What is the energy of a photon whose momentum is \(3.0 \times 10^{-24} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} ?\).

In physics, understanding the properties of light is essential, and one such property is the momentum of a photon. Unlike objects with mass, photons are massless particles, but they do carry momentum. The concept may seem counterintuitive since momentum is typically associated with mass, but in the realm of quantum mechanics, light behaves both as a wave and as a particle, linking it to possessing momentum.

The momentum (\(p\)) of a photon is related to its wavelength (\( \text{λ} \text{m/s}\)) by the equation:\[ p = \frac{h}{\text{λ}}\]where \(h\) is Planck's constant, approximately \(6.626 \times 10^{-34} \text{m}^2\text{kg} / \text{s}\). This relationship shows that despite lacking mass, the photon's momentum is a consequence of its wave properties. For students striving to grasp this phenomenon, visualize how light waves transfer energy and momentum when they interact with surfaces, similar to how a ball transfers energy as it bounces.

The speed of light in a vacuum, symbolized as \(c\), is a fundamental constant in the universe, measuring roughly \(3 \times 10^8 \text{m/s}\). This constant does more than just represent the speed at which light travels; it is a cornerstone of Einstein's theory of relativity and plays a crucial role in various equations and phenomena in physics.

Understanding the speed of light is critical when calculating the behavior of photons and interpreting their interactions with matter. It's also essential for students to recognize that no information or matter can travel faster than \(c\), making it a universal speed limit. In practice, the speed at which light travels enables various technologies like GPS and laser measurements, by allowing precise calculations of distances and times based on the constant speed of light.

The energy-momentum relationship is a fundamental principle that ties the momentum of a particle to its energy. For a photon, this takes a specific form, as photons have no rest mass, and their energy (\(E\)) and momentum (\(p\)) are directly proportional to each other, interconnected by the speed of light (\(c\)), as shown in the equation:\[ E = pc \]This equation is highly significant because it links the concepts of energy and momentum in the context of light and other electromagnetic radiation. A key takeaway for students is that as the momentum of a photon increases, so does its energy in a linear relationship.

Interactions involving photons, such as the photoelectric effect, can be understood and predicted using this relationship. Additionally, it’s a clear illustration of how the principles of quantum mechanics correspond with those of classical mechanics, bridging the gap between the behaviors of subatomic particles and macroscopic objects.