Show that \(p=h / \lambda\) and \(E_{f}=h f\) are consistent with the relativistic formula \(E^{2}=p^{2} c^{2}+m_{0}^{2} c^{2}\).

In relativistic physics, the concept of momentum extends beyond its classical definition to account for the effects of relativity at high speeds, nearing the speed of light. The relativistic momentum formula is expressed as \( p = \frac{h}{\lambda} \), where \( p \) is the momentum, \( h \) is Planck's constant, and \( \lambda \) is the wavelength of a particle, such as a photon.

This formula showcases the wave-particle duality inherent in quantum mechanics, highlighting how particles can exhibit both wave-like and particle-like properties. In contexts where the particle's inherent mass becomes substantially relativistic, the momentum formula encapsulates the necessity to integrate quantum mechanics into our understanding of motion.

By using Planck's constant, the formula bridges the gap between the quantum scale and relativistic speeds, and this equation becomes especially pertinent when describing the behavior of particles in high-energy physics, such as in particle accelerators.

The energy formula in the realm of relativistic physics is expressed as \( E_f = hf \), where \( E_f \) denotes the energy of a photon, \( h \) is Planck's constant, once again, and \( f \) is the frequency of the photon.

This equation is fundamental in describing the energy carried by quantum particles like photons, which are essentially 'packets' of light or other electromagnetic radiation. The formula encapsulates the idea that energy is quantized and directly proportional to frequency—higher-frequency light carries more energy per photon.

The relationship between energy and frequency, articulated through this formula, is essential for understanding phenomena such as the photoelectric effect, where electrons are ejected from materials upon the absorption of light. This principle has practical applications in areas ranging from solar energy conversion to digital imaging.

The relativistic energy-momentum relationship is an equation that unites the concepts of energy and momentum in the framework of special relativity. It is given by \( E^2 = p^2c^2 + m_0^2c^4 \), where \( E \) is the total energy of a particle, \( p \) is its momentum, \( c \) is the speed of light, and \( m_0 \) is the rest mass of the particle.

This profound equation reflects the interconnectedness of energy and momentum while also incorporating the rest mass energy of a particle, expressed by the famous \( E = mc^2 \) when a particle is at rest (i.e., momentum, \( p \) is zero). It shows that the total energy of a particle is not merely the sum of its kinetic energy and mass energy but rather a relationship that must account for the geometric-like addition of these quantities in space-time.

Through this formula, we understand that a particle's total energy increases with its speed, revealing how mass and energy are essentially two sides of the same coin in the relativistic regime. It encapsulates the idea that you cannot consider energy without momentum and vice versa when dealing with particles moving at speeds close to that of light. Understanding this relationship is crucial for accurately describing numerous high-energy processes in physics, such as those occurring in supernovae, particle colliders, or nuclear reactions.