Two equal-energy photons collide head-on and annihilate each other, producing a ${\text{µ}}^{+}{\text{µ}}^{-}$ pair. The muon mass is given in terms of the electron mass in Section $44.1$. Calculate the maximum wavelength of the photons for this to occur. If the photons have this wavelength, describe the motion of the ${\text{µ}}^{+}$ and immediately after they are produced.

Verified step by step guidance

1. Step 1: Begin by understanding the energy conservation principle in this problem. The energy of the two photons must equal the total rest energy of the muon-antimuon pair produced. The rest energy of a particle is given by E = m * c^2, where m is the mass of the particle and c is the speed of light.
2. Step 2: Use the given information about the muon mass in terms of the electron mass. From Section 44.1, the muon mass (m_μ) is approximately 207 times the electron mass (m_e). Therefore, the rest energy of one muon is E_μ = 207 * m_e * c^2.
3. Step 3: Since two muons are produced (µ^+ and µ^-), the total rest energy required is 2 * E_μ = 2 * (207 * m_e * c^2). This total energy must be provided by the two photons. Each photon contributes half of this energy, so the energy of each photon is E_photon = (207 * m_e * c^2).
4. Step 4: Relate the energy of a photon to its wavelength using the equation E = h * c / λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength. Rearrange this equation to solve for the wavelength: λ = h * c / E.
5. Step 5: Substitute the energy of the photon (E_photon = 207 * m_e * c^2) into the wavelength equation. This will give the maximum wavelength of the photons required for the annihilation to occur. After the annihilation, if the photons have this maximum wavelength, the muons will be produced at rest because all the photon energy is converted into the rest energy of the muons.

Key Concepts

Photon Energy and Wavelength

The energy of a photon is inversely related to its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. For two photons to annihilate and produce a particle-antiparticle pair, their combined energy must equal the rest mass energy of the produced particles, which can be calculated using the mass-energy equivalence principle, E = mc².

Particle-Antiparticle Annihilation

When a particle and its corresponding antiparticle collide, they can annihilate each other, converting their mass into energy, typically in the form of photons. In this scenario, the annihilation of two photons results in the creation of a muon (µ^+) and an antimuon (µ^-), which are heavier than electrons and require sufficient energy from the photons to be produced.

Conservation of Momentum

In any collision or interaction, the total momentum before the event must equal the total momentum after, as stated by the law of conservation of momentum. In the case of the head-on collision of two photons, their momenta are equal and opposite, resulting in a net momentum of zero. After the annihilation, the produced muons will also have a combined momentum of zero, indicating they will move in opposite directions with equal momentum.