If E1, E2 and E3 are the kinetic energies of an electron, an α-particle and a proton, with same

step 1 Here the kinetic energy of alpha particle, electron and proton is given. Let E1, E2 and E3 be th

If E1, E2 and E3 are the kinetic energies of an electron, an...

If $E_{1}, E_{2}$ and $E_{3}$ are the kinetic energies of an electron, an $\alpha$ -particle and a proton, with same de-Broglie wavelength, then
(a) $E_{1}>E_{3}>E_{2}$
(b) $E_{2}>E_{3}>E_{1}$
(c) $E_{1}>E_{2}>E_{3}$
(d) $E_{1}=E_{2}=E_{3}$

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Key Concepts

de Broglie Wavelength

This concept refers to the wave-particle duality in quantum mechanics, where every particle is associated with a wavelength given by the de Broglie relation ? = h/p. It links the momentum of a particle to its wave-like properties, and is central to understanding how differences in mass affect other properties like kinetic energy when the wavelength is kept constant.

Momentum-Kinetic Energy Relation

In non-relativistic physics, kinetic energy is related to the momentum and mass of a particle by the equation KE = p²/(2m). For particles with the same de Broglie wavelength (and hence the same momentum), the kinetic energy becomes inversely proportional to the mass. This relation explains how particles with different masses, but identical momenta, acquire different kinetic energies.

Mass Dependence of Kinetic Energy

Since the kinetic energy depends inversely on mass for a given momentum, lighter particles will have higher kinetic energies compared to heavier ones. This principle is crucial when analyzing scenarios involving particles like electrons, alpha particles, and protons that have the same de Broglie wavelength, as their significantly different masses result in distinctly different kinetic energies.

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