In an electron microscope, electrons are accelerated by passing them through a voltage difference. The kinetic energy thus acquired by the electrons is equal to the voltage times the charge on the electron. Thus, a voltage difference of \(1 \mathrm{~V}\) imparts a kinetic energy of \(1.602 \times 10^{-19} \mathrm{C} \times \mathrm{V}\) or \(1.602 \times\) \(10^{-19} \mathrm{~J}\). Calculate the wavelength associated with electrons accelerated by \(5.00 \times 10^{3} \mathrm{~V}\).

An electron microscope uses a beam of accelerated electrons to create an image of a specimen. It exploits the shorter de Broglie wavelengths of electrons, which allows it to achieve a higher resolution than a light microscope. When electrons are passed through a voltage difference in an electron microscope, they gain kinetic energy. This energy is directly proportional to the voltage; hence, a greater voltage difference translates to faster moving electrons and subsequently shorter wavelengths, which improves the microscope's ability to resolve smaller details.

Understanding this principle is crucial for calculating the associated wavelength of the electrons, as in the original exercise. In essence, the electron microscope is a practical application of the wave-particle duality of electrons, where their wave nature is harnessed to observe the world at the microscopic scale.

The kinetic energy of electrons is a fundamental concept when examining the behavior of electrons in different contexts, including electron microscopes. It is given by the equation KE = eV, where KE is the kinetic energy, e is the charge of an electron (\(1.602 \times 10^{-19}\text{ C}\)), and V is the voltage through which the electrons are accelerated. The kinetic energy indicates how fast the electrons are moving; higher kinetic energy means higher velocity.

In the context of the original exercise, calculating the kinetic energy is the first step in determining the de Broglie wavelength of accelerated electrons. The underlying notion is that this energy conversion from electric potential to motion is what later enables us to link to the wave-like characteristic of the electron via the de Broglie relationship.

Wave-particle duality is a fundamental principle of quantum mechanics that posits that every particle or quantic entity exhibits both wave and particle properties. For electrons, this means they can be described not only in terms of particles moving in paths but also as waves spreading out in space. This dual behavior is harnessed in tools like the electron microscope, where the wave nature is particularly significant.

The de Broglie hypothesis is a direct consequence of wave-particle duality, suggesting that all matter has a wave-like nature characterized by a wavelength, given by de Broglie's equation \(λ = h / p\), where \(λ\) is the wavelength, \(h\) is Planck's constant, and \(p\) is momentum. Calculating the de Broglie wavelength of an electron involves understanding both its kinetic energy (particle aspect) and the corresponding wavelength (wave aspect), illustrating this duality.

Planck's constant (denoted as \(h\)) is a physical constant that plays a vital role in the quantum world, including the calculation of the de Broglie wavelength. It is the proportionality factor between the energy (\(E\)) of a photon and the frequency (\(ν\)) of its associated electromagnetic wave, given by the equation \(E = hν\). This constant has a value of approximately \(6.62607015 \times 10^{-34} \text{ J\cdot s}\).

In relation to the original problem, Planck's constant is used to solve for the de Broglie wavelength by expressing the electron's momentum in terms of its kinetic energy and substituting this into the de Broglie equation. The constant serves as the bridge between the energy and wave aspects of particles, enabling us to quantify the wave nature of electrons in practical applications such as the electron microscope.