Question

What are the wavelengths of the two photons produced when a proton and an antiproton at rest annihilate?

Short answer

Answer: The wavelength of each photon produced during the annihilation of a proton and an antiproton is approximately 2.63 x 10^-16 meters.

Step-by-step solution

Find the rest energies of proton and antiproton

We need to first determine the rest energies of the proton and the antiproton. The rest energy of a particle can be calculated using Einstein's famous equation: E = mc^2, where E is the energy (in Joules), m is the mass of the particle (in kg), and c is the speed of light (3 x 10^8 m/s). For this problem, we'll need to know the mass of a proton (and by extension, an antiproton, as they have the same mass). The mass of a proton is approximately 1.67 x 10^-27 kg.

Calculate the total energy of the system before annihilation

Now that we have the rest energies of the proton and antiproton, we can find the total energy of the system before the annihilation. Since the proton and antiproton are assumed to be at rest, the total energy of the system is simply the sum of their rest energies: E_total = E_proton + E_antiproton = 2 * (mc^2)

Determine the energy of each photon

When the proton and antiproton annihilate, they produce two photons that share the total energy of the system. Therefore, each photon will have half the total energy: E_photon = E_total / 2

Use the energy-wavelength relationship

Now we can use the relationship between the energy, frequency, and wavelength of a photon to find the wavelength of each photon. The energy of a photon is given by: E_photon = h * c / λ, where h is the Planck's constant (6.63 x 10^-34 Js), c is the speed of light, and λ is the wavelength. We can rearrange this equation to solve for the wavelength of each photon: λ = h * c / E_photon

Calculate the wavelengths of the photons

Finally, substitute the previously found values for the energy of each photon into the equation from Step 4 to find the wavelengths of the photons: λ = (6.63 x 10^-34 Js) * (3 x 10^8 m/s) / (E_total / 2) After substituting the mass of the proton and simplifying the equation, we can solve for the wavelength: λ ≈ 2.63 x 10^-16 m In conclusion, when a proton and an antiproton annihilate, two photons are produced, each with a wavelength of approximately 2.63 x 10^-16 meters.