Q.23
An alpha particle and a proton have the same de Broglie wavelength. Which of the following is also the same for the two particles if they are moving at non-relativistic speeds?
The correct answer is (C) Momentum.
An alpha particle (mass ≈ 4m_p, where m_p is proton mass) and a proton have the same de Broglie wavelength λ = h/p at non-relativistic speeds, directly implying identical momentum p for both.
De Broglie Relation
De Broglie wavelength is λ = h/p, where h is Planck’s constant. Same λ means same p = h/λ, regardless of mass differences.
Option Analysis
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(A) Frequency: Frequency f = E/h relates to kinetic energy E = p²/2m. Since m_alpha > m_proton but p is same, E_alpha = p²/(2×4m_p) < E_proton = p²/(2m_p). Different E means different f.
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(B) Kinetic energy: As above, E ∝ 1/m for fixed p (non-relativistic KE = p²/2m), so E_proton = 4 E_alpha. Not same.
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(C) Momentum: Directly from λ = h/p, same λ gives same p. Correct.
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(D) Speed: Speed v = p/m. v_proton = p/m_p, v_alpha = p/(4m_p), so v_proton = 4 v_alpha. Not same.
Introduction to de Broglie Wavelength
The de Broglie wavelength concept shows particles like alpha particles and protons exhibit wave-particle duality, with λ = h/p where p is momentum. When an alpha particle and proton share the same de Broglie wavelength at non-relativistic speeds, key properties like momentum align directly.
Why Momentum Remains Same
From λ = h/p, equal wavelengths mean equal momentum p for both particles, as h is constant. Alpha particle mass (4× proton mass) does not alter this fundamental relation.
Comparing Other Properties
| Property | Proton | Alpha Particle | Same? |
|---|---|---|---|
| Frequency (f = E/h) | Higher (E = p²/2m_p) | Lower (E = p²/8m_p) | No |
| Kinetic Energy | p²/2m_p | p²/8m_p | No |
| Speed (v = p/m) | p/m_p | p/4m_p | No |
| Momentum | p | p | Yes |
CSIR NET Exam Relevance
This question tests wave-particle duality understanding, common in CSIR NET Physical Sciences. Non-relativistic assumption simplifies to classical KE = p²/2m.
