Find the de Broglie wavelength of electrons with kinetic energies (a) \(10 \mathrm{eV},\) (b) \(1.0 \mathrm{keV},\) and (c) \(10 \mathrm{keV}\)

Quantum physics is a fundamental theory in physics that describes nature at the smallest scales, such as atomic and subatomic levels. It is in contrast with classical physics, which deals with the macroscopic world we are accustomed to. One of the most revolutionary concepts in quantum physics is the wave-particle duality, which posits that particles like electrons exhibit both particle-like and wave-like properties.

Understanding this duality is essential when studying the nature of electrons, as in the case of de Broglie wavelength calculations. According to Louis de Broglie, a French physicist, matter has an associated wave, and its wavelength is inversely proportional to its momentum. This dual nature of matter is pivotal in explaining various quantum phenomena and is especially important in technologies such as electron microscopy and quantum computing.

Electron kinetic energy refers to the energy that an electron possesses due to its motion. An object's kinetic energy can be expressed using the formula \( K.E = \frac{1}{2}mv^2 \), where \(m\) is the mass of the object and \(v\) is its velocity. In the context of quantum physics, it's often necessary to convert this energy into different units, such as electronvolts (eV), because the energies involved at the quantum level are very small.

Electronvolts are a convenient unit in that they provide a direct measure of energy based on the charge of an electron moving through a potential difference of one volt. The energy provided by electron volts is crucial in understanding how it affects the electron's de Broglie wavelength.

Momentum is a fundamental concept in both classical and quantum physics, representing the quantity of motion an object has. For a particle with mass and velocity, it is calculated using the equation \(p = mv\), where \(p\) is the momentum, \(m\) is the mass, and \(v\) is the velocity of the particle.

In quantum mechanics, the momentum calculation for particles such as electrons can be a bit different, especially when dealing with kinetic energy. This is where the relationship \(p = \sqrt{2mK.E}\) comes into play, where the momentum is derived from the particle's kinetic energy (K.E). It’s an important step in calculating the de Broglie wavelength because it provides one of the two necessary components for the de Broglie equation.

Planck's constant, denoted as \(h\), is a crucial quantity in quantum physics. It relates the energy of photons to their frequency and is a fundamental part of equations that describe the quantum world. With a value of approximately \(6.62607015 \times 10^{-34} m^2 kg/s\), Planck's constant sets a scale for action, or the product of energy and time, in quantum mechanics.

The constant plays a pivotal role in the de Broglie wavelength equation which states that the wavelength (\(\lambda\)) of a particle is equal to Planck's constant divided by the particle's momentum (\(p\)): \(\lambda = \frac{h}{p}\). This equation bridges the gap between the classical and quantum worlds, and Planck's constant is the key to understanding how objects we traditionally think of as particles can also behave like waves.