Question

A stationary muon \(\mu^{-}\) annihilates with a stationary antimuon \(\mu^{+}\) (same mass. \(1.88 \times 10^{\circ} 28 \mathrm{~kg}\), but opposite charge). The two disappear, replaced by electromagnetic radiation. (a) Why is it not possible for a single photon to result? (b) Suppose two photons result. Describe their possible directions of motion and wavelengths.

Short answer

It is not possible for a single photon to result due to the conservation of linear momentum. The two resulting photons must propagate in opposite directions. Their wavelengths can be calculated by substituting the known values in the formula λ = h/mc, derived from the energy-mass equivalence principle and the photon energy formula.

Step-by-step solution

Conservation of Linear Momentum

Firstly, consider the conservation of linear momentum. If a single photon were to result from the annihilation, there would be a net momentum as the photon would carry away the energy and hence, momentum associated with the annihilation. This is inconsistent with the fact that the initial system, being at rest, had zero momentum. Hence, a single photon cannot result from this process.

Two Photon Result

Since a single photon cannot result from the annihilation, consider the possible result of two photons. Their momenta must be equal and opposite in order for the total momentum of the system to remain at zero, as it was initially. Thus, the two photons must propagate in opposite directions.

Calculate the Wavelengths

To calculate the wavelengths of the photons, we use the energy-mass equivalence principle (E=mc^2), where E is the energy, m is the mass, and c is the speed of light. As E = hf for a photon, where h is the Planck's constant and f is the frequency of the photon, we can equate mc^2 = hf. The wavelength, λ, is given by c/f, and therefore, we obtain λ = hc/mc^2 = h/mc. Substituting the known values, we can calculate the wavelength of the resulting photons.

Key concepts

Conservation of Linear Momentum

The concept of conservation of linear momentum is essential in understanding the behavior of particles and systems in physics. It states that if no external forces act on a system, the total momentum of the system remains constant over time. This principle is deeply rooted in classical mechanics and also applies to subatomic processes.

In the context of a stationary muon and antimuon annihilation, initially, both particles are at rest; thus, the total momentum of the system is zero. When these particles annihilate, they must do so in a way that conserves this initial momentum of the system. If a single photon were to be produced, it would have momentum (since photons always move), and there would be no way to balance this momentum out, resulting in a non-zero total momentum which violates the conservation of momentum. Therefore, the production of a single photon is not possible in this scenario.

Thus, when considering the annihilation of a muon pair, it is understood that multiple photons must result—specifically, two photons moving in exactly opposite directions such that their momenta cancel each other out. This ensures that the system's total momentum after annihilation remains zero, thereby satisfying the conservation of linear momentum.

Photon Energy and Momentum

Photons, or particles of light, carry both energy and momentum despite having no mass. The energy of a photon is quantified by the equation E=hf, where 'E' represents energy, 'h' is Planck's constant, and 'f' is the frequency of the photon. The momentum of a photon is given by p=E/c, where 'p' is momentum and 'c' is the speed of light.

In our muon annihilation scenario, the two resulting photons must have equal and opposite momentum to ensure conservation of momentum. Because the photons are massless, their energy and momentum are directly proportional, meaning that the collision that produces higher energy photons will also produce photons with greater momentum.

The relationship between photon energy and momentum is fundamental in particle physics and helps explain phenomena like the Compton effect, where photons transfer part of their energy and momentum to electrons. For the muon annihilation considered in the exercise, this relationship is crucial because it determines the directions the photons must travel in when they are created in order to satisfy conservation laws.

Energy-Mass Equivalence Principle

The energy-mass equivalence principle, encapsulated in the famous equation E=mc^2, where 'E' is energy, 'm' is mass, and 'c' is the speed of light, is a cornerstone of modern physics introduced by Albert Einstein. This principle asserts that mass and energy are two forms of the same thing and can be converted into each other.

In the context of the muon annihilation exercise, the principle implies that the mass of the muon and antimuon can be converted entirely into energy in the form of photons. The total energy of the photons produced will be equivalent to the mass-energy of the original muon pair multiplied by the square of the speed of light (c^2).

To find the wavelength of the photons resulting from the annihilation, we use the equation for photon energy E=hf (where 'h' is Planck's constant and 'f' is frequency) and solve for the wavelength (λ) using the relationship λ=c/f. By equating the mass-energy of the muons to the energy of the photons and rearranging the equations, one can deduce that λ=h/mc, which allows for the calculation of the wavelength of the photons given the mass of the muons. Through this principle, we can understand that the entirety of the muons' mass is converted into the radiant energy of photons, illustrating the profound link between mass and energy.