Photons and Matter Waves Problems

This section provides 100 problems to test your understanding of the quantum nature of light and matter, including calculations of photon energy, momentum, photoelectric effect parameters (e.g., stopping potential, maximum kinetic energy), de Broglie wavelength, and uncertainty principle quantities, as well as applications like electron microscopy and spacecraft sensors. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering modern physics, a key topic for JEE/NEET success.

Numerical Problems

1. A photon has a wavelength $\lambda = 500 , \text{nm}$. Calculate its energy in eV.

   • (a) 2.47 eV
   • (b) 2.48 eV
   • (c) 2.49 eV
   • (d) 2.50 eV

2. A photon has a frequency $\nu = 4 \times 10^{14} , \text{Hz}$. Calculate its momentum in kg·m/s.

   • (a) 8.82 $\times 10^{-28}$
   • (b) 8.83 $\times 10^{-28}$
   • (c) 8.84 $\times 10^{-28}$
   • (d) 8.85 $\times 10^{-28}$

3. A metal has a work function $\phi = 2.0 , \text{eV}$. Light of $\lambda = 300 , \text{nm}$ shines on it. Calculate the maximum kinetic energy of the ejected photoelectrons in eV.

   • (a) 2.12 eV
   • (b) 2.13 eV
   • (c) 2.14 eV
   • (d) 2.15 eV

4. A metal has a threshold frequency $\nu_0 = 5 \times 10^{14} , \text{Hz}$. Light of $\nu = 6 \times 10^{14} , \text{Hz}$ is incident on it. Calculate the stopping potential $V_s$ in volts.

   • (a) 0.41 V
   • (b) 0.42 V
   • (c) 0.43 V
   • (d) 0.44 V

5. An electron ($m = 9.11 \times 10^{-31} , \text{kg}$) moves at $v = 1 \times 10^6 , \text{m/s}$. Calculate its de Broglie wavelength in nm.

   • (a) 0.726 nm
   • (b) 0.727 nm
   • (c) 0.728 nm
   • (d) 0.729 nm

6. A particle has a position uncertainty $\Delta x = 1 \times 10^{-10} , \text{m}$. Calculate the minimum momentum uncertainty $\Delta p$ in kg·m/s.

   • (a) 5.27 $\times 10^{-25}$
   • (b) 5.28 $\times 10^{-25}$
   • (c) 5.29 $\times 10^{-25}$
   • (d) 5.30 $\times 10^{-25}$

7. A photon has energy $E = 3 , \text{eV}$. Calculate its wavelength in nm.

   • (a) 412 nm
   • (b) 413 nm
   • (c) 414 nm
   • (d) 415 nm

8. A metal with $\phi = 2.5 , \text{eV}$ is illuminated by light of $\lambda = 400 , \text{nm}$. Calculate $K_{\text{max}}$ in Joules.

   • (a) 9.59 $\times 10^{-20}$
   • (b) 9.60 $\times 10^{-20}$
   • (c) 9.61 $\times 10^{-20}$
   • (d) 9.62 $\times 10^{-20}$

9. A proton ($m = 1.67 \times 10^{-27} , \text{kg}$) moves at $v = 2 \times 10^5 , \text{m/s}$. Calculate its de Broglie wavelength in nm.

   • (a) 1.97 nm
   • (b) 1.98 nm
   • (c) 1.99 nm
   • (d) 2.00 nm

10. A particle with $\Delta p = 2 \times 10^{-20} , \text{kg·m/s}$. Calculate the minimum position uncertainty $\Delta x$ in meters.

    • (a) 2.63 $\times 10^{-15}$
    • (b) 2.64 $\times 10^{-15}$
    • (c) 2.65 $\times 10^{-15}$
    • (d) 2.66 $\times 10^{-15}$

11. A photon with $\lambda = 200 , \text{nm}$. Calculate its momentum in kg·m/s.

    • (a) 3.31 $\times 10^{-27}$
    • (b) 3.32 $\times 10^{-27}$
    • (c) 3.33 $\times 10^{-27}$
    • (d) 3.34 $\times 10^{-27}$

12. A metal with $\nu_0 = 4 \times 10^{14} , \text{Hz}$ is illuminated by $\nu = 5 \times 10^{14} , \text{Hz}$. Calculate $K_{\text{max}}$ in eV.

