Question

Calculate the maximum kinetic energy of the electron when a muon decays from rest via μ⁻ ⟶ e⁻ + vₑ + vμ. [Hint: In what direction do the two neutrinos move relative to the electron in order to give the electron the maximum kinetic energy? Both energy and momentum are conserved; use relativistic formulas.]

Accepted Answer

Understand the problem: A muon decays into an electron, an electron neutrino, and a muon neutrino. To find the maximum kinetic energy of the electron, we need to consider the conservation of energy and momentum, and the relativistic nature of the particles involved. Step 1: Write the conservation of energy equation. The total energy of the system before and after the decay must be equal. The initial energy is the rest energy of the muon, given by $$ E_{\mu} = m_{\mu}c^2 $$, where $$ m_{\mu}  is $$the mass of the muon and $$ c  is $$the speed of light. After the decay, the total energy is the sum of the relativistic energies of the electron, electron neutrino, and muon neutrino: $$ E_{\mu} = E_e + E_{v_e} + E_{v_\mu} . $$Step 2: Write the conservation of momentum equation. The total momentum of the system before and after the decay must also be equal. Since the muon is initially at rest, its momentum is zero. After the decay, the sum of the momenta of the electron, electron neutrino, and muon neutrino must also be zero: $$ \vec{p}_e + \vec{p}_{v_e} + \vec{p}_{v_\mu} = 0 . $$Step 3: Determine the configuration for maximum kinetic energy of the electron. The electron will have maximum kinetic energy when the two neutrinos move in the same direction, opposite to the electron. This configuration maximizes the electron's energy while conserving momentum. Step 4: Use the relativistic energy-momentum relation to express the electron's energy. The total energy of the electron is given by $$ E_e = \sqrt{(p_e c)^2 + (m_e c^2)^2} $$, where $$ p_e  is $$the momentum of the electron and $$ m_e  is $$its mass. The kinetic energy of the electron is then $$ K_e = E_e - m_e c^2 . $$Solve for $$ p_e $$ and $$ E_e $$ using the conservation equations from Steps 1 and 2, considering the maximum energy configuration.

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Key Concepts

Conservation of Energy

Conservation of Momentum

Relativistic Kinetic Energy