Dual Nature · Chapter 03
Matter Waves And De Broglie Hypothesis
Understand particle waves, electron wavelength, diffraction, wave packets and the experiments that revealed quantum matter.
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→λ = h/p
1. Dual Nature of Matter
Classical Particle Picture
A classical particle has a definite position and momentum and follows a trajectory. This picture works well when quantum wavelengths are tiny compared with the apparatus.
Quantum Wave Picture
Quantum mechanics assigns a complex probability amplitude. Its phase produces interference and diffraction, while measurement outcomes remain particle-like and localized.
Complementarity
Wave and particle descriptions are not competing substances. They are complementary experimental manifestations of one quantum object.
Scale of Observation
Wave effects become visible when λ is comparable with slit width, lattice spacing or another relevant experimental dimension.
2. de Broglie Hypothesis
Louis de Broglie proposed in 1924 that the wave-particle symmetry of radiation should extend to matter. A material particle with momentum p has wavelength λ = h/p. The idea does not mean the particle physically wiggles along a sinusoidal path; the wavelength belongs to its quantum phase.
\(\lambda=\dfrac{h}{p}\)
Momentum Form
\(p=h/\lambda=\hbar k\)Energy-Frequency Form
\(E=h\nu=\hbar\omega\)Non-relativistic Form
\(\lambda=h/(mv)\)Kinetic-Energy Form
\(\lambda=h/\sqrt{2mK}\)3. de Broglie Wavelength
Dependence on Momentum
Wavelength decreases as momentum increases. High-momentum macroscopic bodies therefore have unobservably small wavelengths.
Dependence on Mass
At equal speed, the lighter particle has the larger wavelength. At equal momentum, all particles have the same wavelength.
Dependence on Kinetic Energy
For non-relativistic motion λ ∝ 1/√K. Four times the kinetic energy halves the wavelength.
Relativistic Momentum
At high speed use p = γmv or the energy-momentum relation rather than p = mv.
\(\lambda=h/(\gamma mv)\)4. Mathematical Derivations
From photon relations
For a photon, E = hν = hc/λ.
Relativistic photon energy is E = pc.
Equating pc = hc/λ gives p = h/λ.
de Broglie postulated the same momentum-wavelength relation for matter.
From kinetic energy
For a non-relativistic particle K = p²/2m.
Therefore p = √(2mK).
Insert this momentum into λ = h/p.
Electron through voltage
An electron accelerated from rest gains K = eV.
Then p = √(2mₑeV).
Insert into λ = h/p and evaluate constants.
General charged particle
A particle of charge magnitude |q| gains K = |q|V.
Its non-relativistic momentum is √(2m|q|V).
Apply λ = h/p.
5. Electron Wavelength
Electrons are ideal for matter-wave experiments because their small mass gives a measurable wavelength at modest accelerating voltage. For V up to a few kilovolts, the standard non-relativistic relation is highly useful.
SI Form
\(\lambda=h/\sqrt{2m_e eV}\)Angstrom Form
\(\lambda(\text{Å})=12.27/\sqrt V\)Nanometre Form
\(\lambda(\text{nm})=1.227/\sqrt V\)Relativistic Correction
\(\lambda=\dfrac{h}{\sqrt{2m_e eV(1+eV/2m_ec^2)}}\)Use when accelerating voltage is large.
6. Charged Particle Wavelength
General Formula
\(\lambda=h/\sqrt{2m|q|V}\)Same Voltage
At equal V and equal charge magnitude, λ ∝ 1/√m. Electrons have much longer wavelength than protons.
Same Momentum
All particles with the same p have the same λ, independent of mass and charge.
Same Kinetic Energy
At equal K, λ ∝ 1/√m, so the lighter particle has the longer wavelength.
7. Applications
Electron Microscope
Short electron wavelengths permit much finer resolution than visible light. Electromagnetic lenses focus the beam.
Electron Diffraction
Crystal diffraction reveals lattice spacing and electron-wave coherence.
Neutron Diffraction
Neutrons probe crystal and magnetic structures and penetrate deeply because they are neutral.
Quantum Devices
Tunnelling microscopes, semiconductor devices and electron interferometers depend on matter-wave behaviour.
8. Experimental Verification and Davisson-Germer
Davisson and Germer directed electrons of controlled energy onto a nickel crystal and measured scattered intensity versus angle. A pronounced maximum appeared because crystal planes acted as a diffraction grating. Bragg’s law, nλ = 2d sinθ, gave a wavelength agreeing with λ = h/√(2mₑeV). This quantitative agreement was decisive evidence for electron waves.
Bragg Condition
\(n\lambda=2d\sin\theta\)de Broglie Prediction
\(\lambda=h/\sqrt{2m_e eV}\)Observed Quantity
Scattered electron current as a function of detector angle.
Conclusion
Electrons show diffraction and therefore possess wave nature.
9. Wave Packet, Phase Velocity and Group Velocity
Wave Packet
A localized packet requires a spread of wave numbers. Narrower localization means a broader momentum spread, consistent with uncertainty.
Group Velocity
\(v_g=d\omega/dk\)For a free non-relativistic particle, vᵍ = p/m = particle speed v.
Phase Velocity
\(v_p=\omega/k=E/p\)Using non-relativistic kinetic energy E = p²/2m gives vₚ = v/2. Using total relativistic energy gives vₚ = c²/v.
No Superluminal Signal
Phase velocity can exceed c in the relativistic expression, but it carries no localized information. Signal transport follows group velocity.
10. Important Graphs
11. 100 Conceptual Questions
C1. Why do electrons show diffraction? (core concept)
Answer: Their de Broglie wavelength is comparable with atomic-plane spacing, so scattered probability amplitudes interfere.
Explanation: Their de Broglie wavelength is comparable with atomic-plane spacing, so scattered probability amplitudes interfere. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C2. Why do cricket balls not show observable diffraction? (core concept)
Answer: Their enormous momentum makes λ = h/p fantastically small compared with any practical aperture.
Explanation: Their enormous momentum makes λ = h/p fantastically small compared with any practical aperture. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C3. Why are matter waves not electromagnetic waves? (core concept)
Answer: They describe a quantum probability amplitude associated with particles and do not consist of oscillating electric and magnetic fields.
Explanation: They describe a quantum probability amplitude associated with particles and do not consist of oscillating electric and magnetic fields. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C4. Why does wavelength decrease when momentum increases? (core concept)
Answer: de Broglie’s relation λ = h/p is inverse.