    • (a) 0.41 eV
    • (b) 0.42 eV
    • (c) 0.43 eV
    • (d) 0.44 eV

13. An electron is accelerated through a potential difference of $V = 50 , \text{V}$. Calculate its de Broglie wavelength in nm.

    • (a) 1.72 nm
    • (b) 1.73 nm
    • (c) 1.74 nm
    • (d) 1.75 nm

14. A particle with $\Delta x = 5 \times 10^{-12} , \text{m}$ and $m = 1 \times 10^{-30} , \text{kg}$. Calculate the minimum velocity uncertainty $\Delta v$ in m/s.

    • (a) 1.05 $\times 10^{7}$
    • (b) 1.06 $\times 10^{7}$
    • (c) 1.07 $\times 10^{7}$
    • (d) 1.08 $\times 10^{7}$

15. A photon with $E = 4 , \text{eV}$. Calculate its frequency in Hz.

    • (a) 9.63 $\times 10^{14}$
    • (b) 9.64 $\times 10^{14}$
    • (c) 9.65 $\times 10^{14}$
    • (d) 9.66 $\times 10^{14}$

16. A metal with $\phi = 3.0 , \text{eV}$ is illuminated by $\lambda = 250 , \text{nm}$. Calculate $V_s$ in volts.

    • (a) 1.95 V
    • (b) 1.96 V
    • (c) 1.97 V
    • (d) 1.98 V

17. A neutron ($m = 1.67 \times 10^{-27} , \text{kg}$) moves at $v = 1 \times 10^4 , \text{m/s}$. Calculate its de Broglie wavelength in nm.

    • (a) 39.6 nm
    • (b) 39.7 nm
    • (c) 39.8 nm
    • (d) 39.9 nm

18. A photon with $\nu = 3 \times 10^{14} , \text{Hz}$. Calculate its energy in Joules.

    • (a) 1.98 $\times 10^{-19}$
    • (b) 1.99 $\times 10^{-19}$
    • (c) 2.00 $\times 10^{-19}$
    • (d) 2.01 $\times 10^{-19}$

19. A metal with $\phi = 2.2 , \text{eV}$ is illuminated by $\lambda = 350 , \text{nm}$. Calculate $K_{\text{max}}$ in eV.

    • (a) 1.33 eV
    • (b) 1.34 eV
    • (c) 1.35 eV
    • (d) 1.36 eV

20. An electron with $\Delta p = 1 \times 10^{-24} , \text{kg·m/s}$. Calculate the minimum $\Delta x$ in meters.

    • (a) 5.27 $\times 10^{-11}$
    • (b) 5.28 $\times 10^{-11}$
    • (c) 5.29 $\times 10^{-11}$
    • (d) 5.30 $\times 10^{-11}$

21. A photon with $\lambda = 600 , \text{nm}$. Calculate its energy in Joules.

    • (a) 3.30 $\times 10^{-19}$
    • (b) 3.31 $\times 10^{-19}$
    • (c) 3.32 $\times 10^{-19}$
    • (d) 3.33 $\times 10^{-19}$

22. A metal with $\nu_0 = 6 \times 10^{14} , \text{Hz}$ is illuminated by $\nu = 7 \times 10^{14} , \text{Hz}$. Calculate $V_s$ in volts.

