Show that the following process is dynamically inpossible:
A single photon in empty space is transformed into an electron and a positron, each with rest mass m(e).Explain in detail, How can this this process be made possible?
In this case, what is the minimum photon energy required? m(e)=511Mev
To determine whether a process is dynamically possible, we need to consider the conservation laws of energy and momentum. In this case, we need to examine whether the process of transforming a single photon into an electron and a positron violates these conservation laws.
First, let's look at the conservation of momentum. Since the photon is massless, it carries momentum p = E/c, where E is the energy of the photon and c is the speed of light. After the transformation, an electron and a positron are produced. Each particle has a rest mass m(e), so each will have a momentum of p = sqrt(E^2 - m(e)^2c^2).
The initial momentum of the photon must be equal to the sum of the momenta of the electron and positron:
p_photon = p_electron + p_positron
E_photon/c = sqrt(E_electron^2 - m(e)^2c^2) + sqrt(E_positron^2 - m(e)^2c^2)
To simplify this equation, we can assume that the electron and positron have equal energies, denoted as E_electron = E_positron = E. Now we have:
E_photon/c = sqrt(E^2 - m(e)^2c^2) + sqrt(E^2 - m(e)^2c^2)
Simplifying further:
E_photon/c = 2*sqrt(E^2 - m(e)^2c^2)
E_photon = 2*sqrt(E^2 - m(e)^2c^2)*c
From this equation, we can see that the energy of the photon must be greater than or equal to the minimum threshold energy (E_minimum) required for this process to be possible. To find this minimum energy, we can set E_photon = E_minimum and solve for E_minimum:
E_minimum = 2*sqrt(E_minimum^2 - m(e)^2c^2)*c
This is a non-linear equation, and solving it can be challenging. However, we know that the rest mass of the electron, m(e), is equal to 511 MeV/c^2. By substituting this value into the equation, we can find the minimum threshold photon energy.
E_minimum = 2*sqrt(E_minimum^2 - (511 MeV/c^2)^2)*c
To solve this equation numerically, we can use various computational techniques or software.
So, to determine the minimum photon energy required for this process to be possible, we need to solve the equation E_minimum = 2*sqrt(E_minimum^2 - (511 MeV/c^2)^2)*c numerically.