A proton, an electron and an alpha particle have the same energies. Their de-Broglie wavelengths will be compared as :

Solution

Physics sir · 2 steps

Same energy means wavelength depends only on mass: the lighter particle has the longer wavelength.

1. Momentum from energy. Kinetic energy is E = p²/2m, so p = √2mE. Put this into λ = h/p.

   λ = h/√2mE ⇒ λ ∝ 1/√m (same E)

   Particle	Mass	Wavelength
   Electron	mₑ, lightest	longest
   Proton	mₚ ≈ 1836 mₑ	middle
   Alpha	m_α ≈ 4 mₚ, heaviest	shortest

2. So. λ_α < λₚ < λₑ, which is option (b).

The idea to keep. For equal kinetic energy, use λ = h/√2mE. Then λ ∝ 1/√m, so a heavier particle always has a shorter wavelength.

Answer from the NTA final answer key. Solution by PrepWiser.