91Ó°ÊÓ

Relativistic energy is crucial when dealing with particles moving at speeds close to the speed of light. In such cases, classical physics fails to accurately describe the particle's behavior, and we must use Einstein's theory of relativity instead. The relativistic energy formula is given by:\[ E = \sqrt{(pc)^2 + (mc^2)^2} \]where:

• \(E\) is the total energy of the particle,
• \(p\) is the momentum,
• \(c\) is the speed of light,
• \(m\) is the rest mass of the particle.

This equation accounts for the increase in mass that occurs as a particle approaches the speed of light.
When calculating the de Broglie wavelength for particles like electrons, which can move at such high velocities, this formula helps us determine the momentum by solving for \(p\). Once we have \(p\), we can easily use de Broglie’s relation, \(\lambda = \frac{h}{p}\), to find the wavelength. This ensures accuracy when dealing with high-energy or fast-moving particles like the free electron with 2 MeV energy in our problem.

The kinetic energy of a particle is the energy it possesses due to its motion. For non-relativistic speeds, which are much slower than the speed of light, the kinetic energy \(E_k\) is given by:\[ E_k = \frac{1}{2}mv^2 \]where:

• \(m\) is the mass,
• \(v\) is the velocity.

In the context of de Broglie’s theory, knowing the kinetic energy allows us to find the velocity required for subsequent calculations of a particle's wavelength using the de Broglie relation.
For instance, when we consider an electron with a kinetic energy of 20 eV, its velocity can be calculated using the kinetic energy formula. This situation involves speeds where non-relativistic equations apply, simplifying the computations. Once the velocity is determined, it can be plugged into de Broglie’s equation, \(\lambda = \frac{h}{mv}\), to find the wavelength.

Planck's constant \(h\) is one of the pivotal constants in quantum mechanics. It is defined as \(6.626 \times 10^{-34} \mathrm{Js}\) and serves as the bridge between the wave and particle nature of matter. In de Broglie’s framework, Planck's constant helps connect a particle's momentum to its wavelength using the famous relation:\[ \lambda = \frac{h}{p} \]This relation implies that even massive objects have a wavelength, though it is typically extremely small and hard to detect for macroscopic objects, like the jogging woman in the problem.
Planck's constant ensures that we can calculate the de Broglie wavelength for any particle once we know its momentum. It emphasizes the duality nature, showing that matter has both particle and wave characteristics. This duality is especially pronounced and observable for small particles like electrons, which exhibit significant wavelike behavior due to their relatively small mass and large associated wavelengths. In the exercise, Planck's constant remains a crucial part of each wavelength calculation loop.