∙(a) What is the de Broglie wavelength of an electron...

$\bullet$ (a) What is the de Broglie wavelength of an electron accelerated through 800 $\mathrm{V} ?$ (b) What is the de Broglie wavelength of a proton accelerated through the same potential difference?

Key Concepts

de Broglie Hypothesis

The de Broglie hypothesis posits that every particle exhibits wave-like properties, characterized by a wavelength that is inversely related to its momentum. This concept is central to quantum mechanics, as it bridges the gap between classical and quantum descriptions of matter, indicating that matter can display interference and diffraction phenomena similar to those of light.

Kinetic Energy from Potential Difference

When a charged particle is accelerated through a potential difference, the electric potential energy is converted into kinetic energy. This kinetic energy is given by the product of the charge of the particle and the voltage, establishing a direct link between the applied potential and the particle's speed, which in turn affects its momentum and de Broglie wavelength.

Momentum and the de Broglie Relation

The de Broglie wavelength of a particle is determined by the relation ? = h/p, where h is Planck's constant and p is the momentum of the particle. For non-relativistic speeds, momentum can be related to the kinetic energy (derived from the potential difference) using the formula p = ?(2mE), with E representing the kinetic energy, highlighting the dependence of the wavelength on both the particle's mass and the accelerating voltage.

Mass Dependence in de Broglie Wavelength

The mass of a particle plays a critical role in determining its de Broglie wavelength when accelerated through a given potential difference. A lighter particle, having a smaller mass, will have a larger velocity and hence a smaller momentum compared to a heavier particle; this results in a longer wavelength for the lighter particle and a shorter wavelength for the heavier one, illustrating the inverse relationship between momentum (which depends on mass) and the wavelength.

Transcript

Please give Ace some feedback