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Photon energy is a fundamental concept in quantum mechanics. It quantifies the energy carried by a photon, which is a particle of light. This energy depends on the photon's wavelength, as described by the formula:

• \( E = \frac{hc}{\lambda} \)

Here, \( E \) is the energy, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \; \text{Js}) \), \( c \) is the speed of light \( (3 \times 10^8 \; \text{m/s}) \), and \( \lambda \) is the wavelength. The shorter the wavelength, the higher the energy of the photon. In this calculation, for a 0.10 µm wavelength, the energy approximates to \( 1.986 \times 10^{-18} \; \text{J} \).
Photon energy plays a crucial role in various applications such as solar power, where photons are transformed into electrical energy. Understanding this energy is also key in fields like photonics and quantum computing, where light's discrete nature is explored.

The de Broglie wavelength reflects the wave-particle duality of matter, a centerpiece of quantum mechanics. Louis de Broglie postulated that not just light has wave properties, but all matter does, too. This wavelength is defined as:

• \( \lambda = \frac{h}{p} \)

Where \( \lambda \) is the de Broglie wavelength, \( h \) is Planck's constant, and \( p \) is the momentum of the particle. For a particle with velocity \( v \), momentum \( p = mv \).
In applications like electron microscopy and quantum physics, the wave properties of particles allow for phenomena like interference and diffraction. For an electron in this exercise, expressing wave nature requires acceleration through a voltage so its de Broglie wavelength corresponds to a certain physical size (like a 0.10 µm pinhole). Calculating this yields an acceleration potential difference of around 150.4 volts.

Electron acceleration involves increasing the speed of electrons using an electric field, accomplished by applying a potential difference, \( V \). This involves converting electric potential energy to kinetic energy

• \( eV = \frac{1}{2}mv^2 \)

where \( e \) is the charge of an electron, \( m \) is its mass, and \( v \) is its velocity. Rearranging for \( v \) allows us to find how fast an electron moves after being accelerated through a potential \( V \).
In this example, applying a potential of 150.4 volts accelerates the electrons to speeds of approximately \( 5.78 \times 10^6 \; \text{m/s} \). Electron acceleration is pivotal in devices like cathode ray tubes and particle accelerators, showing how energy transformations manifest across various technologies.

Wave-particle duality is a core concept of quantum mechanics, suggesting that all particles exhibit both wave and particle characteristics. This paradigm shift from classical mechanics revolutionizes our understanding of fundamental processes.

• Light behaves as particles (photons) in phenomena like the photoelectric effect, but as waves in interference and diffraction.
• Matter, such as electrons, shows wave properties under certain conditions, evidenced when they pass through small apertures.

Understanding this duality helps explain experimental results that one could not rationalize using classical physics alone. For example, electrons demonstrating interference patterns after passing through a pinhole highlight their wave nature, contributing to technologies like quantum computing and tunneling in semiconductors. Explaining everyday world observations goes beyond tangible perceptions, accounting for quantum oddities described by duality.