(a) Calculate the wavelength of a photon that has the same momentum as a proton moving with \(1 \%\) of the speed of light in a vacuum. (b) What is the energy of this photon in MeV? (c) What is the kinetic energy of the proton in MeV?

Understanding photon momentum is essential when delving into the quantum world. It tells us that even particles of light, also known as photons, carry momentum despite having no mass. According to quantum physics, the momentum of a photon can be calculated using the equation
\( p = \frac{h}{\lambda} \)
where \( h \) is the Planck constant and \( \lambda \) is the photon's wavelength. It's interesting to note that momentum here is purely due to the photon's movement and energy, showcasing the wave-particle duality of light. When comparing a photon to a proton with the same momentum, we see a wonderful intersection of classical and quantum physics. This equality allows us to calculate a photon's De Broglie wavelength, a concept that bridges particles and waves.

The kinetic energy of a proton, or any other particle, tells us how much work it can do based on its motion. For a particle with mass, like a proton, the kinetic energy is derived from its velocity and can be calculated with
\( KE = \frac{1}{2} m v^2 \)
where \( m \) is the mass of the proton and \( v \) is its velocity. In the case of a proton moving at 1% of the speed of light, such kinetic energy becomes an indicator of its dynamic state. This kinetic energy forms the basis for a variety of fundamental physics calculations, including those involving collisions and particle interactions, and is pivotal in both theoretical and practical applications like particle accelerators.

Einstein's famous equation \( E = mc^2 \) describes the energy-mass equivalence principle, which states that energy and mass are two forms of the same thing and can be converted into each other. In this context, 'mass' refers to the rest mass, or the intrinsic mass of an object at rest.
When looking at particles like protons and photons, this principle manifests in fascinating ways. Photons, for instance, have energy but no rest mass, whereas protons have both mass and energy. When converting energies (e.g., from Joules to MeV) we are essentially using this principle. Understanding this concept is key to grasping the full implications of the exercise that compares the momentum and energies of a photon and a proton.