Question

A particle of rest mass \(m\) decays from rest into a particle of rest mass \(m\) ' and a photon. Find the separate energies of these end products. [Answer: \(c^{2}\left(m^{2} \pm m^{\prime 2}\right) / 2 m\). Hint: use a four-vector argument.]

Short answer

Answer: The separate energies of the end products are given by: E_m' = \frac{c^2(m^2 - m'^2)}{2m} E_photon = \frac{c^2(m^2 + m'^2)}{2m}

Step-by-step solution

Understand the 4-vectors for particles and photons.

The four-vector used for a relativistic particle is given as P = (E/c, p) where E is the energy of the particle, c is the speed of light, and p is the momentum of the particle. For a photon, the four-vector is given as Q = (E/c, p) where E is the energy of the photon and p is the momentum of the photon. Note that the mass of a photon is zero, which is why there is no mass term in its four-vector.

Apply the conservation of energy and momentum.

The conservation of energy and momentum states that the energy and momentum before the decay is equal to the energy and momentum after the decay. Using the four-vector notation, the equation for the conservation of both quantities is given by: Initial 4-vector = Final 4-vector P_particle = P_m' + Q_photon

Write down the initial 4-vector.

Given that the particle is initially at rest, its energy is equal to its rest mass energy mc^2 (Einstein's famous equation). Therefore, the initial four-vector for the particle is: P_particle = (mc^2/c, 0)

Write down the final 4-vectors.

After the decay, the particle of rest mass m' has an energy E_m' and momentum p_m', and the photon has an energy E_photon and momentum p_photon. The final four-vectors are given by: P_m' = (E_m'/c, p_m') Q_photon = (E_photon/c, p_photon)

Use the conservation equation to find the energy and momentum of the end products.

Applying the conservation equation we derived in step 2 and substituting in the four-vectors from steps 3 and 4: (mc^2/c, 0) = (E_m'/c, p_m') + (E_photon/c, p_photon) We are only concerned with finding the energies, so we can look solely at the energy part of the four-vector equation: mc^2 = E_m' + E_photon

Solve for E_m' and E_photon using four-vector conservation equations and hint.

We want to find E_m' and E_photon. We can use the square of the conservation of four-vector magnitude: P_particle^2 = (P_m' + Q_photon)^2 (mc^2)^2 = (E_m' + E_photon)^2 Solving for E_m' and E_photon, we get: E_m' = \frac{c^2(m^2 - m'^2)}{2m} E_photon = \frac{c^2(m^2 + m'^2)}{2m} So, the separate energies of the end products are given by: E_m' = \frac{c^2(m^2 - m'^2)}{2m} E_photon = \frac{c^2(m^2 + m'^2)}{2m}

Key concepts

Four-Vector Conservation

In the realms of Special Relativity, the concept of four-vector conservation is central to understanding the behavior of particles during events like decay. A four-vector is a mathematical vector that comprises four components: the time component, which is energy divided by the speed of light, and three spatial components representing momentum.

During particle decay, as in the given exercise, the sum of the four-vectors before the decay must equal the sum after the decay. This principle ensures that energy and momentum are conserved in all directions. Therefore, when a particle of rest mass m decays to a particle of rest mass m' and a photon, the total four-vector remains unchanged. This concept is pivotal in calculating the separate energies of the decay products by equating the initial and final four-vector components.

Energy-Momentum Conservation

Energy-momentum conservation is another cornerstone of Special Relativity closely related to four-vector conservation. Specifically, energy-momentum conservation refers to the conservation of energy and momentum separately in the context of relativistic physics.

In the exercise, the conservation of energy and momentum is utilized to find the energies of the decay products. Initially, the particle has energy equivalent to its rest mass energy and zero momentum since it's at rest. Following decay, the combined energy and momentum of the new particle and the photon must match the initial state.

Applying the Conservation

By creating equations that reflect energy and momentum conservation, and recognizing that photons have energy but no rest mass, students can use algebraic manipulation to solve for the unknowns. This elegant symmetry between energy and momentum in relativistic physics is what allows us to predict and understand particle behavior after decays and collisions.

Rest Mass Energy

The concept of rest mass energy is an integral part of understanding particle physics within the framework of Special Relativity. Rest mass energy is the energy that an object possesses due to its mass when it is not in motion.

In Einstein's famous equation, E = mc^2, E represents the rest mass energy, m is the rest mass, and c is the speed of light. The equation implies that mass can be converted into energy, and this energy is significant even for small amounts of mass because the speed of light is a large number.

Decay and Rest Mass Energy

When the particle decays, the rest mass of the original particle is converted into the rest mass energy of the new particle and the energy of the photon emitted. Understanding how the rest mass energy is distributed among the decay products enables students to solve for their respective energies, as demonstrated in the step-by-step solution.

Einstein's Famous Equation

Einstein's famous equation, E = mc^2, connects mass and energy in a way that revolutionized physics. This concise but profound equation indicates that energy (E) and mass (m) are interchangeable; they are different forms of the same thing. The speed of light (c) squared in the equation acts as a conversion factor between these two entities.

This relationship is pivotal in particle physics. When a stationary particle decays as in the provided exercise, the initial energy is purely the rest mass energy of the particle, calculated by multiplying the rest mass by the speed of light squared. This idea is essential for understanding how, in the decay process, the rest mass is converted into energy of motion for the resulting particles.

The Role in Decays

Therefore, Einstein's equation allows us to determine the energy of the particle before decay, setting the stage for energy conservation which then leads to finding the energies of the decay products once the particle has decayed. The equation is fundamental to the laws of physics and underpins much of modern physics, from the workings of the sun to the principles of atomic energy.