Question

When a positron and an electron collide, they annihilate each other and produce two gamma photons, which carry the same amount of energy. What is the wavelength (in nanometers) of these photons?

Short answer

Answer: First, calculate the total rest mass energy (E_total) of the electron-positron pair using their combined mass and the speed of light. Then, find the energy of a single gamma photon (E_photon) by dividing E_total by 2. Use Planck's constant to calculate the frequency (f) of the gamma photons and, finally, find the wavelength (λ) using the wave equation. Convert the wavelength to nanometers for the final answer. Using the provided values: E_total = 2 * (9.109 x 10^{-31} kg) * (3 x 10^8 m/s)^2 E_photon = E_total / 2 f = E_photon / (6.626 x 10^{-34} J*s) λ = (3 x 10^8 m/s) / f λ in nm = λ * 10^9 The wavelength of the gamma photons produced is calculated using these steps.

Step-by-step solution

Find the rest mass energy of the electron and positron

First, we need to determine the total rest mass energy (E) of the electron-positron pair. The rest mass energy of an electron (or positron) can be calculated using the equation E = mc^2, where m is the mass and c is the speed of light. Since an electron and a positron have the same mass, we can denote their combined rest mass energy as E_total = 2 * E_electron. The mass of an electron is me = 9.109 x 10^{-31} kg, and the speed of light is c = 3 x 10^8 m/s. Using these values, calculate the total rest mass energy. E_total = 2 * me * c^2

Find the energy of a single gamma photon

As given in the exercise, the two gamma photons carry the same amount of energy, so we can divide the total energy by 2 to find the energy of a single gamma photon. E_photon = E_total / 2

Calculate the wavelength using Planck's constant and the wave equation

Now we can use Planck's constant (h) and the photon energy (E_photon) to find the frequency (f) of the gamma photons using the equation E_photon = h * f. Planck's constant h is 6.626 x 10^{-34} J*s. f = E_photon / h Finally, we can find the wavelength (λ) of the gamma photons using the wave equation, which is: λ = c / f Convert the wavelength to nanometers (1 m = 10^9 nm) to get the final answer.

Putting it all together

Calculate the total rest mass energy (E_total), find the energy of a single gamma photon (E_photon), and then use these values to calculate the frequency (f) and finally the wavelength (λ) of the gamma photons. Don't forget to convert the wavelength to nanometers for the final answer. E_total = 2* (9.109 x 10^{-31} kg) * (3 x 10^8 m/s)^2 E_photon = E_total / 2 f = E_photon / (6.626 x 10^{-34} J*s) λ = (3 x 10^8 m/s) / f λ in nm = λ * 10^9 This will give the wavelength in nanometers of the gamma photons produced from the annihilation of the electron and positron.