    • (a) 0.41 V
    • (b) 0.42 V
    • (c) 0.43 V
    • (d) 0.44 V

23. A proton accelerated through $V = 100 , \text{V}$. Calculate its de Broglie wavelength in nm.

    • (a) 0.0285 nm
    • (b) 0.0286 nm
    • (c) 0.0287 nm
    • (d) 0.0288 nm

24. A particle with $\Delta x = 1 \times 10^{-9} , \text{m}$. Calculate the minimum $\Delta p$ in kg·m/s.

    • (a) 5.27 $\times 10^{-26}$
    • (b) 5.28 $\times 10^{-26}$
    • (c) 5.29 $\times 10^{-26}$
    • (d) 5.30 $\times 10^{-26}$

25. A photon with $E = 5 , \text{eV}$. Calculate its wavelength in nm.

    • (a) 247 nm
    • (b) 248 nm
    • (c) 249 nm
    • (d) 250 nm

26. A metal with $\phi = 1.8 , \text{eV}$ is illuminated by $\lambda = 500 , \text{nm}$. Calculate $K_{\text{max}}$ in eV.

    • (a) 0.67 eV
    • (b) 0.68 eV
    • (c) 0.69 eV
    • (d) 0.70 eV

27. An electron ($m = 9.11 \times 10^{-31} , \text{kg}$) moves at $v = 5 \times 10^5 , \text{m/s}$. Calculate its de Broglie wavelength in nm.

    • (a) 1.45 nm
    • (b) 1.46 nm
    • (c) 1.47 nm
    • (d) 1.48 nm

28. A photon with $\nu = 8 \times 10^{14} , \text{Hz}$. Calculate its momentum in kg·m/s.

    • (a) 1.76 $\times 10^{-27}$
    • (b) 1.77 $\times 10^{-27}$
    • (c) 1.78 $\times 10^{-27}$
    • (d) 1.79 $\times 10^{-27}$

29. A metal with $\phi = 2.8 , \text{eV}$ is illuminated by $\lambda = 300 , \text{nm}$. Calculate $V_s$ in volts.

    • (a) 1.32 V
    • (b) 1.33 V
    • (c) 1.34 V
    • (d) 1.35 V

30. A neutron with $\Delta p = 3 \times 10^{-22} , \text{kg·m/s}$. Calculate the minimum $\Delta x$ in meters.

    • (a) 1.75 $\times 10^{-13}$
    • (b) 1.76 $\times 10^{-13}$
    • (c) 1.77 $\times 10^{-13}$
    • (d) 1.78 $\times 10^{-13}$

31. A spacecraft emits photons with $\lambda = 400 , \text{nm}$ for propulsion. Calculate the energy of each photon in eV.

    • (a) 3.09 eV
    • (b) 3.10 eV
    • (c) 3.11 eV
    • (d) 3.12 eV

32. A metal with $\nu_0 = 3 \times 10^{14} , \text{Hz}$ is illuminated by $\nu = 4 \times 10^{14} , \text{Hz}$. Calculate $K_{\text{max}}$ in Joules.

    • (a) 6.62 $\times 10^{-20}$
    • (b) 6.63 $\times 10^{-20}$
    • (c) 6.64 $\times 10^{-20}$
    • (d) 6.65 $\times 10^{-20}$

33. An electron accelerated through $V = 200 , \text{V}$. Calculate its de Broglie wavelength in nm.

    • (a) 0.865 nm
    • (b) 0.866 nm
    • (c) 0.867 nm
    • (d) 0.868 nm

34. A particle with $\Delta x = 2 \times 10^{-11} , \text{m}$ and $m = 2 \times 10^{-30} , \text{kg}$. Calculate the minimum $\Delta v$ in m/s.