Explanation: de Broglie’s relation λ = h/p is inverse. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C5. Can a stationary particle have a finite de Broglie wavelength? (core concept)
Answer: The simple λ = h/p tends to infinity as p tends to zero; a perfectly sharp zero momentum state is completely delocalized.
Explanation: The simple λ = h/p tends to infinity as p tends to zero; a perfectly sharp zero momentum state is completely delocalized. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C6. Does a charged particle alone have matter waves? (core concept)
Answer: No. Every quantum particle, charged or neutral, has wave behaviour.
Explanation: No. Every quantum particle, charged or neutral, has wave behaviour. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C7. Why is an electron microscope powerful? (core concept)
Answer: Accelerated electrons can have wavelengths much shorter than visible light.
Explanation: Accelerated electrons can have wavelengths much shorter than visible light. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C8. What does the Davisson-Germer experiment prove? (core concept)
Answer: It verifies electron diffraction and quantitatively agrees with de Broglie wavelength.
Explanation: It verifies electron diffraction and quantitatively agrees with de Broglie wavelength. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C9. Why is group velocity physically important? (core concept)
Answer: It describes motion of the wave-packet envelope, information and the particle’s probability distribution.
Explanation: It describes motion of the wave-packet envelope, information and the particle’s probability distribution. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C10. Why can phase velocity exceed particle velocity? (core concept)
Answer: Phase velocity tracks crests rather than signal or energy transport; it is not itself the particle speed.
Explanation: Phase velocity tracks crests rather than signal or energy transport; it is not itself the particle speed. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C11. How does accelerating voltage affect electron wavelength? (core concept)
Answer: λ is proportional to 1/√V in the non-relativistic range.
Explanation: λ is proportional to 1/√V in the non-relativistic range. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C12. How does mass affect wavelength at equal speed? (core concept)
Answer: Greater mass gives greater momentum and therefore smaller wavelength.
Explanation: Greater mass gives greater momentum and therefore smaller wavelength. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C13. How does mass affect wavelength at equal kinetic energy? (core concept)
Answer: λ = h/√(2mK), so greater mass gives smaller wavelength.
Explanation: λ = h/√(2mK), so greater mass gives smaller wavelength. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C14. How does charge affect wavelength at equal voltage? (core concept)
Answer: λ = h/√(2m|q|V), so larger charge magnitude gives smaller wavelength.
Explanation: λ = h/√(2m|q|V), so larger charge magnitude gives smaller wavelength. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C15. What is a wave packet? (core concept)
Answer: A localized superposition of waves with nearby wavelengths and momenta.
Explanation: A localized superposition of waves with nearby wavelengths and momenta. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C16. Why is a single sinusoidal matter wave not a localized particle? (core concept)
Answer: It extends through all space and has perfectly definite momentum but completely indefinite position.
Explanation: It extends through all space and has perfectly definite momentum but completely indefinite position. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C17. What connects momentum and wave number? (core concept)
Answer: p = ħk.
Explanation: p = ħk. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C18. What connects energy and angular frequency? (core concept)
Answer: E = ħω.
Explanation: E = ħω. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C19. Is de Broglie wavelength a physical path drawn by a particle? (core concept)
Answer: No. It characterizes phase variation of the particle’s quantum state.
Explanation: No. It characterizes phase variation of the particle’s quantum state. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C20. When is the non-relativistic voltage formula invalid? (core concept)
Answer: When kinetic energy is no longer negligible compared with mc².
Explanation: When kinetic energy is no longer negligible compared with mc². In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C21. Why do neutrons diffract? (core concept)
Answer: They are quantum particles with momentum and hence de Broglie wavelength despite having no charge.
Explanation: They are quantum particles with momentum and hence de Broglie wavelength despite having no charge. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C22. Can atoms and molecules interfere? (core concept)
Answer: Yes, if coherence is maintained and experimental dimensions match their de Broglie wavelength.
Explanation: Yes, if coherence is maintained and experimental dimensions match their de Broglie wavelength. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C23. What sets diffraction-angle scale? (core concept)
Answer: The ratio of wavelength to aperture or lattice spacing.
Explanation: The ratio of wavelength to aperture or lattice spacing. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C24. Why does measurement disturb matter waves? (core concept)
Answer: A measurement interaction changes the quantum state and generally its momentum or phase relations.
Explanation: A measurement interaction changes the quantum state and generally its momentum or phase relations. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C25. What is the uncertainty connection? (core concept)
Answer: A localized packet needs a spread in wave number, giving ΔxΔp ≥ ħ/2.
Explanation: A localized packet needs a spread in wave number, giving ΔxΔp ≥ ħ/2. In core concept, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C26. Why do electrons show diffraction? (NEET viewpoint)
Answer: Their de Broglie wavelength is comparable with atomic-plane spacing, so scattered probability amplitudes interfere.
Explanation: Their de Broglie wavelength is comparable with atomic-plane spacing, so scattered probability amplitudes interfere. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C27. Why do cricket balls not show observable diffraction? (NEET viewpoint)
Answer: Their enormous momentum makes λ = h/p fantastically small compared with any practical aperture.
Explanation: Their enormous momentum makes λ = h/p fantastically small compared with any practical aperture. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C28. Why are matter waves not electromagnetic waves? (NEET viewpoint)
Answer: They describe a quantum probability amplitude associated with particles and do not consist of oscillating electric and magnetic fields.
Explanation: They describe a quantum probability amplitude associated with particles and do not consist of oscillating electric and magnetic fields. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C29. Why does wavelength decrease when momentum increases? (NEET viewpoint)
Answer: de Broglie’s relation λ = h/p is inverse.
Explanation: de Broglie’s relation λ = h/p is inverse. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C30. Can a stationary particle have a finite de Broglie wavelength? (NEET viewpoint)
Answer: The simple λ = h/p tends to infinity as p tends to zero; a perfectly sharp zero momentum state is completely delocalized.
Explanation: The simple λ = h/p tends to infinity as p tends to zero; a perfectly sharp zero momentum state is completely delocalized. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C31. Does a charged particle alone have matter waves? (NEET viewpoint)
Answer: No. Every quantum particle, charged or neutral, has wave behaviour.
Explanation: No. Every quantum particle, charged or neutral, has wave behaviour. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C32. Why is an electron microscope powerful? (NEET viewpoint)
Answer: Accelerated electrons can have wavelengths much shorter than visible light.
Explanation: Accelerated electrons can have wavelengths much shorter than visible light. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C33. What does the Davisson-Germer experiment prove? (NEET viewpoint)
Answer: It verifies electron diffraction and quantitatively agrees with de Broglie wavelength.
