-lived particles like the \(\pi^0\). By understanding these fleeting particles, scientists can probe the fundamental forces and rules that govern our universe.