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

An α-particle and a deuteron are moving with velocities v and 2v respectively. What will be the ratio of their de Broglie wavelengths?

Asked in GUJCET 2008 · Matter waves in other settings

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

Step-by-step solution

Why the other options are wrong

• (1) This uses the equal-energy rule λ ∝ 1/(√m) and ignores the speeds: √(m_d)/(m_α) = √2/4 = 1/(√2). The kinetic energies are not equal here, and λ = h/(mv) needs both mass and speed.
• (2) This ignores the masses and uses only λ ∝ 1/v, giving (2v)/v = 2. The α-particle is twice as heavy as the deuteron, which exactly cancels its lower speed.
• (4) This uses λ ∝ 1/(√K) and forgets the mass inside λ = h/(√2mK): the deuteron's kinetic energy is twice the α-particle's, giving √2. With the masses included, mK is the same for both.

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