Explanation: It verifies electron diffraction and quantitatively agrees with de Broglie wavelength. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C34. Why is group velocity physically important? (NEET viewpoint)
Answer: It describes motion of the wave-packet envelope, information and the particle’s probability distribution.
Explanation: It describes motion of the wave-packet envelope, information and the particle’s probability distribution. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C35. Why can phase velocity exceed particle velocity? (NEET viewpoint)
Answer: Phase velocity tracks crests rather than signal or energy transport; it is not itself the particle speed.
Explanation: Phase velocity tracks crests rather than signal or energy transport; it is not itself the particle speed. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C36. How does accelerating voltage affect electron wavelength? (NEET viewpoint)
Answer: λ is proportional to 1/√V in the non-relativistic range.
Explanation: λ is proportional to 1/√V in the non-relativistic range. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C37. How does mass affect wavelength at equal speed? (NEET viewpoint)
Answer: Greater mass gives greater momentum and therefore smaller wavelength.
Explanation: Greater mass gives greater momentum and therefore smaller wavelength. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C38. How does mass affect wavelength at equal kinetic energy? (NEET viewpoint)
Answer: λ = h/√(2mK), so greater mass gives smaller wavelength.
Explanation: λ = h/√(2mK), so greater mass gives smaller wavelength. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C39. How does charge affect wavelength at equal voltage? (NEET viewpoint)
Answer: λ = h/√(2m|q|V), so larger charge magnitude gives smaller wavelength.
Explanation: λ = h/√(2m|q|V), so larger charge magnitude gives smaller wavelength. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C40. What is a wave packet? (NEET viewpoint)
Answer: A localized superposition of waves with nearby wavelengths and momenta.
Explanation: A localized superposition of waves with nearby wavelengths and momenta. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C41. Why is a single sinusoidal matter wave not a localized particle? (NEET viewpoint)
Answer: It extends through all space and has perfectly definite momentum but completely indefinite position.
Explanation: It extends through all space and has perfectly definite momentum but completely indefinite position. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C42. What connects momentum and wave number? (NEET viewpoint)
Answer: p = ħk.
Explanation: p = ħk. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C43. What connects energy and angular frequency? (NEET viewpoint)
Answer: E = ħω.
Explanation: E = ħω. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C44. Is de Broglie wavelength a physical path drawn by a particle? (NEET viewpoint)
Answer: No. It characterizes phase variation of the particle’s quantum state.
Explanation: No. It characterizes phase variation of the particle’s quantum state. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C45. When is the non-relativistic voltage formula invalid? (NEET viewpoint)
Answer: When kinetic energy is no longer negligible compared with mc².
Explanation: When kinetic energy is no longer negligible compared with mc². In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C46. Why do neutrons diffract? (NEET viewpoint)
Answer: They are quantum particles with momentum and hence de Broglie wavelength despite having no charge.
Explanation: They are quantum particles with momentum and hence de Broglie wavelength despite having no charge. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C47. Can atoms and molecules interfere? (NEET viewpoint)
Answer: Yes, if coherence is maintained and experimental dimensions match their de Broglie wavelength.
Explanation: Yes, if coherence is maintained and experimental dimensions match their de Broglie wavelength. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C48. What sets diffraction-angle scale? (NEET viewpoint)
Answer: The ratio of wavelength to aperture or lattice spacing.
Explanation: The ratio of wavelength to aperture or lattice spacing. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C49. Why does measurement disturb matter waves? (NEET viewpoint)
Answer: A measurement interaction changes the quantum state and generally its momentum or phase relations.
Explanation: A measurement interaction changes the quantum state and generally its momentum or phase relations. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C50. What is the uncertainty connection? (NEET viewpoint)
Answer: A localized packet needs a spread in wave number, giving ΔxΔp ≥ ħ/2.
Explanation: A localized packet needs a spread in wave number, giving ΔxΔp ≥ ħ/2. In NEET viewpoint, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C51. Why do electrons show diffraction? (JEE reasoning)
Answer: Their de Broglie wavelength is comparable with atomic-plane spacing, so scattered probability amplitudes interfere.
Explanation: Their de Broglie wavelength is comparable with atomic-plane spacing, so scattered probability amplitudes interfere. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C52. Why do cricket balls not show observable diffraction? (JEE reasoning)
Answer: Their enormous momentum makes λ = h/p fantastically small compared with any practical aperture.
Explanation: Their enormous momentum makes λ = h/p fantastically small compared with any practical aperture. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C53. Why are matter waves not electromagnetic waves? (JEE reasoning)
Answer: They describe a quantum probability amplitude associated with particles and do not consist of oscillating electric and magnetic fields.
Explanation: They describe a quantum probability amplitude associated with particles and do not consist of oscillating electric and magnetic fields. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C54. Why does wavelength decrease when momentum increases? (JEE reasoning)
Answer: de Broglie’s relation λ = h/p is inverse.
Explanation: de Broglie’s relation λ = h/p is inverse. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C55. Can a stationary particle have a finite de Broglie wavelength? (JEE reasoning)
Answer: The simple λ = h/p tends to infinity as p tends to zero; a perfectly sharp zero momentum state is completely delocalized.
Explanation: The simple λ = h/p tends to infinity as p tends to zero; a perfectly sharp zero momentum state is completely delocalized. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C56. Does a charged particle alone have matter waves? (JEE reasoning)
Answer: No. Every quantum particle, charged or neutral, has wave behaviour.
Explanation: No. Every quantum particle, charged or neutral, has wave behaviour. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C57. Why is an electron microscope powerful? (JEE reasoning)
Answer: Accelerated electrons can have wavelengths much shorter than visible light.
Explanation: Accelerated electrons can have wavelengths much shorter than visible light. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C58. What does the Davisson-Germer experiment prove? (JEE reasoning)
Answer: It verifies electron diffraction and quantitatively agrees with de Broglie wavelength.
Explanation: It verifies electron diffraction and quantitatively agrees with de Broglie wavelength. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C59. Why is group velocity physically important? (JEE reasoning)
Answer: It describes motion of the wave-packet envelope, information and the particle’s probability distribution.
Explanation: It describes motion of the wave-packet envelope, information and the particle’s probability distribution. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C60. Why can phase velocity exceed particle velocity? (JEE reasoning)
Answer: Phase velocity tracks crests rather than signal or energy transport; it is not itself the particle speed.