    • (a) 2.63 $\times 10^{6}$
    • (b) 2.64 $\times 10^{6}$
    • (c) 2.65 $\times 10^{6}$
    • (d) 2.66 $\times 10^{6}$

35. A photon with $E = 2.5 , \text{eV}$. Calculate its frequency in Hz.

    • (a) 6.02 $\times 10^{14}$
    • (b) 6.03 $\times 10^{14}$
    • (c) 6.04 $\times 10^{14}$
    • (d) 6.05 $\times 10^{14}$

Conceptual Problems

36. What is the energy of a photon proportional to?

    • (a) Wavelength
    • (b) Frequency
    • (c) Speed
    • (d) Momentum

37. What does the photoelectric effect demonstrate?

    • (a) Wave nature of light
    • (b) Particle nature of light
    • (c) Both wave and particle nature
    • (d) Neither wave nor particle nature

38. What is the unit of photon momentum $p$ in SI units?

    • (a) kg·m/s
    • (b) Joule
    • (c) Hertz
    • (d) Watt

39. What happens if the frequency of light is below the threshold frequency in the photoelectric effect?

    • (a) Electrons are ejected with high energy
    • (b) No electrons are ejected
    • (c) Electrons are ejected with low energy
    • (d) Electrons are ejected instantly

40. What does the de Broglie wavelength depend on?

    • (a) Mass only
    • (b) Velocity only
    • (c) Momentum
    • (d) Energy

41. What is the unit of the de Broglie wavelength $\lambda$?

    • (a) Meter
    • (b) Joule
    • (c) Hertz
    • (d) Watt

42. What does a smaller de Broglie wavelength indicate?

    • (a) Lower momentum
    • (b) Higher momentum
    • (c) No momentum
    • (d) Constant momentum

43. What happens to the de Broglie wavelength as the velocity of a particle increases?

    • (a) Increases
    • (b) Decreases
    • (c) Remains the same
    • (d) Becomes zero

44. What does the uncertainty principle state?

    • (a) Position and momentum can be known exactly
    • (b) Position and momentum cannot be known exactly simultaneously
    • (c) Position and energy are related
    • (d) Momentum and energy are related

45. What is the dimension of $h \nu$?

    • (a) $[\text{M} \text{L}^2 \text{T}^{-2}]$
    • (b) $[\text{M} \text{L} \text{T}^{-1}]$
    • (c) $[\text{L} \text{T}^{-2}]$
    • (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

46. What does the stopping potential $V_s$ depend on in the photoelectric effect?

    • (a) Intensity of light
    • (b) Frequency of light
    • (c) Wavelength of light only
    • (d) Work function only

47. What is the significance of $\frac{h}{p}$?

    • (a) Photon energy
    • (b) De Broglie wavelength
    • (c) Stopping potential
    • (d) Uncertainty principle

48. What happens to $K_{\text{max}}$ as the frequency of incident light increases?

    • (a) Decreases
    • (b) Increases
    • (c) Remains the same
    • (d) Becomes zero

49. What does wave-particle duality apply to?

    • (a) Light only
    • (b) Matter only
    • (c) Both light and matter
    • (d) Neither light nor matter

50. How does wave-particle duality assist in spacecraft quantum sensors?

    • (a) Increases energy
    • (b) Enables precision through matter wave interference
    • (c) Reduces momentum
    • (d) Increases wavelength

