Compared to the electron in Problem 69, the wavelength of an...

Compared to the electron in Problem 69, the wavelength of an electron accelerated from rest across a 10,000-V potential difference would be
(a) the same length (b) 10 times longer
(c) 1000 times longer (d) 1>10 times as long
(e) 1/1000 times as long

Key Concepts

de Broglie Wavelength

This concept describes how particles such as electrons exhibit wave-like behavior. The wavelength associated with a particle is given by ? = h/p, where h is Planck's constant and p is the momentum of the particle. Understanding the de Broglie wavelength is fundamental in quantum mechanics and explains phenomena like electron diffraction.

Kinetic Energy from Potential Difference

When a charged particle such as an electron is accelerated from rest through a potential difference, its gain in kinetic energy is equal to the product of its charge and the potential difference (KE = eV). This energy determines the electron's velocity and momentum, directly influencing its de Broglie wavelength.

Relationship Between Energy and Momentum

For non-relativistic particles, kinetic energy and momentum are related by the formula KE = p²/(2m). This relationship shows that as the potential difference increases, the kinetic energy and consequently the momentum of the electron increase, which leads to a decrease in its de Broglie wavelength.

Scaling Laws in Quantum Mechanics

In the context of accelerating electrons, scaling laws reveal that the de Broglie wavelength is inversely proportional to the square root of the kinetic energy (or the potential difference). That is, an increase in the voltage by a factor changes the wavelength by the reciprocal of the square root of that factor, which is essential for comparing changes in de Broglie wavelengths under different conditions.

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