Explanation: Phase velocity tracks crests rather than signal or energy transport; it is not itself the particle speed. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C61. How does accelerating voltage affect electron wavelength? (JEE reasoning)
Answer: λ is proportional to 1/√V in the non-relativistic range.
Explanation: λ is proportional to 1/√V in the non-relativistic range. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C62. How does mass affect wavelength at equal speed? (JEE reasoning)
Answer: Greater mass gives greater momentum and therefore smaller wavelength.
Explanation: Greater mass gives greater momentum and therefore smaller wavelength. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C63. How does mass affect wavelength at equal kinetic energy? (JEE reasoning)
Answer: λ = h/√(2mK), so greater mass gives smaller wavelength.
Explanation: λ = h/√(2mK), so greater mass gives smaller wavelength. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C64. How does charge affect wavelength at equal voltage? (JEE reasoning)
Answer: λ = h/√(2m|q|V), so larger charge magnitude gives smaller wavelength.
Explanation: λ = h/√(2m|q|V), so larger charge magnitude gives smaller wavelength. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C65. What is a wave packet? (JEE reasoning)
Answer: A localized superposition of waves with nearby wavelengths and momenta.
Explanation: A localized superposition of waves with nearby wavelengths and momenta. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C66. Why is a single sinusoidal matter wave not a localized particle? (JEE reasoning)
Answer: It extends through all space and has perfectly definite momentum but completely indefinite position.
Explanation: It extends through all space and has perfectly definite momentum but completely indefinite position. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C67. What connects momentum and wave number? (JEE reasoning)
Answer: p = ħk.
Explanation: p = ħk. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C68. What connects energy and angular frequency? (JEE reasoning)
Answer: E = ħω.
Explanation: E = ħω. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C69. Is de Broglie wavelength a physical path drawn by a particle? (JEE reasoning)
Answer: No. It characterizes phase variation of the particle’s quantum state.
Explanation: No. It characterizes phase variation of the particle’s quantum state. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C70. When is the non-relativistic voltage formula invalid? (JEE reasoning)
Answer: When kinetic energy is no longer negligible compared with mc².
Explanation: When kinetic energy is no longer negligible compared with mc². In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C71. Why do neutrons diffract? (JEE reasoning)
Answer: They are quantum particles with momentum and hence de Broglie wavelength despite having no charge.
Explanation: They are quantum particles with momentum and hence de Broglie wavelength despite having no charge. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C72. Can atoms and molecules interfere? (JEE reasoning)
Answer: Yes, if coherence is maintained and experimental dimensions match their de Broglie wavelength.
Explanation: Yes, if coherence is maintained and experimental dimensions match their de Broglie wavelength. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C73. What sets diffraction-angle scale? (JEE reasoning)
Answer: The ratio of wavelength to aperture or lattice spacing.
Explanation: The ratio of wavelength to aperture or lattice spacing. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C74. Why does measurement disturb matter waves? (JEE reasoning)
Answer: A measurement interaction changes the quantum state and generally its momentum or phase relations.
Explanation: A measurement interaction changes the quantum state and generally its momentum or phase relations. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C75. What is the uncertainty connection? (JEE reasoning)
Answer: A localized packet needs a spread in wave number, giving ΔxΔp ≥ ħ/2.
Explanation: A localized packet needs a spread in wave number, giving ΔxΔp ≥ ħ/2. In JEE reasoning, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C76. Why do electrons show diffraction? (experimental interpretation)
Answer: Their de Broglie wavelength is comparable with atomic-plane spacing, so scattered probability amplitudes interfere.
Explanation: Their de Broglie wavelength is comparable with atomic-plane spacing, so scattered probability amplitudes interfere. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C77. Why do cricket balls not show observable diffraction? (experimental interpretation)
Answer: Their enormous momentum makes λ = h/p fantastically small compared with any practical aperture.
Explanation: Their enormous momentum makes λ = h/p fantastically small compared with any practical aperture. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C78. Why are matter waves not electromagnetic waves? (experimental interpretation)
Answer: They describe a quantum probability amplitude associated with particles and do not consist of oscillating electric and magnetic fields.
Explanation: They describe a quantum probability amplitude associated with particles and do not consist of oscillating electric and magnetic fields. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C79. Why does wavelength decrease when momentum increases? (experimental interpretation)
Answer: de Broglie’s relation λ = h/p is inverse.
Explanation: de Broglie’s relation λ = h/p is inverse. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C80. Can a stationary particle have a finite de Broglie wavelength? (experimental interpretation)
Answer: The simple λ = h/p tends to infinity as p tends to zero; a perfectly sharp zero momentum state is completely delocalized.
Explanation: The simple λ = h/p tends to infinity as p tends to zero; a perfectly sharp zero momentum state is completely delocalized. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C81. Does a charged particle alone have matter waves? (experimental interpretation)
Answer: No. Every quantum particle, charged or neutral, has wave behaviour.
Explanation: No. Every quantum particle, charged or neutral, has wave behaviour. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C82. Why is an electron microscope powerful? (experimental interpretation)
Answer: Accelerated electrons can have wavelengths much shorter than visible light.
Explanation: Accelerated electrons can have wavelengths much shorter than visible light. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C83. What does the Davisson-Germer experiment prove? (experimental interpretation)
Answer: It verifies electron diffraction and quantitatively agrees with de Broglie wavelength.
Explanation: It verifies electron diffraction and quantitatively agrees with de Broglie wavelength. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C84. Why is group velocity physically important? (experimental interpretation)
Answer: It describes motion of the wave-packet envelope, information and the particle’s probability distribution.
Explanation: It describes motion of the wave-packet envelope, information and the particle’s probability distribution. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C85. Why can phase velocity exceed particle velocity? (experimental interpretation)
Answer: Phase velocity tracks crests rather than signal or energy transport; it is not itself the particle speed.
Explanation: Phase velocity tracks crests rather than signal or energy transport; it is not itself the particle speed. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C86. How does accelerating voltage affect electron wavelength? (experimental interpretation)
Answer: λ is proportional to 1/√V in the non-relativistic range.
Explanation: λ is proportional to 1/√V in the non-relativistic range. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C87. How does mass affect wavelength at equal speed? (experimental interpretation)
Answer: Greater mass gives greater momentum and therefore smaller wavelength.
Explanation: Greater mass gives greater momentum and therefore smaller wavelength. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C88. How does mass affect wavelength at equal kinetic energy? (experimental interpretation)
Answer: λ = h/√(2mK), so greater mass gives smaller wavelength.
