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Dual nature, de Broglie relation and uncertainty principle: NEET MCQs with answers

89 questions on this topic of Atomic Structure (NEET 2027 syllabus). Here are 12; tap one to see its answer and explanation.

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easy

1. Which of the following experimental observations provided the first direct experimental confirmation of the wave nature of electrons proposed by de Broglie?

  1. Diffraction of electrons by a nickel crystal in the Davisson-Germer experiment
  2. Blackbody radiation spectrum observed by Max Planck
  3. Emission of photoelectrons in the photoelectric effect
  4. Discrete spectral lines in the emission spectrum of atomic hydrogen
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Answer: (A) Diffraction of electrons by a nickel crystal in the Davisson-Germer experiment

The Davisson and Germer experiment (1927) demonstrated that electrons undergo diffraction by a nickel crystal just like X-rays, providing conclusive direct proof of the wave nature of electrons.

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2. For a moving particle of mass m, if a plot of log λ versus log v is drawn (where λ is the de Broglie wavelength and v is its velocity), the slope and y-intercept of the resulting straight line are respectively:

  1. -1 and log(m/h)
  2. 1 and log(h/m)
  3. -1 and log(h/m)
  4. 1 and log(m/h)
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Answer: (C) -1 and log(h/m)

From the de Broglie relation, λ = h / (m v). Taking logarithm on both sides yields log λ = log(h/m) - log v = (-1) log v + log(h/m). Comparing this with the standard equation y = mx + c shows that the slope is -1 and the intercept is log(h/m).

hard

3. A proton and an electron both have kinetic energy. If the de Broglie wavelength of the electron is 100 times that of the proton, what is the ratio of the kinetic energy of the proton to that of the electron (E_proton / E_electron)? (Assume mass of proton ≈ 1840 times mass of electron)

  1. 0.184
  2. 1.84
  3. 5.43
  4. 54.3
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Answer: (C) 5.43

Since λ = h / √(2mE), we have E = h² / (2m λ²). Thus, E_p / E_e = (m_e / m_p) × (λ_e / λ_p)². Substituting the given values: E_p / E_e = (1 / 1840) × (100)² = 10000 / 1840 ≈ 5.43.

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4. Heisenberg's uncertainty principle is expressed as:

  1. Δx · Δp ≥ h/4π
  2. Δx · Δp = 0
  3. Δx/Δp ≥ h
  4. Δx + Δp ≥ h/2π
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Answer: (A) Δx · Δp ≥ h/4π

It is impossible to determine simultaneously the exact position and momentum of a microscopic particle: Δx · Δp ≥ h/4π.

NCERT Class 11

easy

5. Consider the following statements regarding the physical significance of Heisenberg's Uncertainty Principle: Statement I: It rules out the existence of definite paths or trajectories of electrons in atoms. Statement II: The effect of Heisenberg's Uncertainty Principle is significant only for microscopic entities and negligible for macroscopic bodies. In the light of the above statements, choose the most appropriate answer:

  1. Both Statement I and Statement II are correct
  2. Statement I is correct but Statement II is incorrect
  3. Statement I is incorrect but Statement II is correct
  4. Both Statement I and Statement II are incorrect
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Answer: (A) Both Statement I and Statement II are correct

The simultaneous determination of precise position and momentum is forbidden by the uncertainty principle, ruling out classical definite trajectories. Because h is extremely small (6.626 × 10⁻³⁴ J s), the uncertainty product is significant only for submicroscopic objects like electrons and completely negligible for everyday macroscopic bodies.

hard

6. If the uncertainty in the position of a moving particle is equal to its de Broglie wavelength, what is the minimum fractional uncertainty in its velocity (Δv / v)?

  1. 1 / (2π)
  2. 1 / (4π)
  3. 4π
  4. π / 2
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Answer: (B) 1 / (4π)

Heisenberg's principle gives Δx · Δp ≥ h / (4π). Here, Δx = λ = h / p. Substituting this into the inequality: (h / p) · (m · Δv) ≥ h / (4π). Since p = m · v, this simplifies to (h / (m · v)) · (m · Δv) ≥ h / (4π) ⇒ Δv / v ≥ 1 / (4π).

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7. According to Bohr's postulate, the angular momentum of an electron in a stationary orbit is mvr = nh / (2π). How did de Broglie explain this postulate?

