A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon (b) if the p and p collide head-on, each with an initial kinetic energy of 620 MeV.

Verified step by step guidance

1. First, understand that when a proton and an antiproton annihilate, their rest mass energy is converted into the energy of the photons. The rest mass energy of a proton (or antiproton) is given by Einstein's equation: <math xmlns='http://www.w3.org/1998/Math/MathML'>E = mc^2</math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'>m</math> is the rest mass of the proton and <math xmlns='http://www.w3.org/1998/Math/MathML'>c</math> is the speed of light.
2. Calculate the total energy of each particle before the collision. This includes both the rest mass energy and the kinetic energy. The total energy <math xmlns='http://www.w3.org/1998/Math/MathML'>E_{total}</math> for each particle is: <math xmlns='http://www.w3.org/1998/Math/MathML'>E_{total} = E_{rest} + E_{kinetic}</math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'>E_{kinetic} = 620 \, \text{MeV}</math>.
3. Since the proton and antiproton collide head-on, the total energy available for the photons is the sum of the total energies of both particles. Therefore, the energy of the two photons combined is: <math xmlns='http://www.w3.org/1998/Math/MathML'>E_{photons} = 2 \times E_{total}</math>.
4. Each photon will have half of the total energy calculated in the previous step. Use the energy of one photon to find its frequency using the relation: <math xmlns='http://www.w3.org/1998/Math/MathML'>E = h \nu</math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'>h</math> is Planck's constant and <math xmlns='http://www.w3.org/1998/Math/MathML'>\nu</math> is the frequency.
5. Finally, find the wavelength of each photon using the speed of light equation: <math xmlns='http://www.w3.org/1998/Math/MathML'>c = \lambda \nu</math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'>\lambda</math> is the wavelength. Rearrange this equation to solve for <math xmlns='http://www.w3.org/1998/Math/MathML'>\lambda</math>.

Key Concepts

Conservation of Energy

In particle physics, the conservation of energy principle states that the total energy before and after a reaction must remain constant. When a proton and an antiproton annihilate, their rest mass energy and kinetic energy are converted into the energy of the resulting photons. This principle allows us to calculate the energy of each photon produced in the annihilation process.

Photon Energy and Frequency Relationship

The energy of a photon is directly related to its frequency through the equation E = hf, where E is the energy, h is Planck's constant, and f is the frequency. This relationship is crucial for determining the frequency of the photons produced in the annihilation, given their energy. Understanding this concept allows us to connect the energy of the photons to their frequency and subsequently their wavelength.

Relativistic Kinetic Energy

Relativistic kinetic energy accounts for the effects of special relativity when particles move at speeds close to the speed of light. For a proton with significant kinetic energy, such as 620 MeV, relativistic effects must be considered. The total energy of the proton includes both its rest mass energy and its relativistic kinetic energy, which is essential for accurately calculating the energy available for photon production in the annihilation.