Explanation: λ = h/√(2mK), so greater mass gives smaller wavelength. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C89. How does charge affect wavelength at equal voltage? (experimental interpretation)
Answer: λ = h/√(2m|q|V), so larger charge magnitude gives smaller wavelength.
Explanation: λ = h/√(2m|q|V), so larger charge magnitude gives smaller wavelength. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C90. What is a wave packet? (experimental interpretation)
Answer: A localized superposition of waves with nearby wavelengths and momenta.
Explanation: A localized superposition of waves with nearby wavelengths and momenta. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C91. Why is a single sinusoidal matter wave not a localized particle? (experimental interpretation)
Answer: It extends through all space and has perfectly definite momentum but completely indefinite position.
Explanation: It extends through all space and has perfectly definite momentum but completely indefinite position. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C92. What connects momentum and wave number? (experimental interpretation)
Answer: p = ħk.
Explanation: p = ħk. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C93. What connects energy and angular frequency? (experimental interpretation)
Answer: E = ħω.
Explanation: E = ħω. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C94. Is de Broglie wavelength a physical path drawn by a particle? (experimental interpretation)
Answer: No. It characterizes phase variation of the particle’s quantum state.
Explanation: No. It characterizes phase variation of the particle’s quantum state. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C95. When is the non-relativistic voltage formula invalid? (experimental interpretation)
Answer: When kinetic energy is no longer negligible compared with mc².
Explanation: When kinetic energy is no longer negligible compared with mc². In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C96. Why do neutrons diffract? (experimental interpretation)
Answer: They are quantum particles with momentum and hence de Broglie wavelength despite having no charge.
Explanation: They are quantum particles with momentum and hence de Broglie wavelength despite having no charge. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C97. Can atoms and molecules interfere? (experimental interpretation)
Answer: Yes, if coherence is maintained and experimental dimensions match their de Broglie wavelength.
Explanation: Yes, if coherence is maintained and experimental dimensions match their de Broglie wavelength. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C98. What sets diffraction-angle scale? (experimental interpretation)
Answer: The ratio of wavelength to aperture or lattice spacing.
Explanation: The ratio of wavelength to aperture or lattice spacing. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C99. Why does measurement disturb matter waves? (experimental interpretation)
Answer: A measurement interaction changes the quantum state and generally its momentum or phase relations.
Explanation: A measurement interaction changes the quantum state and generally its momentum or phase relations. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
C100. What is the uncertainty connection? (experimental interpretation)
Answer: A localized packet needs a spread in wave number, giving ΔxΔp ≥ ħ/2.
Explanation: A localized packet needs a spread in wave number, giving ΔxΔp ≥ ħ/2. In experimental interpretation, connect the statement directly to λ = h/p, quantum superposition or the experimental length scale.
12. 50 Solved Numericals
1. Numerical
Find the de Broglie wavelength of an electron moving at 1.00 × 10⁶ m s⁻¹.
Answer: 7.274 Å
Step-by-step solution: Use λ = h/(mₑv). Substitution gives 7.274e-10 m = 7.274 Å.
2. Numerical
An electron is accelerated from rest through 30 V. Find its wavelength.
Answer: 2.240 Å
Step-by-step solution: For non-relativistic electrons λ(Å) = 12.27/√V = 12.27/√30 = 2.240 Å.
3. Numerical
A particle has momentum 2.20e-24 kg m s⁻¹. Find λ.
Answer: 3.012e-10 m
Step-by-step solution: Directly apply λ = h/p = 6.626 × 10⁻³⁴ / 2.20e-24 = 3.012e-10 m.
4. Numerical
Find the electron wavelength when kinetic energy is 23 eV.
Answer: 2.557 Å
Step-by-step solution: Convert K to joules and use λ = h/√(2mₑK). The result is 2.557 Å.
5. Numerical
Electron voltage rises from 54 V to 216 V. Find λ₂/λ₁.
Answer: 1/2
Step-by-step solution: Since λ ∝ 1/√V, λ₂/λ₁ = √(V₁/V₂) = √(1/4) = 1/2.
6. Numerical
At the same speed, compare proton and electron wavelengths.
Answer: λₚ/λₑ = mₑ/mₚ ≈ 1/1836
Step-by-step solution: At equal v, λ = h/(mv). The proton’s mass is about 1836 electron masses, so its wavelength is 1836 times smaller.
7. Numerical
Electrons produce first-order Bragg reflection from planes 0.091 nm apart at 25°. Find wavelength.
Answer: 0.077 nm
Step-by-step solution: Use nλ = 2d sinθ with n = 1: λ = 2(0.091)sin25° = 0.077 nm.
8. Numerical
Find momentum corresponding to wavelength 0.100 nm.
Answer: 6.626e-24 kg m s⁻¹
Step-by-step solution: p = h/λ = 6.626 × 10⁻³⁴ / 1.00 × 10⁻¹⁰ = 6.626e-24 kg m s⁻¹.
9. Numerical
A neutron and proton have equal momenta. Compare their de Broglie wavelengths.
Answer: Equal
Step-by-step solution: de Broglie wavelength depends only on momentum: λ = h/p. Equal momenta imply equal wavelengths.
10. Numerical
A free non-relativistic particle moves at speed v. Compare group velocity and phase velocity.
Answer: vᵍ = v and vₚ = v/2
Step-by-step solution: With E = p²/2m, ω = ħk²/2m. Thus dω/dk = ħk/m = v, while ω/k = ħk/2m = v/2.
11. Numerical
Find the de Broglie wavelength of an electron moving at 1.30 × 10⁶ m s⁻¹.
Answer: 5.595 Å
Step-by-step solution: Use λ = h/(mₑv). Substitution gives 5.595e-10 m = 5.595 Å.
12. Numerical
An electron is accelerated from rest through 80 V. Find its wavelength.
Answer: 1.372 Å
Step-by-step solution: For non-relativistic electrons λ(Å) = 12.27/√V = 12.27/√80 = 1.372 Å.
13. Numerical
A particle has momentum 3.20e-24 kg m s⁻¹. Find λ.
Answer: 2.071e-10 m
Step-by-step solution: Directly apply λ = h/p = 6.626 × 10⁻³⁴ / 3.20e-24 = 2.071e-10 m.
14. Numerical
Find the electron wavelength when kinetic energy is 33 eV.
Answer: 2.135 Å
Step-by-step solution: Convert K to joules and use λ = h/√(2mₑK). The result is 2.135 Å.
15. Numerical
Electron voltage rises from 64 V to 256 V. Find λ₂/λ₁.
