(a) Find the momentum of a 100 -keV X-ray photon. (b) Find the velocity of a neutron with the same momentum. (c) What is the neutron's kinetic energy in eV?

Understanding the momentum of X-ray photons can be a bit counterintuitive because we're dealing with particles that have no mass—photons. The momentum of a photon is directly related to its energy; the higher the energy, the more momentum it carries.

To calculate the momentum of an X-ray photon, we use the formula \(p = \frac{E}{c}\), where \(p\) is the momentum, \(E\) is the energy, and \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8\) meters per second. Since the energy of an X-ray photon is typically given in electron volts (eV), it is crucial to convert it into Joules (J) using the conversion factor \(1\text{eV} = 1.6 \times 10^{-19} J\) before using it in the momentum formula.

For example, a 100-keV X-ray photon has significant momentum, but because photons are massless, this momentum is a result of its high energy and the intrinsic properties of light as described by quantum mechanics.

When it comes to comparing subatomic particles like neutrons to photons, the concept of momentum bridges the gap between their different natures. To find the velocity of a neutron with the same momentum as an X-ray photon, we leverage the formula for momentum \(p = mv\), where \(m\) is the mass of the particle and \(v\) is its velocity.

For neutrons, which have a rest mass, determining velocity is straightforward once you've established momentum: \(v = \frac{p}{m}\). Given that a neutron's mass is approximately \(1.675 \times 10^{-27}\) kilograms, calculating the velocity with a known momentum involves a basic algebraic rearrangement of the momentum equation. This calculation gives us the neutron's velocity, which, despite its mass, can reach very high values due to the smallness of its mass on the atomic scale. However, these speeds are substantially slower than the speed of light, reflecting the limits imposed by special relativity on particles with mass.

Kinetic energy is a measure of the energy associated with an object's motion. For particles at the atomic or subatomic level, such as neutrons, this energy can be expressed in electron volts (eV), a unit commonly used in the realm of particle physics and quantum mechanics.

To convert kinetic energy in Joules to eV, we use the conversion factor \(1\text{eV} = 1.6 \times 10^{-19} J\), simplifying the comparison of energy levels across instances. The formula for kinetic energy, \(K = \frac{1}{2}mv^2\), relates the mass (\(m\)) of a particle to its velocity (\(v\)), indicating that faster-moving particles will possess greater kinetic energy. After finding the kinetic energy in Joules, the conversion gives us the energy in eV. This converted energy is especially relevant in contexts like nuclear and atomic physics, where eV provides a more intuitive scale for the energies involved.