What is the de Broglie wavelength of an electron traveling at \(1.35 \times 10^{5} \mathrm{m} / \mathrm{s}\) ?

Quantum Mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It introduces counterintuitive concepts such as wave-particle duality, in which particles such as electrons exhibit both wave-like and particle-like properties. This dual nature is central to understanding phenomena that cannot be explained by classical physics.

For instance, the de Broglie wavelength is a concept derived from quantum mechanics, which suggests that every moving particle or object has an associated wave. The wavelength of this wave for a moving particle is inversely proportional to its momentum, a concept known as the de Broglie hypothesis. It is this very idea that bridges the classical and quantum worlds, allowing us to calculate the wavelength of particles like electrons using Planck's constant and the particle's momentum.

Electrons are subatomic particles with several intrinsic properties such as charge, mass, and spin. They play a vital role in chemistry, electricity, and quantum mechanics.

One of the key characteristics of an electron is its rest mass, which is approximately \(9.109 \times 10^{-31} \mathrm{kg}\). Even though this mass is extremely small, it is essential for calculating the momentum of electrons in motion, which in turn is used to determine their de Broglie wavelength. The charge of an electron is equally vital in understanding its interactions with other particles and electromagnetic fields, influencing how electrons bind to nuclei to form atoms.

In classical mechanics, the momentum of a particle is a product of its mass and velocity, given by the equation \(p = m \times v\). This relationship also holds true in the context of quantum mechanics for calculating the momentum of microscopic particles such as electrons.

The concept of momentum is especially important when we delve into the realm of quantum mechanics, where the momentum of a particle is directly related to its wave-like properties as postulated by de Broglie. Understanding an electron's momentum enables us to utilize de Broglie’s equation to find its wavelength, thus revealing the quantum nature of the electron. The ability to compute an electron's de Broglie wavelength can provide insight into its behavior in various scientific applications, from electron microscopy to quantum computing.