Answer: 1/2
Step-by-step solution: Since λ ∝ 1/√V, λ₂/λ₁ = √(V₁/V₂) = √(1/4) = 1/2.
16. Numerical
At the same speed, compare proton and electron wavelengths.
Answer: λₚ/λₑ = mₑ/mₚ ≈ 1/1836
Step-by-step solution: At equal v, λ = h/(mv). The proton’s mass is about 1836 electron masses, so its wavelength is 1836 times smaller.
17. Numerical
Electrons produce first-order Bragg reflection from planes 0.091 nm apart at 25°. Find wavelength.
Answer: 0.077 nm
Step-by-step solution: Use nλ = 2d sinθ with n = 1: λ = 2(0.091)sin25° = 0.077 nm.
18. Numerical
Find momentum corresponding to wavelength 0.100 nm.
Answer: 6.626e-24 kg m s⁻¹
Step-by-step solution: p = h/λ = 6.626 × 10⁻³⁴ / 1.00 × 10⁻¹⁰ = 6.626e-24 kg m s⁻¹.
19. Numerical
A neutron and proton have equal momenta. Compare their de Broglie wavelengths.
Answer: Equal
Step-by-step solution: de Broglie wavelength depends only on momentum: λ = h/p. Equal momenta imply equal wavelengths.
20. Numerical
A free non-relativistic particle moves at speed v. Compare group velocity and phase velocity.
Answer: vᵍ = v and vₚ = v/2
Step-by-step solution: With E = p²/2m, ω = ħk²/2m. Thus dω/dk = ħk/m = v, while ω/k = ħk/2m = v/2.
21. Numerical
Find the de Broglie wavelength of an electron moving at 1.60 × 10⁶ m s⁻¹.
Answer: 4.546 Å
Step-by-step solution: Use λ = h/(mₑv). Substitution gives 4.546e-10 m = 4.546 Å.
22. Numerical
An electron is accelerated from rest through 130 V. Find its wavelength.
Answer: 1.076 Å
Step-by-step solution: For non-relativistic electrons λ(Å) = 12.27/√V = 12.27/√130 = 1.076 Å.
23. Numerical
A particle has momentum 4.20e-24 kg m s⁻¹. Find λ.
Answer: 1.578e-10 m
Step-by-step solution: Directly apply λ = h/p = 6.626 × 10⁻³⁴ / 4.20e-24 = 1.578e-10 m.
24. Numerical
Find the electron wavelength when kinetic energy is 43 eV.
Answer: 1.870 Å
Step-by-step solution: Convert K to joules and use λ = h/√(2mₑK). The result is 1.870 Å.
25. Numerical
Electron voltage rises from 74 V to 296 V. Find λ₂/λ₁.
Answer: 1/2
Step-by-step solution: Since λ ∝ 1/√V, λ₂/λ₁ = √(V₁/V₂) = √(1/4) = 1/2.
26. Numerical
At the same speed, compare proton and electron wavelengths.
Answer: λₚ/λₑ = mₑ/mₚ ≈ 1/1836
Step-by-step solution: At equal v, λ = h/(mv). The proton’s mass is about 1836 electron masses, so its wavelength is 1836 times smaller.
27. Numerical
Electrons produce first-order Bragg reflection from planes 0.091 nm apart at 25°. Find wavelength.
Answer: 0.077 nm
Step-by-step solution: Use nλ = 2d sinθ with n = 1: λ = 2(0.091)sin25° = 0.077 nm.
28. Numerical
Find momentum corresponding to wavelength 0.100 nm.
Answer: 6.626e-24 kg m s⁻¹
Step-by-step solution: p = h/λ = 6.626 × 10⁻³⁴ / 1.00 × 10⁻¹⁰ = 6.626e-24 kg m s⁻¹.
29. Numerical
A neutron and proton have equal momenta. Compare their de Broglie wavelengths.
Answer: Equal
Step-by-step solution: de Broglie wavelength depends only on momentum: λ = h/p. Equal momenta imply equal wavelengths.
30. Numerical
A free non-relativistic particle moves at speed v. Compare group velocity and phase velocity.
Answer: vᵍ = v and vₚ = v/2
Step-by-step solution: With E = p²/2m, ω = ħk²/2m. Thus dω/dk = ħk/m = v, while ω/k = ħk/2m = v/2.
31. Numerical
Find the de Broglie wavelength of an electron moving at 1.90 × 10⁶ m s⁻¹.
Answer: 3.828 Å
Step-by-step solution: Use λ = h/(mₑv). Substitution gives 3.828e-10 m = 3.828 Å.
32. Numerical
An electron is accelerated from rest through 180 V. Find its wavelength.
Answer: 0.915 Å
Step-by-step solution: For non-relativistic electrons λ(Å) = 12.27/√V = 12.27/√180 = 0.915 Å.
33. Numerical
A particle has momentum 5.20e-24 kg m s⁻¹. Find λ.
Answer: 1.274e-10 m
Step-by-step solution: Directly apply λ = h/p = 6.626 × 10⁻³⁴ / 5.20e-24 = 1.274e-10 m.
34. Numerical
Find the electron wavelength when kinetic energy is 53 eV.
Answer: 1.685 Å
Step-by-step solution: Convert K to joules and use λ = h/√(2mₑK). The result is 1.685 Å.
35. Numerical
Electron voltage rises from 84 V to 336 V. Find λ₂/λ₁.
Answer: 1/2
Step-by-step solution: Since λ ∝ 1/√V, λ₂/λ₁ = √(V₁/V₂) = √(1/4) = 1/2.
36. Numerical
At the same speed, compare proton and electron wavelengths.
Answer: λₚ/λₑ = mₑ/mₚ ≈ 1/1836
Step-by-step solution: At equal v, λ = h/(mv). The proton’s mass is about 1836 electron masses, so its wavelength is 1836 times smaller.
37. Numerical
Electrons produce first-order Bragg reflection from planes 0.091 nm apart at 25°. Find wavelength.
Answer: 0.077 nm
Step-by-step solution: Use nλ = 2d sinθ with n = 1: λ = 2(0.091)sin25° = 0.077 nm.
38. Numerical
Find momentum corresponding to wavelength 0.100 nm.
Answer: 6.626e-24 kg m s⁻¹
Step-by-step solution: p = h/λ = 6.626 × 10⁻³⁴ / 1.00 × 10⁻¹⁰ = 6.626e-24 kg m s⁻¹.
39. Numerical
A neutron and proton have equal momenta. Compare their de Broglie wavelengths.
