Find the ratios of de Broglie wavelengths of a \(100-\mathrm{MeV}\) proton to a 100 - MeV electron.

Planck's constant, represented by the symbol \( h \), is a fundamental parameter in quantum mechanics. It has a value of approximately \( 6.626 \times 10^{-34} \text{ Js} \) (joule seconds) and it is the quantum of electromagnetic action that relates the energy carried by a photon to its frequency. More formally, Planck's constant defines the relationship between energy \( E \) and the frequency \( u \) of a photon: \[ E = hu \].

This constant is also essential in explaining the particle-wave duality concept, which is exemplified in the de Broglie hypothesis. According to de Broglie, every particle exhibits wave-like properties, with the de Broglie wavelength \( \text{(}\text{λ}\text{)} \) of the particle being inversely proportional to its momentum \( p \), as defined by the formula \( \text{λ} = \frac{h}{p} \). Understanding Planck's constant offers insight into the quantized nature of light and matter, providing a bridge between classical and quantum theories.

In classical mechanics, momentum of a particle is defined as the product of its mass \( m \) and velocity \( v \), represented by \( p = mv \). However, in the context of quantum mechanics and relativity, the concept of momentum is more nuanced. For particles with mass, momentum becomes more significant at high velocities (near the speed of light, \( c \)), where relativistic effects must be considered.

The de Broglie wavelength's dependence on momentum means that as the momentum of a particle increases, its wavelength decreases. This inverse relationship is crucial in understanding the behaviour of particles at atomic and subatomic scales. Since particles at these scales can have very high velocities, their momentum cannot be accurately described using classical mechanics alone, which leads to the inclusion of relativistic modifications for high energy particles, such as those encountered in particle accelerators or cosmic rays.

The relativistic energy-momentum relationship is a cornerstone of Einstein's theory of relativity, enriching our understanding of how energy, momentum, and mass relate at high velocities. It is given by the equation \[ E = \text{└}\!\!\!└ sqrt{(pc)^2 + (mc^2)^2} \], where \( E \) is the total energy of the particle, \( p \) is its momentum, \( m \) its rest mass, and \( c \) the speed of light in vacuum. It shows that the energy of a particle grows rapidly as its velocity approaches the speed of light, reflecting the necessity to include the energy contribution of mass as well as the kinetic energy.

When dealing with high-energy particles like protons and electrons (as in the given textbook problem), the energies are significant enough that relativistic effects cannot be ignored. Hence, finding the particle momentum is not as straightforward as in classical physics, and we must use this relationship to compute momentum correctly for high-speed particles. This calculation is crucial for predicting observed phenomena, such as particle speeds in particle accelerators or the decay products of high-energy collisions in particle physics experiments.