The wavelength of an electron moving with velocity of \(10^{7} \mathrm{~m} \mathrm{~s}^{-1}\) is (a) \(7.27 \times 10^{-11} \mathrm{~m}\) (b) \(3.55 \times 10^{-11} \mathrm{~m}\) (c) \(8.25 \times 10^{-4} \mathrm{~m}\) (d) \(1.05 \times 10^{-16} \mathrm{~m}\)

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics, including quantum chemistry, quantum field theory, quantum technology, and quantum information science. At its core, quantum mechanics deals with phenomena that classical physics cannot explain, such which explains the behavior and interactions of particles like electrons and photons.

One of the hallmarks of quantum mechanics is the dual nature of matter and light, which means they can exhibit both wave-like and particle-like characteristics. The de Broglie hypothesis, which posits that all matter has a wave associated with it, is a classic example of quantum mechanics in action. This concept is vital in understanding how electrons move around an atom and play a key role in technologies like electron microscopy and quantum computing.

To calculate the wavelength of an electron, we can invoke the de Broglie hypothesis. The formula to find the wavelength \(\lambda\) of an electron is given by \(\lambda = \frac{h}{p}\), where \(h\) is the Planck constant and \(p\) is the momentum of the particle. The Planck constant \(h\) is a physical constant that is essential in the realm of quantum mechanics and its value is approximately \(6.62607015 \times 10^{-34}\) J·s.

In practice, to find an electron's wavelength, you must know its momentum. If only the electron's velocity is known, then the momentum can be found first. Once you have the electron's momentum, you can apply de Broglie's equation to find its wavelength. This wavelength is not something you can see with the naked eye, but it can be detected with scientific instruments and is fundamental to the understanding of electron behavior in quantum systems.

In physics, momentum of a particle is a vector quantity that is the product of the mass \(m\) and velocity \(v\) of the particle. It is described by the second law of motion, which states that the rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the direction of that force. The formula to calculate the momentum \(p\) of a particle is given by \(p = mv\), with mass measured in kilograms and velocity in meters per second.

For subatomic particles like electrons, momentum plays a pivotal role in understanding their quantum behavior. Since these particles have very small masses, their momentum can have a significant impact on their overall properties, such as energy and stability. The momentum is also a crucial component in the determination of the de Broglie wavelength, which helps to reveal the wave-like behavior of particles and has implications across quantum mechanics, from explaining atomic orbitals to the operation of lasers.