Answer: Equal
Step-by-step solution: de Broglie wavelength depends only on momentum: λ = h/p. Equal momenta imply equal wavelengths.
40. Numerical
A free non-relativistic particle moves at speed v. Compare group velocity and phase velocity.
Answer: vᵍ = v and vₚ = v/2
Step-by-step solution: With E = p²/2m, ω = ħk²/2m. Thus dω/dk = ħk/m = v, while ω/k = ħk/2m = v/2.
41. Numerical
Find the de Broglie wavelength of an electron moving at 2.20 × 10⁶ m s⁻¹.
Answer: 3.306 Å
Step-by-step solution: Use λ = h/(mₑv). Substitution gives 3.306e-10 m = 3.306 Å.
42. Numerical
An electron is accelerated from rest through 230 V. Find its wavelength.
Answer: 0.809 Å
Step-by-step solution: For non-relativistic electrons λ(Å) = 12.27/√V = 12.27/√230 = 0.809 Å.
43. Numerical
A particle has momentum 6.20e-24 kg m s⁻¹. Find λ.
Answer: 1.069e-10 m
Step-by-step solution: Directly apply λ = h/p = 6.626 × 10⁻³⁴ / 6.20e-24 = 1.069e-10 m.
44. Numerical
Find the electron wavelength when kinetic energy is 63 eV.
Answer: 1.545 Å
Step-by-step solution: Convert K to joules and use λ = h/√(2mₑK). The result is 1.545 Å.
45. Numerical
Electron voltage rises from 94 V to 376 V. Find λ₂/λ₁.
Answer: 1/2
Step-by-step solution: Since λ ∝ 1/√V, λ₂/λ₁ = √(V₁/V₂) = √(1/4) = 1/2.
46. Numerical
At the same speed, compare proton and electron wavelengths.
Answer: λₚ/λₑ = mₑ/mₚ ≈ 1/1836
Step-by-step solution: At equal v, λ = h/(mv). The proton’s mass is about 1836 electron masses, so its wavelength is 1836 times smaller.
47. Numerical
Electrons produce first-order Bragg reflection from planes 0.091 nm apart at 25°. Find wavelength.
Answer: 0.077 nm
Step-by-step solution: Use nλ = 2d sinθ with n = 1: λ = 2(0.091)sin25° = 0.077 nm.
48. Numerical
Find momentum corresponding to wavelength 0.100 nm.
Answer: 6.626e-24 kg m s⁻¹
Step-by-step solution: p = h/λ = 6.626 × 10⁻³⁴ / 1.00 × 10⁻¹⁰ = 6.626e-24 kg m s⁻¹.
49. Numerical
A neutron and proton have equal momenta. Compare their de Broglie wavelengths.
Answer: Equal
Step-by-step solution: de Broglie wavelength depends only on momentum: λ = h/p. Equal momenta imply equal wavelengths.
50. Numerical
A free non-relativistic particle moves at speed v. Compare group velocity and phase velocity.
Answer: vᵍ = v and vₚ = v/2
Step-by-step solution: With E = p²/2m, ω = ħk²/2m. Thus dω/dk = ħk/m = v, while ω/k = ħk/2m = v/2.
13. 120 PYQ-Pattern Questions
These are original questions based on recurring exam patterns from the listed examinations, not verbatim reproductions of copyrighted past papers.
Q1. NEET MCQ pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q2. AIPMT Assertion-Reason pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q3. JEE Main Match the Column pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q4. JEE Advanced Integer Type pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q5. IIT-JEE Multi Correct pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q6. CBSE Subjective pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q7. NEET Numerical Value pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q8. AIPMT MCQ pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q9. JEE Main Assertion-Reason pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q10. JEE Advanced Match the Column pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q11. IIT-JEE Integer Type pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q12. CBSE Multi Correct pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q13. NEET Subjective pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q14. AIPMT Numerical Value pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q15. JEE Main MCQ pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q16. JEE Advanced Assertion-Reason pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q17. IIT-JEE Match the Column pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q18. CBSE Integer Type pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q19. NEET Multi Correct pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q20. AIPMT Subjective pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q21. JEE Main Numerical Value pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q22. JEE Advanced MCQ pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q23. IIT-JEE Assertion-Reason pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q24. CBSE Match the Column pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q25. NEET Integer Type pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q26. AIPMT Multi Correct pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q27. JEE Main Subjective pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q28. JEE Advanced Numerical Value pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q29. IIT-JEE MCQ pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q30. CBSE Assertion-Reason pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q31. NEET Match the Column pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q32. AIPMT Integer Type pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q33. JEE Main Multi Correct pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q34. JEE Advanced Subjective pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q35. IIT-JEE Numerical Value pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q36. CBSE MCQ pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q37. NEET Assertion-Reason pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q38. AIPMT Match the Column pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q39. JEE Main Integer Type pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q40. JEE Advanced Multi Correct pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q41. IIT-JEE Subjective pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q42. CBSE Numerical Value pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q43. NEET MCQ pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q44. AIPMT Assertion-Reason pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q45. JEE Main Match the Column pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q46. JEE Advanced Integer Type pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q47. IIT-JEE Multi Correct pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q48. CBSE Subjective pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q49. NEET Numerical Value pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q50. AIPMT MCQ pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q51. JEE Main Assertion-Reason pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q52. JEE Advanced Match the Column pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q53. IIT-JEE Integer Type pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q54. CBSE Multi Correct pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q55. NEET Subjective pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q56. AIPMT Numerical Value pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q57. JEE Main MCQ pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q58. JEE Advanced Assertion-Reason pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q59. IIT-JEE Match the Column pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q60. CBSE Integer Type pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q61. NEET Multi Correct pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q62. AIPMT Subjective pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q63. JEE Main Numerical Value pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q64. JEE Advanced MCQ pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q65. IIT-JEE Assertion-Reason pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q66. CBSE Match the Column pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q67. NEET Integer Type pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q68. AIPMT Multi Correct pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q69. JEE Main Subjective pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q70. JEE Advanced Numerical Value pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q71. IIT-JEE MCQ pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q72. CBSE Assertion-Reason pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q73. NEET Match the Column pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q74. AIPMT Integer Type pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q75. JEE Main Multi Correct pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q76. JEE Advanced Subjective pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q77. IIT-JEE Numerical Value pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q78. CBSE MCQ pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q79. NEET Assertion-Reason pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q80. AIPMT Match the Column pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q81. JEE Main Integer Type pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q82. JEE Advanced Multi Correct pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q83. IIT-JEE Subjective pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q84. CBSE Numerical Value pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q85. NEET MCQ pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q86. AIPMT Assertion-Reason pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q87. JEE Main Match the Column pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q88. JEE Advanced Integer Type pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q89. IIT-JEE Multi Correct pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q90. CBSE Subjective pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q91. NEET Numerical Value pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q92. AIPMT MCQ pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q93. JEE Main Assertion-Reason pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q94. JEE Advanced Match the Column pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q95. IIT-JEE Integer Type pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q96. CBSE Multi Correct pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q97. NEET Subjective pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q98. AIPMT Numerical Value pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q99. JEE Main MCQ pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q100. JEE Advanced Assertion-Reason pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q101. IIT-JEE Match the Column pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q102. CBSE Integer Type pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q103. NEET Multi Correct pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q104. AIPMT Subjective pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q105. JEE Main Numerical Value pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q106. JEE Advanced MCQ pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q107. IIT-JEE Assertion-Reason pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q108. CBSE Match the Column pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q109. NEET Integer Type pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q110. AIPMT Multi Correct pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
Q111. JEE Main Subjective pattern: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p.
