Understanding the Photon Frequency from Electron-Positron Annihilation

When an electron and its antiparticle, the positron, annihilate, the mass converts into pure energy, yielding two photons. This process is a prime example of Einstein's mass-energy equivalence principle, represented by the equation E mc2.

Step-by-Step Calculation

Here, we will go through the detailed steps to calculate the photon frequency resulting from the annihilation of an electron and a positron.

Step 1: Calculating the Total Mass

The mass of an electron and a positron is approximately:

[ m_e approx 9.11 times 10^{-31} , text{kg} ]

Therefore, the total mass during annihilation is:

[ m_{text{total}} m_e m_e 2m_e approx 2 times 9.11 times 10^{-31} , text{kg} approx 1.82 times 10^{-30} , text{kg} ]

Step 2: Calculating the Energy Released

Using the equation E mc2 where c is the speed of light in a vacuum (approximately 3.00 x 108 m/s), we can calculate the kinetic energy released:

[ E (1.82 times 10^{-30} , text{kg}) times (3.00 times 10^8 , text{m/s})^2 approx (1.82 times 10^{-30} , text{kg}) times 9 times 10^{16} , text{m}^2/text{s}^2 ]

Performing the multiplication:

[ E approx 1.64 times 10^{-13} , text{J} ]

Step 3: Calculating the Frequency of the Photons

The energy of a photon is given by the equation:

[ E h f ]

Where h is Planck's constant (6.626 x 10-34 J·s). Since there are two photons, each photon's energy is:

[ E_{text{photon}} frac{E}{2} frac{1.64 times 10^{-13} , text{J}}{2} approx 8.20 times 10^{-14} , text{J} ]

Now substituting into the equation for photon energy to find the frequency:

[ f frac{E}{h} frac{8.20 times 10^{-14} , text{J}}{6.626 times 10^{-34} , text{J·s}} approx 1.24 times 10^{20} , text{Hz} ]

Conclusion

The frequency of each resulting photon from the annihilation of an electron and a positron is approximately:

1.24 × 1020 Hz

Alternative Calculation: Rest Mass Energy

Another approach to understanding the energy involved in electron-positron annihilation involves the rest mass energy of the electron or positron. The rest mass energy of an electron is about 511.0 KeV, which simplifies the calculation of the photon energy and subsequently the frequency. Here's the simplified calculation:

The energy of each gamma ray from the annihilation can be calculated as:

[ E_{text{photon}} ge 511.0 , text{KeV} ]

Using E hf, and converting the energy to joules:

[ f ge frac{511.0 times 10^3 , text{eV}}{h} approx 1.2356 times 10^{20} , text{Hz} ]

This result is consistent with our previous calculation and demonstrates the equivalence of rest mass energy and photon frequency in quantum physics.

Note that in other reference frames, the resulting gamma rays could be redshifted or blueshifted to different frequencies due to relativistic effects.