  1. By assuming that the electron behaves strictly as a localized corpuscle in orbit
  2. By postulating that the radius of the orbit must equal half of the de Broglie wavelength
  3. By showing that destructive interference occurs around the entire orbit
  4. By requiring the circumference of the orbit to be an integral multiple of the electron's de Broglie wavelength (2πr = nλ)
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Answer: (D) By requiring the circumference of the orbit to be an integral multiple of the electron's de Broglie wavelength (2πr = nλ)

de Broglie explained Bohr's quantization by showing that stationary orbits correspond to circular standing waves. For constructive interference of the electron matter wave, the orbit circumference must contain an integral number of wavelengths: 2πr = nλ. Substituting λ = h / (mv) yields mvr = nh / (2π).

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8. Two particles A and B are in motion. The de Broglie wavelength of particle A is 2.0 × 10⁻¹⁰ m. If particle B has half the mass of A and its velocity is twice that of A, the de Broglie wavelength of particle B is:

  1. 1.0 × 10⁻¹⁰ m
  2. 2.0 × 10⁻¹⁰ m
  3. 4.0 × 10⁻¹⁰ m
  4. 8.0 × 10⁻¹⁰ m
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Answer: (B) 2.0 × 10⁻¹⁰ m

The de Broglie relation is λ = h / (m v). For particle B, mB = mA / 2 and vB = 2 vA. Therefore, the momentum of B is pB = mB vB = (mA / 2) × (2 vA) = mA vA = pA. Since their momenta are identical, their de Broglie wavelengths are also equal: λB = λA = 2.0 × 10⁻¹⁰ m.

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9. If the uncertainty in momentum of an electron is equal to twice the uncertainty in its position, what is the minimum value of uncertainty in position (Δx)?

  1. √(h / (2π))
  2. √(h / (8π))
  3. √(h / (4π))
  4. √(h / (16π))
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Answer: (B) √(h / (8π))

According to Heisenberg's uncertainty principle, Δx · Δp ≥ h / (4π). Given Δp = 2Δx, we substitute to get Δx · (2Δx) ≥ h / (4π), which yields (Δx)² ≥ h / (8π). Thus, the minimum uncertainty is Δx = √(h / (8π)).

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10. An electron is moving in the 3rd orbit of a hydrogen atom. According to de Broglie's relation, how many complete electron waves are associated with this orbit?

  1. 6
  2. 1
  3. 3
  4. 9
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Answer: (C) 3

Bohr's quantization postulate is mvr = n h / (2π). Rearranging this gives 2πr = n · (h / mv) = nλ. For the n = 3 orbit, the circumference equals exactly 3 de Broglie wavelengths (n = 3), meaning there are 3 complete electron standing waves.

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11. Given below are two statements: Statement I: Bohr's model of hydrogen atom violates Heisenberg's uncertainty principle because it assumes precise simultaneous determination of both the position and momentum of the electron. Statement II: The wave packet representation of an electron implies that a perfectly monochromatic wave has a completely known momentum and a completely undetermined position. In the light of the above statements, choose the most appropriate answer from the options given below:

  1. Both Statement I and Statement II are correct
  2. Statement I is correct but Statement II is incorrect
  3. Statement I is incorrect but Statement II is correct
  4. Both Statement I and Statement II are incorrect
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Answer: (A) Both Statement I and Statement II are correct

Bohr's model specifies fixed orbits with definite radii (position) and definite velocities (momentum), which directly contradicts Heisenberg's uncertainty principle. A pure sinusoidal (monochromatic) wave has a precise wavelength λ and hence precise momentum p (Δp = 0), which by Δx ≥ h/(4πΔp) makes the position completely uncertain (Δx → ∞). Both statements are correct.

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12. A proton and an α-particle have identical de Broglie wavelengths. What is the ratio of their velocities (v_proton : v_α)? (Assume mass of α-particle = 4 × mass of proton)

  1. 4 : 1
  2. 1 : 4
  3. 1 : 2
  4. 2 : 1
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Answer: (A) 4 : 1

The de Broglie wavelength is given by λ = h / (m v). Since λ is the same for both, m_p v_p = m_α v_α. Therefore, v_p / v_α = m_α / m_p = 4 / 1.

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