Q112. JEE Advanced Numerical Value pattern: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV).
Q113. IIT-JEE MCQ pattern: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes.
Q114. CBSE Assertion-Reason pattern: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass.
Q115. NEET Match the Column pattern: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v.
Q116. AIPMT Integer Type pattern: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction.
Q117. JEE Main Multi Correct pattern: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p.
Q118. JEE Advanced Subjective pattern: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal.
Q119. IIT-JEE Numerical Value pattern: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases.
Q120. CBSE MCQ pattern: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK).
14. International Curriculum Banks
IB Physics
Q1. IB Physics exam-style: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q2. IB Physics exam-style: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV). State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q3. IB Physics exam-style: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q4. IB Physics exam-style: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q5. IB Physics exam-style: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q6. IB Physics exam-style: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q7. IB Physics exam-style: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q8. IB Physics exam-style: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal. State the governing relation, substitute SI quantities where required, and interpret the physical result.
IGCSE Physics
Q1. IGCSE Physics exam-style: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV). State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q2. IGCSE Physics exam-style: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q3. IGCSE Physics exam-style: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q4. IGCSE Physics exam-style: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q5. IGCSE Physics exam-style: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q6. IGCSE Physics exam-style: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q7. IGCSE Physics exam-style: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q8. IGCSE Physics exam-style: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases. State the governing relation, substitute SI quantities where required, and interpret the physical result.
ICSE Physics
Q1. ICSE Physics exam-style: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q2. ICSE Physics exam-style: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q3. ICSE Physics exam-style: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q4. ICSE Physics exam-style: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q5. ICSE Physics exam-style: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q6. ICSE Physics exam-style: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q7. ICSE Physics exam-style: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q8. ICSE Physics exam-style: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK). State the governing relation, substitute SI quantities where required, and interpret the physical result.
A-Level Physics
Q1. A-Level Physics exam-style: At equal momentum, proton and electron wavelengths are:
Answer: A. Equal
Explanation: λ depends on momentum, not separately on mass. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q2. A-Level Physics exam-style: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q3. A-Level Physics exam-style: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q4. A-Level Physics exam-style: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q5. A-Level Physics exam-style: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q6. A-Level Physics exam-style: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q7. A-Level Physics exam-style: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK). State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q8. A-Level Physics exam-style: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p. State the governing relation, substitute SI quantities where required, and interpret the physical result.
AP Physics
Q1. AP Physics exam-style: The group velocity of a free non-relativistic matter wave equals:
Answer: A. v
Explanation: vᵍ = dE/dp = p/m = v. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q2. AP Physics exam-style: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q3. AP Physics exam-style: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q4. AP Physics exam-style: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q5. AP Physics exam-style: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q6. AP Physics exam-style: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK). State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q7. AP Physics exam-style: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q8. AP Physics exam-style: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV). State the governing relation, substitute SI quantities where required, and interpret the physical result.
British Curriculum Physics
Q1. British Curriculum Physics exam-style: The experiment verifying electron waves is:
Answer: B. Davisson-Germer
Explanation: Nickel-crystal electron scattering showed Bragg diffraction. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q2. British Curriculum Physics exam-style: If momentum doubles, wavelength becomes:
Answer: B. Half
Explanation: λ = h/p. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q3. British Curriculum Physics exam-style: Matter waves are associated with:
Answer: C. All moving particles
Explanation: de Broglie’s hypothesis is universal. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q4. British Curriculum Physics exam-style: Electron microscope resolution improves mainly because:
Answer: B. Electron wavelength is short
Explanation: Resolution scale improves as wavelength decreases. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q5. British Curriculum Physics exam-style: For equal kinetic energy, the lighter particle has:
Answer: B. Longer λ
Explanation: λ = h/√(2mK). State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q6. British Curriculum Physics exam-style: The de Broglie wavelength is inversely proportional to:
Answer: A. Momentum
Explanation: λ = h/p. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q7. British Curriculum Physics exam-style: For an electron accelerated through V, wavelength varies as:
Answer: D. 1/√V
Explanation: λ = h/√(2mₑeV). State the governing relation, substitute SI quantities where required, and interpret the physical result.
Q8. British Curriculum Physics exam-style: Electron diffraction directly supports:
Answer: B. Wave nature of matter
Explanation: A diffraction pattern requires coherent wave amplitudes. State the governing relation, substitute SI quantities where required, and interpret the physical result.
Exam Revision Centre
Most Important Formula Sheet
λ = h/p
λ = h/mv
λ = h/√(2mK)
λ = h/√(2m|q|V)
λₑ(Å) = 12.27/√V
nλ = 2d sinθ
Top 50 Revision Questions
Revise momentum comparisons, equal-energy cases, voltage ratios, Bragg diffraction, electron microscope resolution and graph shapes.
Top 25 JEE Advanced Concepts
Wave packets, uncertainty, relativistic correction, mixed particle comparisons, Bragg geometry, group velocity and multi-step voltage problems.
Top 25 NEET Concepts
λ = h/p, voltage formula, Davisson-Germer result, electron microscope, same momentum and same kinetic-energy comparisons.
Common Mistakes
Do not confuse particle trajectory with wave shape. Use charge magnitude. Convert eV correctly. Check whether a relativistic correction is required.
One Day Revision Notes
Memorise the five core formulas, practise ratio methods, identify the controlled quantity, and connect every diffraction result to wavelength comparable with spacing.