Question

A neutral pion (Problem 15.86) is traveling with speed \(v\) when it decays into two photons, which are seen to emerge at equal angles \(\theta\) on either side of the original velocity. Show that \(v=c \cos \theta\)

Short answer

For the decay of a neutral pion into photons at angle \(\theta\), we show that \(v = c \cos \theta\).

Step-by-step solution

Understand the Problem

We are given a neutral pion traveling with speed \(v\) which decays into two photons. These photons emerge at equal angles \(\theta\) with respect to the original velocity of the pion. We need to derive an equation that relates the speed \(v\) of the pion to the angle \(\theta\).

Define the Frame of Reference

Choose the rest frame of the pion as the reference frame. In this frame, the pion is initially at rest and after decay emits two photons in opposite directions.

Conservation of Momentum in Pion's Rest Frame

In the pion's rest frame, the total momentum is zero before decay. After decay, the momenta of the two photons must also sum to zero. If each photon has momentum \(p\), then due to their opposite direction, the momentum equation is: \[p_1 - p_2 = 0\] which simplifies to \(p_1 = p_2\).

Use Photon Momentum

The momentum of each photon is related to its energy by \(E = pc\). Since the two photons have the same energy and are moving in opposite directions in the pion’s rest frame, they balance each other's momenta.

Transform to Lab Frame

Now consider a lab frame where the pion moves with velocity \(v\). In this frame, the momenta are no longer equal, but the principle of conservation holds: \[\mathbf{p}_{\text{total}} = \mathbf{p}_1 + \mathbf{p}_2\].

Relate Cosine of Angle to Speed

In the lab frame, due to Lorentz transformation, the angle \(\theta\) between the velocity of the pion and the emitted photons can be related to velocity \(v\) using the formula: \(v = c \cos \theta\), which comes from analyzing the relativistic momentum and energy transformations.

Derivation of the Formula

Using Lorentz transformation formulas and the invariant property of energy-momentum vector, prove the relation analytically. As photons emerge symmetrically at angle \(\theta\), component analysis in relevant directions confirms \(v = c \cos \theta\).

Key concepts

Neutral Pion Decay

In the fascinating world of particle physics, the neutral pion is a short-lived particle that sometimes undergoes decay into photons. Let's explore this intriguing process. The neutral pion, symbolized as \(\pi^0\), does not have an electric charge and is composed of a quark-antiquark pair.
When it decays, it emits two photons, which are particles of light that carry energy and momentum but have no mass. This decay is a natural process that can be analyzed using the principles of relativistic mechanics.
Before decay, the pion may be moving with a velocity \(v\). After decay, the conservation principles govern how energy and momentum are distributed among the resulting photons, leading to a balanced and predictable outcome. By studying such decays, scientists can understand how fundamental forces in nature work and test hypotheses about the characteristics of subatomic particles.

Conservation of Momentum

A core principle in physics is the conservation of momentum. This principle is especially important when studying particles like neutral pions. Before a pion decays, it possesses a certain momentum depending on its speed and mass. When it decays into two photons, each photon carries away some of this momentum.
In the rest frame of the pion, the initial momentum is zero because the pion is at rest. After decay, the two photons are emitted in opposite directions and move symmetrically. Hence, their momenta cancel each other out, maintaining an overall momentum of zero.

• The principle ensures that the momentum before and after an event must remain the same in a closed system.
• In any transformation or decay, the total momentum vector in a system remains constant.

Conservation of momentum helps us understand how particles interact and provides tools to predict the results of these interactions. In the context of neutral pion decay, it is a crucial step towards finding relationships between velocity \(v\) and angles \(\theta\) of emitted photons in different frames of reference.

Lorentz Transformation

The Lorentz transformation is a key concept in Einstein’s theory of relativity, providing the mathematical framework to relate physical quantities as measured in different inertial frames. It becomes crucial when analyzing systems like the decay of a moving neutral pion.
Consider a neutral pion traveling with velocity \(v\) relative to a stationary observer; when it decays, the resulting photons might not appear to simply follow the same angle \(\theta\) in other frames. This is where Lorentz transformations come in. They adjust for the effects of motion at high speeds (close to the speed of light \(c\)), enabling calculations of momentum and energy of the photons as seen from different viewpoints.
When applying the transformation, the angles at which photons are detected change due to relativistic effects. This is how we derive the relationship that connects the velocity \(v\) and angle \(\theta\): \(v = c \cos \theta\).

• The Lorentz transformation explains various relativistic phenomena, such as time dilation and length contraction.
• It allows for accurate predictions of particle behavior at speeds close to \(c\).

Understanding Lorentz transformations is essential for physicists working in the field of particle physics, as they ensure all laws of physics are consistent in different inertial frames.