Practice portal › Dual Nature of Matter and Radiation › Comparing de Broglie Wavelengths of Particles

The de Broglie wavelengths of a proton and an α-particle are the same. The ratio of their velocities (vₚ:v_α) will be ______. (An α-particle is a He nucleus having two protons and two neutrons, so m_α = 4mₚ.)

Asked in RS Academy GUJCET booklet · Equal wavelengths and equal speeds

• (1) 1:4
• (2) 1:2
• (3) 2:1
• (4) 4:1

Step-by-step solution

Why the other options are wrong

• (1) This is the ratio upside down, as if speed rose with mass. For equal momentum the lighter proton must move faster.
• (2) No single slip gives 1:2. Sanity check: equal wavelengths mean equal momenta, and the proton is the lighter particle, so it must be the faster one: vₚ:v_α is greater than 1.
• (3) This treats equal wavelength as equal kinetic energy, which gives v ∝ 1/(√m) and √4 = 2. Equal wavelength means equal momentum, so v ∝ 1/m.

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