Consider the following statements about the de Broglie wavelength of material particles:
(A) The de Broglie wavelength of a moving particle is related to its momentum p as \( \lambda=h/p \).
(B) For the same kinetic energy, a heavier particle has a longer de Broglie wavelength.
(C) The de Broglie wavelength of a macroscopic object of everyday life is too small to be measured.
(D) For the same momentum, the de Broglie wavelength of a particle depends on its charge.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
(A) The de Broglie wavelength of a moving particle is related to its momentum p as \( \lambda=h/p \). — True. This is the de Broglie relation. It links a wave property, the wavelength, to a particle property, the momentum.
(B) For the same kinetic energy, a heavier particle has a longer de Broglie wavelength. — False. A particle of mass \( m \) and kinetic energy \( K \) has momentum \( p=\sqrt{2mK} \), so
\( \displaystyle \lambda=\frac{h}{\sqrt{2mK}} \)
For the same \( K \), the heavier particle has the larger momentum, so it has the shorter wavelength.
(C) The de Broglie wavelength of a macroscopic object of everyday life is too small to be measured. — True. Everyday objects have large masses and so large momenta. A ball of mass 0.12 kg moving at \( 20\,\mathrm{m\,s^{-1}} \) has \( \lambda\approx2.76\times10^{-34}\,\mathrm{m} \). This is far too small to be measured. So the wave nature of such objects is never observed.
(D) For the same momentum, the de Broglie wavelength of a particle depends on its charge. — False. The relation \( \lambda=h/p \) contains only the momentum. Two particles of the same momentum have the same wavelength whatever their charge, or whether they are charged at all.
NCERT (2026-27):
Summary – “9. The de Broglie wavelength (λ) associated with a moving particle is related to its momentum p as: λ = h/p. The dualism of matter is inherent in the de Broglie relation which contains a wave concept (λ) and a particle concept (p). The de Broglie wavelength is independent of the charge and nature of the material particle. It is significantly measurable (of the order of the atomic-planes spacing in crystals) only in case of sub-atomic particles like electrons, protons, etc. (due to smallness of their masses and hence, momenta). However, it is indeed very small, quite beyond measurement, in case of macroscopic objects, commonly encountered in everyday life.”