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ASSIGNMENT

Section A: Very Short Answer Questions

1. State de Broglie’s equation and explain the terms involved.
2. What is the significance of de Broglie’s hypothesis for macroscopic and microscopic particles?
3. Write the mathematical form of Heisenberg’s Uncertainty Principle.
4. Why does Heisenberg’s principle not apply to large objects like a cricket ball?
5. What is meant by the term “wave-particle duality”?

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Section B: Short Answer Questions (2 Marks Each)

6. An electron and a proton have the same kinetic energy. Which one will have the longer de Broglie wavelength and why?
7. Why can’t electrons in an atom be assigned fixed paths or orbits like planets around the sun?
8. How does Heisenberg’s uncertainty principle contradict Bohr’s model of the atom?
9. If the position of an electron is known within an accuracy of 0.01 nm, calculate the uncertainty in its momentum.
   (Use h = 6.626 × 10⁻³⁴ J·s)

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Section C: Numericals (Show Complete Calculations)

10. Calculate the de Broglie wavelength of an electron moving with a speed of 1.6 × 10⁶ m/s.
    (mass of electron = 9.1 × 10⁻³¹ kg, h = 6.626 × 10⁻³⁴ J·s)
11. A bullet of mass 0.02 kg is moving with a velocity of 1000 m/s. Calculate its de Broglie wavelength.
12. Calculate the uncertainty in the velocity of an electron if the uncertainty in its position is 0.002 nm.
    (mass of electron = 9.1 × 10⁻³¹ kg)
13. A microscope can measure position up to an accuracy of 1 × 10⁻¹⁰ m. What will be the minimum uncertainty in the velocity of an electron observed using it?

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KEY

Section A: Very Short Answer Questions

1. de Broglie equation:
   λ = h / p = h / (mv)
   where λ = wavelength, h = Planck’s constant, m = mass, v = velocity
2. Significance:
   de Broglie’s hypothesis is significant for microscopic particles like electrons (wavelength is measurable), but negligible for macroscopic bodies.
3. Heisenberg’s Principle:
   Δx · Δp ≥ h / (4π)
   where Δx = uncertainty in position, Δp = uncertainty in momentum
4. For large objects, the product of uncertainties is very small compared to their size and speed, hence negligible and not observable.
5. Wave-particle duality:
   The concept that every moving particle has both wave-like and particle-like properties.

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Section B: Short Answer Questions

6. Answer:
   Electron will have a longer de Broglie wavelength because wavelength is inversely proportional to momentum, and electron has much smaller mass than proton.
7. Due to uncertainty principle, exact position and momentum of electrons cannot be known simultaneously. Thus, fixed paths are not possible.
8. Bohr’s model assumes precise orbits for electrons, but Heisenberg’s principle denies simultaneous exact knowledge of position and momentum.
9. Δx = 0.01 nm = 1 × 10⁻¹¹ m
   Δp ≥ h / (4πΔx) = (6.626 × 10⁻³⁴) / (4π × 1 × 10⁻¹¹)
   ≈ 5.28 × 10⁻²⁴ kg·m/s

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Section C: Numericals

10. λ = h / mv = (6.626 × 10⁻³⁴) / (9.1 × 10⁻³¹ × 1.6 × 10⁶)
    ≈ 4.55 × 10⁻¹⁰ m
11. λ = h / mv = (6.626 × 10⁻³⁴) / (0.02 × 1000)
    = 3.31 × 10⁻³⁸ m
    → Negligible for macroscopic object
12. Δx = 0.002 nm = 2 × 10⁻¹² m
    Δv ≥ h / (4πmΔx)
    = (6.626 × 10⁻³⁴) / (4π × 9.1 × 10⁻³¹ × 2 × 10⁻¹²)
    ≈ 2.89 × 10⁴ m/s
13. Δx = 1 × 10⁻¹⁰ m
    Δv ≥ h / (4πmΔx)
    = (6.626 × 10⁻³⁴) / (4π × 9.1 × 10⁻³¹ × 1 × 10⁻¹⁰)
    ≈ 5.78 × 10³ m/s

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