Question

What is the meaning of matter-wave of Louis de Broglie? Deduce the relation for wavelength of de Broglie in terms of kinetic energy.

Hint:

According to de Broglie, matter exhibits wave-particle duality. This is the foundation for quantum mechanics and was confirmed experimentally by the Davisson-Germer experiment.

Answer 1

Louis de Broglie proposed that every moving particle is associated with a wave. This wave is known as the matter wave or de Broglie wave. He suggested that particles such as electrons, protons, and even macroscopic objects exhibit both wave-like and particle-like behavior, which is a fundamental principle of quantum mechanics.
de Broglie Hypothesis:
The wavelength \( \lambda \) associated with a particle of momentum \( p \) is given by:
\[ \lambda = \frac{h}{p} \] where:
\( h \) = Planck’s constant \( (6.626 \times 10^{-34} \, \text{Js}) \)
\( p \) = momentum of the particle = \( mv \)
Now, in terms of kinetic energy:
Kinetic energy \( K.E. = \frac{1}{2}mv^2 \)
Solving for velocity:
\[ v = \sqrt{\frac{2K.E.}{m}} \] Now, \( p = mv = m \cdot \sqrt{\frac{2K.E.}{m}} = \sqrt{2mK.E.} \)
Substitute into de Broglie equation:
\[ \lambda = \frac{h}{\sqrt{2mK.E.}} \] Final Result:
\[ \boxed{\lambda = \frac{h}{\sqrt{2mK.E.}}} \]