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When particles travel at high speeds, close to the speed of light (\(c\)), we can't use the regular formula for momentum due to the effects of relativity. Traditional momentum is calculated as \(p = mv\), where \(m\) is mass and \(v\) is velocity. However, at speeds close to the speed of light, we must account for relativistic effects using the formula: \(p = \gamma m v\).
Relativistic momentum is crucial when dealing with particles moving at a significant fraction of light speed, such as electrons in this problem. The reason is their mass effectively increases due to relativistic effects.
The "\(\gamma\)" in the equation stands for the Lorentz factor, which plays a pivotal role in ensuring the momentum accounts for these high-speed tweaks. By using this adjusted momentum, physicists can accurately predict behaviors and interactions at tiny scales and high velocities.

Planck's constant is fundamental in linking the world of the tiny particles to wave phenomena. Represented by \(h\), its value is approximately \(6.626 \times 10^{-34} \, \text{m}^2 \text{kg/s}\). This constant appears in the de Broglie equation \(\lambda = h/p\), connecting wave properties with particle momentum.

• It's a bridge between classical physics and quantum mechanics.
• Quantifies the action required to shift the energy level for particles such as photons and electrons.

In this exercise, Planck’s constant helps us find the wavelength of electrons moving at relativistic speeds. By dividing Planck’s constant by the momentum calculated using the relativistic formula, we can determine the de Broglie wavelength, revealing crucial insights into wave-particle dual nature.

The Lorentz factor, symbolized by \(\gamma\), modifies calculations of time, length, and other physical quantities for particles moving close to the speed of light. It's calculated as \(\gamma = 1 / \sqrt{1 - (v/c)^2}\), where \(v\) is velocity and \(c\) is speed of light.
In the realm of special relativity:

• The Lorentz factor quantifies how much relativistic effects alter physical parameters.
• For example, it shows how time dilation and length contraction happen as \(v\) approaches \(c\).

In this problem, \(\gamma\) ensures the momentum values are accurate at high velocities. For velocities such as \(0.480c\) and \(0.960c\), using the Lorentz factor helps account for the difference between everyday Newtonian expectations and what really happens at near-light speeds.