Matter Waves And Kinetic Energy

For a particle of kinetic energy $K$, NCERT gives the de Broglie wavelength as $\lambda =\frac{h}{\sqrt{2mK}}$, since momentum relates to kinetic energy by $p=\sqrt{2mK}$. When two particles share the same kinetic energy, the wavelength depends inversely on the square root of the mass, so the lighter particle has the longer wavelength. Because an electron is roughly 1836 times lighter than a proton, at equal kinetic energy the electron's wavelength is about $\sqrt{1836}\approx 43$ times longer. The option favouring the proton is wrong because although the proton does move slower, its far greater mass gives it larger momentum, hence a shorter wavelength. The option claiming identical wavelengths is wrong because equal energy does not imply equal momentum when masses differ. The option denying a defined wavelength is wrong because every moving particle possesses a definite de Broglie wavelength. As stated in NCERT Class 12, Chapter 11, matter-wave wavelength decreases with increasing mass at fixed energy. This mass dependence is why electrons, being the lightest common charged particle, are the preferred probe for diffraction studies at a given energy, since their longer wavelength produces more widely spaced and thus more measurable fringes. Neutrons, being far heavier, require correspondingly lower energies to achieve comparable wavelengths for neutron diffraction. A consistency check confirms that the lighter electron, with smaller momentum for the same energy, correctly has the longer wavelength.