The de Broglie wavelength of an electron that has been...

The de Broglie wavelength of an electron that has been accelerated through a potential difference of $\phi$ is given by
\[
\lambda=\left(\frac{h^{2}}{2 m e \phi}\right)^{1 / 2}
\]
Derive this equation and verify that it is correct to write equation 24.12

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Key Concepts

de Broglie Hypothesis and Wave-Particle Duality

The de Broglie hypothesis asserts that every particle exhibits wave-like properties, with the wavelength being inversely proportional to its momentum (? = h/p). This concept, which unifies the notions of particles and waves, is essential in understanding phenomena such as electron diffraction and the broader implications of quantum mechanics.

Electron Acceleration and Energy Gain

When an electron is accelerated through a potential difference, it gains kinetic energy equal to the product of its charge and the potential difference. This conversion of electric potential energy to kinetic energy is a fundamental concept in both classical and quantum physics, explaining how external electric fields influence the motion of charged particles.

Momentum from Kinetic Energy

For a non-relativistic electron, the kinetic energy is related to its momentum by the equation K.E. = p²/(2m). This relationship allows one to express the momentum in terms of the kinetic energy acquired during acceleration, serving as a critical link between energy concepts and particle dynamics.

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