Derivation Problems

51. Derive the photon momentum $p = \frac{h}{\lambda}$.

52. Derive the photoelectric equation $K_{\text{max}} = h \nu - \phi$.

53. Derive the stopping potential $V_s = \frac{h \nu - \phi}{e}$.

54. Derive the de Broglie wavelength $\lambda = \frac{h}{p}$.

55. Derive the uncertainty principle $\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$.

56. Derive the photon energy $E = h \nu$.

57. Derive the de Broglie wavelength of an electron accelerated through a potential difference $V$.

58. Derive the momentum uncertainty $\Delta p$ for a given $\Delta x$.

59. Derive the photon wavelength $\lambda$ given its energy $E$.

60. Derive the maximum kinetic energy $K_{\text{max}}$ for a given $\nu$ and $\phi$.

61. Derive the de Broglie wavelength of a proton moving at a given velocity.

62. Derive the position uncertainty $\Delta x$ for a given $\Delta p$.

63. Derive the photon frequency $\nu$ given its wavelength $\lambda$.

64. Derive the stopping potential $V_s$ for a given $\lambda$ and $\phi$.

65. Derive the minimum velocity uncertainty $\Delta v$ for a particle with given $\Delta x$ and mass.

NEET-style Conceptual Problems

66. What is the unit of photon energy $E$ in SI units?

    • (a) Joule
    • (b) Radian
    • (c) Hertz
    • (d) Watt

67. What does a photon’s momentum depend on?

    • (a) Frequency
    • (b) Wavelength
    • (c) Speed
    • (d) Mass

68. What is the relationship between photon energy $E$ and wavelength $\lambda$?

    • (a) $E \propto \lambda$
    • (b) $E \propto \frac{1}{\lambda}$
    • (c) $E$ is independent of $\lambda$
    • (d) $E \propto \lambda^2$

69. What happens to the number of photoelectrons if the intensity of light increases?

    • (a) Decreases
    • (b) Increases
    • (c) Remains the same
    • (d) Becomes zero

70. What is the dimension of Planck’s constant $h$?

    • (a) $[\text{M} \text{L}^2 \text{T}^{-1}]$
    • (b) $[\text{M} \text{L} \text{T}^{-1}]$
    • (c) $[\text{L} \text{T}^{-2}]$
    • (d) $[\text{M} \text{L}^2 \text{T}^{-2}]$

71. What does the work function $\phi$ represent in the photoelectric effect?

    • (a) Maximum kinetic energy
    • (b) Minimum energy to eject an electron
    • (c) Stopping potential
    • (d) Photon energy

72. What is the role of wave-particle duality in electron microscopy?

    • (a) Increases energy
    • (b) Uses electron waves for high-resolution imaging
    • (c) Reduces momentum
    • (d) Increases wavelength

73. What happens to the de Broglie wavelength of a particle if its mass increases?

    • (a) Increases
    • (b) Decreases
    • (c) Remains the same
    • (d) Becomes zero

74. Why does the photoelectric effect occur instantly above the threshold frequency?

    • (a) Due to wave nature of light
    • (b) Due to particle nature of light
    • (c) Due to interference
    • (d) Due to diffraction

75. What is the unit of stopping potential $V_s$?

    • (a) Volt
    • (b) Joule
    • (c) Hertz
    • (d) Watt

76. What does a large uncertainty in position $\Delta x$ indicate?

    • (a) Large uncertainty in momentum
    • (b) Small uncertainty in momentum
    • (c) No uncertainty in momentum
    • (d) Constant momentum

77. Which particles exhibit de Broglie waves?

    • (a) Photons only
    • (b) Electrons only
    • (c) All particles with momentum
    • (d) Stationary particles

78. What is the effect of wave-particle duality in a double-slit experiment?

    • (a) No interference
    • (b) Interference pattern for electrons
    • (c) Diffraction only
    • (d) Refraction only

79. What does a pseudo-force do in a non-inertial frame for quantum calculations?

    • (a) Affects perceived de Broglie wavelength
    • (b) Affects energy
    • (c) Creates photons
    • (d) Reduces momentum

80. What is the dimension of $\frac{h}{\lambda}$?

    • (a) $[\text{M} \text{L} \text{T}^{-1}]$
    • (b) $[\text{M} \text{L}^2 \text{T}^{-2}]$
    • (c) $[\text{L} \text{T}^{-2}]$
    • (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

81. What is the role of photons in spacecraft propulsion?

    • (a) Increases mass
    • (b) Provides thrust through momentum transfer
    • (c) Reduces energy
    • (d) Increases wavelength

82. What happens to $K_{\text{max}}$ if the work function $\phi$ increases?

    • (a) Increases
    • (b) Decreases
    • (c) Remains the same
    • (d) Becomes zero

83. Why does a photon have zero rest mass?

    • (a) Due to $E = pc$
    • (b) Due to $\lambda = \frac{h}{p}$
    • (c) Due to interference
    • (d) Due to diffraction

84. What is the significance of $h \nu_0$?

    • (a) Maximum kinetic energy
    • (b) Work function in the photoelectric effect
    • (c) Stopping potential
    • (d) De Broglie wavelength

