Problem 4 - An Introduction to Mechanics

Question

Particle decay A particle of rest mass \(M\) spontaneously decays from rest into two particles with rest masses \(m_{1}\) and \(m_{2}\). Show that the energies of the particles are \(E_{1}=\left(M^{2}+m_{1}^{2}-\right.\) \(\left.m_{2}^{2}\right) c^{2} / 2 M\) and \(E_{2}=\left(M^{2}-m_{1}^{2}+m_{2}^{2}\right) c^{2} / 2 M .\)

Step-by-Step Solution

Answer

The energies are \( E_1 = \frac{(M^2 + m_1^2 - m_2^2)c^2}{2M} \) and \( E_2 = \frac{(M^2 - m_1^2 + m_2^2)c^2}{2M} \).

Step 1: Understand the Principle of Energy Conservation

In a decay process, the total energy before and after the decay must be equal, based on the conservation of energy principle. For the particle at rest with mass \(M\), its rest energy is \(E = Mc^2\). When it decays, the total energy is distributed as the kinetic energy plus the rest energy of the decay products, \(E_1 + E_2\).

Step 2: Solve for Momentum Conservation

Since the initial particle is at rest, the total momentum before the decay is zero. Using conservation of momentum, the momenta of the decay products, \(p_1\) and \(p_2\), must satisfy \(p_1 = -p_2\). This implies that \(p_1^2 = p_2^2\).

Step 3: Apply Energy-Momentum Relation

The energy-momentum relation for each particle is given by \(E_i^2 = p_i^2c^2 + m_i^2c^4\). For particle 1, we have \(E_1^2 = p_1^2c^2 + m_1^2c^4\), and for particle 2, \(E_2^2 = p_2^2c^2 + m_2^2c^4\).

Step 4: Express Energies Using Terms of Masses and M

Since \(p_1 = -p_2\), substituting into the energy momentum relation gives us the relations for energy:\[E_1^2 = (E - E_2)^2 = M^2c^4 - 2ME_2c^2 + E_2^2 = p_1^2c^2 + m_1^2c^4\]\[E_2^2 = (E - E_1)^2 = M^2c^4 - 2ME_1c^2 + E_1^2 = p_2^2c^2 + m_2^2c^4\]

Step 5: Solve the System of Equations for Energies

Solve the simultaneous equations from Step 4 for \(E_1\) and \(E_2\) by eliminating \(p_1^2\) and \(p_2^2\). First, add the two energy equations:\[M^2c^4 - 2ME_2c^2 + E_2^2 + M^2c^4 - 2ME_1c^2 + E_1^2 = m_1^2c^4 + m_2^2c^4\]Simplify terms and solve for the energies:\(E_1 = \frac{M^2 + m_1^2 - m_2^2}{2M}c^2\) and \(E_2 = \frac{M^2 - m_1^2 + m_2^2}{2M}c^2\).

Step 6: Verify Conservation of Energy and Momentum

Check that the solutions satisfy conservation laws. Total energy after decay should equal initial energy \( Mc^2 \), and momentum calculations in previous steps should confirm conservation \( p_1 = -p_2 \) implying energies calculated correctly balance the equations.

Key Concepts

Energy Conservation

When discussing particle decay, a fundamental concept to grasp is **Energy Conservation**. In physics, the principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. When a particle of rest mass \(M\) decays from rest into two smaller particles, the initial energy is fully contained as its rest energy \( E = Mc^2 \). This energy must be equally present in the system after decay.

After decay, the energy is shared between the rest energy and kinetic energy of the two resulting particles. Therefore, the sum of the energies \(E_1\) and \(E_2\) of the particles must equal the initial energy \(Mc^2\). The exercise reveals these energies as \( E_1 = \frac{(M^2 + m_1^2 - m_2^2)c^2}{2M} \) and \( E_2 = \frac{(M^2 - m_1^2 + m_2^2)c^2}{2M} \). This confirms that:

• The total energy before and after the decay remains constant.
• Individual energies depend on respective rest masses \(m_1\) and \(m_2\) of the decay products.

Momentum Conservation

Another vital principle in particle decay is **Momentum Conservation**. Momentum conservation dictates that the total momentum of a system remains constant if no external forces act on it. For a particle at rest initially, its momentum is zero. After decay, the decayed products must have momenta that add up to this initial total of zero.

As the exercise states, if the initial particle is at rest, the momenta of the two decay particles \( p_1 \) and \( p_2 \) must equal in magnitude but opposite in direction, which can be mathematically represented as \( p_1 = -p_2 \). This means:

• The momentum conservation implies no net motion after decay, balancing the forces.
• It ensures that changes during decay only affect energy distribution, not the system's net motion.

Energy-Momentum Relation

In decay processes, the **Energy-Momentum Relation** provides essential insight into how energy and momentum connect. This relation bridges the gap between energy conservation and momentum conservation. It is expressed for each particle as:\[ E_i^2 = p_i^2c^2 + m_i^2c^4 \] For each decay product, this equation links the kinetic energy, momentum, and rest mass. This relation reaffirms how kinetic energy and momentum contribute to the total energy of each particle.

After the decay of particle \(M\), the energies \(E_1\) and \(E_2\), derived through steps, involve solving two energy equations in terms of \(p_i^2\)'s and masses \(m_i\). The solution verifies the consistency of each particle's energy has satisfied both conservation laws:

• Each equation describes how a particle's energy isn't just from its mass but also its movement and interaction momentum.
• This link helps verify, mathematically, that energy and momentum have been properly conserved in solutions, ensuring they hold true in physical outcomes.