Question

The neutral pion \(\pi^{0}\) is an unstable particle (mass \(m=135\ \mathrm{MeV} / c^{2}\) ) that can decay into two photons, \(\pi^{0} \rightarrow \gamma+\gamma .\) (a) If the pion is at rest, what is the energy of each photon? (b) Suppose instead that the pion is traveling along the \(x\) axis and that the photons are observed also traveling along the \(x\) axis, one forward and one backward. If the first photon has three times the energy of the second, what was the pion's original speed \(v ?\)

Short answer

(a) 67.5 MeV per photon. (b) Pion's speed was \( \frac{c}{2} \).

Step-by-step solution

Analyze the Decay Process at Rest

For part (a), when the neutral pion \( \pi^0 \) is at rest and decays into two photons, the total energy of the pion is its rest mass energy \( E = m c^2 \), where \( m = 135 \text{ MeV}/c^2 \). Each photon carries half of this energy since the initial system is symmetric and both photons share equal energy.

Calculate Photon Energy at Rest

The energy of each photon is given by \( E_{\gamma} = \frac{E}{2} = \frac{m c^2}{2} \). Substituting the given mass, we find \( E_{\gamma} = \frac{135 \text{ MeV}}{2} = 67.5 \text{ MeV} \). Thus, each photon has an energy of 67.5 MeV.

Analyze the Moving Pion Scenario

For part (b), when the pion is moving along the x-axis, let's consider the conservation of energy and momentum. If one photon moves in the forward direction with energy \( E_{1} \) and the other in the opposite direction with energy \( E_{2} \), and \( E_{1} = 3E_{2} \), then the initial energy \( E_{\pi} \) of the pion equals \( E_{1} + E_{2} \).

Set Up Equations for Conservation of Momentum

Similarly, conservation of momentum dictates that \( p_{1} - p_{2} = p_{\pi} \), where \( p_{1} \) and \( p_{2} \) are the magnitudes of momentum for the photons. Since energy \( E = pc \) for photons, we have \( p_{1} - p_{2} = \frac{E_{\pi}}{c} \).

Substitute Energies and Solve for Velocity

Given that \( E_{1} = 3E_{2} \), substitute \( E_{1} = 3E_{2} \) into the energy equation \( E_{\pi} = E_{1} + E_{2} = 4E_{2} \) and momentum equation \( \frac{E_{\pi}}{c} = \frac{3E_{2}}{c} - \frac{E_{2}}{c} = 2\frac{E_{2}}{c} \). Solving these equations, we find \( E_{\pi} = 4E_{2} \) and \( p_{\pi}c = 2E_{2} \), which gives \( v = \frac{p_{\pi}c^2}{E_{\pi}} = \frac{2E_{2}c^2 / c}{4E_{2}} = \frac{1}{2}c \). Thus, the original velocity of the pion was \( v = \frac{c}{2} \).

Key concepts

Photon Energy

When a neutral pion (\(\pi^0\)) decays while at rest, it transforms into two photons. The energy of each photon can be found using the principle of energy conservation. Initially, the pion has an energy equal to its rest mass energy, given by \(E = mc^2\). For a pion with mass \(m = 135 \mathrm{MeV}/c^2 \), this energy is 135 \mathrm{MeV}.Since the pion is at rest and decays evenly into two photons, the energy is shared equally between the two photons. Thus, each photon obtains half of the original energy, calculated as:

• \(E_{\gamma} = \frac{mc^2}{2} = \frac{135\ \mathrm{MeV}}{2} = 67.5\ \mathrm{MeV} \)

Consequently, each photon has an energy of 67.5 MeV. This simple yet vital calculation demonstrates the symmetry and balanced energy sharing in particle decay when at rest. Understanding energy distribution is crucial in particle physics, especially in scenarios involving conservation laws.

Conservation of Momentum

In scenarios where the pion (\(\pi^0\)) is moving, it's key to consider both energy and momentum conservation to understand how the particles interact post-decay.Momentum conservation plays a critical role. For instance, when the pion is originally moving along the x-axis, it decays into two photons, one moving forward and the other backward. The energies of these photons are different, where one photon has three times the energy of the other. Let these energies be \(E_1 = 3E_2\) and \(E_2\). The total original energy \(E_{\pi}\) of the pion is thus:

• \( E_{\pi} = E_1 + E_2 = 4E_2 \)

For momentum conservation, since photons have energy \(E = pc\), the relation is:

• \(p_1 - p_2 = \frac{E_{\pi}}{c} = p_{\pi}\)

These equations capture the entirety of energy and momentum before and after the decay, maintaining the conservation laws integral in physics that hold true in all inertial frames. Recognizing these concepts allows for deeper analysis and understanding of particle interactions and transformations.

Relativistic Velocity Calculation

To find the original velocity of a moving pion (\(\pi^0\)) before it decays, we delve into relativistic physics.Given the scenario where the energy of the first photon is three times the energy of the second (\(E_1 = 3E_2\)), we need to use the relationship between energy and momentum. The equations drawn from energy and momentum conservation provide the foundation for calculating the velocity.The equations are:

• \(E_{\pi} = 4E_2 \)
• \(\frac{E_{\pi}}{c} = 2\frac{E_{2}}{c} = p_{\pi} \)

By solving for velocity, we equate momentum :

• \(v = \frac{p_{\pi}c^2}{E_{\pi}} = \frac{2E_2c^2 / c}{4E_2} = \frac{1}{2}c \)

This reveals that the pion was originally moving at half the speed of light \(v = \frac{c}{2}\). Relativistic calculations are essential when dealing with high-speed particles, ensuring our theoretical predictions match observed behaviors.