85. What is the unit of position uncertainty $\Delta x$?

    • (a) Meter
    • (b) Joule
    • (c) Hertz
    • (d) Watt

86. What does a high $K_{\text{max}}$ indicate in the photoelectric effect?

    • (a) Low frequency
    • (b) High frequency above threshold
    • (c) No frequency
    • (d) Constant frequency

87. What is the physical significance of $\frac{\hbar}{2}$?

    • (a) Photon energy
    • (b) Minimum uncertainty product
    • (c) De Broglie wavelength
    • (d) Stopping potential

88. Why does the de Broglie wavelength of a macroscopic object become negligible?

    • (a) Due to large momentum
    • (b) Due to small momentum
    • (c) Due to interference
    • (d) Due to diffraction

89. What is the dimension of $\Delta x \cdot \Delta p$?

    • (a) $[\text{M} \text{L}^2 \text{T}^{-1}]$
    • (b) $[\text{M} \text{L} \text{T}^{-1}]$
    • (c) $[\text{L} \text{T}^{-2}]$
    • (d) $[\text{M} \text{L}^2 \text{T}^{-2}]$

90. How does the photoelectric effect support the particle nature of light?

    • (a) Through interference patterns
    • (b) Through photon energy $h \nu$
    • (c) Through diffraction
    • (d) Through refraction

91. What is the role of the threshold frequency $\nu_0$?

    • (a) Determines maximum kinetic energy
    • (b) Minimum frequency for electron ejection
    • (c) Determines stopping potential
    • (d) Determines de Broglie wavelength

92. What does a small de Broglie wavelength indicate?

    • (a) Low momentum
    • (b) High momentum
    • (c) No momentum
    • (d) Constant momentum

93. What is the physical significance of $\frac{\sin \alpha}{\alpha}$?

    • (a) Photon energy
    • (b) Intensity factor in diffraction (not applicable here)
    • (c) Incorrect context
    • (d) De Broglie wavelength

94. What is the dimension of $\frac{h c}{\lambda}$?

    • (a) $[\text{M} \text{L}^2 \text{T}^{-2}]$
    • (b) $[\text{M} \text{L} \text{T}^{-1}]$
    • (c) $[\text{L} \text{T}^{-2}]$
    • (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

95. Why does wave-particle duality apply to both light and matter?

    • (a) Due to $\lambda = \frac{h}{p}$
    • (b) Due to interference only
    • (c) Due to diffraction only
    • (d) Due to refraction

NEET-style Numerical Problems

96. A photon with $\lambda = 300 , \text{nm}$. Calculate its energy in eV.

    • (a) 4.12 eV
    • (b) 4.13 eV
    • (c) 4.14 eV
    • (d) 4.15 eV

97. A metal with $\phi = 2.0 , \text{eV}$ is illuminated by $\lambda = 400 , \text{nm}$. Calculate $K_{\text{max}}$ in eV.

    • (a) 1.09 eV
    • (b) 1.10 eV
    • (c) 1.11 eV
    • (d) 1.12 eV

98. An electron ($m = 9.11 \times 10^{-31} , \text{kg}$) moves at $v = 2 \times 10^6 , \text{m/s}$. Calculate its de Broglie wavelength in nm.

    • (a) 0.363 nm
    • (b) 0.364 nm
    • (c) 0.365 nm
    • (d) 0.366 nm

99. A particle with $\Delta x = 1 \times 10^{-8} , \text{m}$. Calculate the minimum $\Delta p$ in kg·m/s.

    • (a) 5.27 $\times 10^{-27}$
    • (b) 5.28 $\times 10^{-27}$
    • (c) 5.29 $\times 10^{-27}$
    • (d) 5.30 $\times 10^{-27}$

100. A photon with $E = 2 , \text{eV}$. Calculate its wavelength in nm.
     - (a) 619 nm
     - (b) 620 nm
     - (c) 621 nm
     - (d) 622 nm

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