Find the Lorentz factor \(\gamma\) and de Broglie's wavelength for a \(50-\mathrm{GeV}\) electron in a particle accelerator.

The concept of de Broglie wavelength is a fundamental principle in quantum mechanics, which marries the classical and quantum world. It is named after the French physicist Louis de Broglie, who proposed that any moving particle can be described by a wave. The wavelength of this wave, known as the de Broglie wavelength, is given by the relation \( \lambda = \frac{h}{p} \) where \( h \) is Planck's constant, and \( p \) is the momentum of the particle.

Planck's constant (\( h \) is a very small number indicating that the wave nature becomes significant at a microscopic scale, typically for particles like electrons, protons, or molecules. In the exercise, the de Broglie wavelength is sought for an electron accelerated to high energies, making it a pertinent concept in understanding particle behavior at subatomic scales in environments such as particle accelerators.

It's essential to realize that while these particles have wave-like characteristics, they still possess particle attributes, which is the core of wave-particle duality. To calculate the de Broglie wavelength in practical scenarios, you need the momentum, which for relativistic particles like electrons in a particle accelerator, would require relativistic momentum calculations.

Relativistic momentum is a concept derived from Einstein’s theory of special relativity, which states that the laws of physics are the same for all non-accelerating observers, and that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer. The momentum (\( p \) of a particle at high velocities approaches an infinite magnitude as the speed approaches that of light (\( c \). The relativistic formula for momentum is given by \( p = \frac{mv}{\sqrt{1 - v^2/c^2}} \), where \( m \) is the rest mass of the particle and \( v \) is its velocity.

This equation reveals that at low speeds, where \( v \) is much less than \( c \) the momentum reduces to its classical form \( p = mv \). However, at speeds approaching \( c \), the denominator becomes significant—indicating a considerable deviation from classical mechanics. In the context of the exercise, understanding relativistic momentum is crucial as it is used in the calculation of both the Lorentz factor and the de Broglie wavelength. Unlike classical momentum, the relativistic version explains why particles cannot reach or exceed light speed, as such an increase in velocity would require infinite energy due to an ever-increasing mass.

Particle accelerators are sophisticated machines that use electric fields to propel charged subatomic particles, such as electrons and protons, to high speeds and contain them in well-defined beams. These high-speed particles are often used in various fields, including fundamental physics research, medical therapies, and materials science.

In particle accelerator physics, particles are accelerated to relativistic speeds—that is, speeds that are a significant fraction of the speed of light. This is where concepts like the Lorentz factor and relativistic momentum become pivotal. The Lorentz factor (\( \gamma \) determines the amount of time dilation, length contraction, and mass increase experienced by objects as their speed grows towards the speed of light.

As applied to the exercise, calculating the Lorentz factor for an electron within a particle accelerator is key to understanding the kinematics and dynamics of the particle. The increase in speed also leads to a change in de Broglie wavelength, which affects how these particles interact with one another and with other forms of matter. Thus, grasp of particle accelerator physics is fundamental for those working in both theoretical and applied aspects of particle physics.