More About Matter Waves Problems

This section provides 100 problems to test your understanding of advanced matter wave concepts, including calculations of wave function probabilities, energy quantization in a box, tunneling probabilities, and uncertainty principle quantities, as well as applications like quantum tunneling in nanotechnology and quantum sensors in spacecraft systems. 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 quantum mechanics, a key topic for JEE/NEET success.

Numerical Problems

1. A particle has a normalized wave function $\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{\pi x}{a}\right)$ for $0 \leq x \leq a$. Calculate the probability of finding the particle between $x = 0$ and $x = \frac{a}{2}$.

   • (a) 0.49
   • (b) 0.50
   • (c) 0.51
   • (d) 0.52

2. An electron ($m = 9.11 \times 10^{-31} , \text{kg}$) is confined in a box of length $L = 1 , \text{nm}$. Calculate the energy $E_1$ for $n=1$ in eV.

   • (a) 0.375 eV
   • (b) 0.376 eV
   • (c) 0.377 eV
   • (d) 0.378 eV

3. An electron tunnels through a barrier with $V_0 - E = 1 , \text{eV}$ and width $a = 0.5 , \text{nm}$. Calculate the tunneling probability $T$.

   • (a) 0.076
   • (b) 0.077
   • (c) 0.078
   • (d) 0.079

4. A particle has a position uncertainty $\Delta x = 0.2 , \text{nm}$. Calculate the minimum momentum uncertainty $\Delta p$ in kg·m/s.

   • (a) 2.63 $\times 10^{-25}$
   • (b) 2.64 $\times 10^{-25}$
   • (c) 2.65 $\times 10^{-25}$
   • (d) 2.66 $\times 10^{-25}$

5. A particle in a box has $E_1 = 0.2 , \text{eV}$ for $n=1$. Calculate $E_3$ in eV.

   • (a) 1.7 eV
   • (b) 1.8 eV
   • (c) 1.9 eV
   • (d) 2.0 eV

6. A proton ($m = 1.67 \times 10^{-27} , \text{kg}$) tunnels through a barrier with $V_0 - E = 0.5 , \text{eV}$, $a = 0.1 , \text{nm}$. Calculate $T$.

   • (a) 0.766
   • (b) 0.767
   • (c) 0.768
   • (d) 0.769

7. An electron in a box ($L = 0.5 , \text{nm}$). Calculate $E_2$ in eV.

   • (a) 0.599 eV
   • (b) 0.600 eV
   • (c) 0.601 eV
   • (d) 0.602 eV

8. A particle with $\Delta p = 5 \times 10^{-20} , \text{kg·m/s}$. Calculate the minimum $\Delta x$ in meters.

   • (a) 1.05 $\times 10^{-15}$
   • (b) 1.06 $\times 10^{-15}$
   • (c) 1.07 $\times 10^{-15}$
   • (d) 1.08 $\times 10^{-15}$

9. A wave function $\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{2\pi x}{a}\right)$ for $0 \leq x \leq a$. Calculate the probability between $x = 0$ and $x = \frac{a}{4}$.

   • (a) 0.24
   • (b) 0.25
   • (c) 0.26
   • (d) 0.27

10. A particle in a box ($L = 2 , \text{nm}$, $m = 9.11 \times 10^{-31} , \text{kg}$). Calculate $E_1$ in Joules.

    • (a) 1.50 $\times 10^{-20}$
    • (b) 1.51 $\times 10^{-20}$
    • (c) 1.52 $\times 10^{-20}$
    • (d) 1.53 $\times 10^{-20}$

11. An electron tunnels with $V_0 - E = 0.8 , \text{eV}$, $a = 0.3 , \text{nm}$. Calculate $T$.

    • (a) 0.145
    • (b) 0.146
    • (c) 0.147
    • (d) 0.148

12. A particle with $\Delta x = 1 \times 10^{-11} , \text{m}$, $m = 1 \times 10^{-30} , \text{kg}$. Calculate the minimum velocity uncertainty $\Delta v$ in m/s.

    • (a) 5.27 $\times 10^{6}$
    • (b) 5.28 $\times 10^{6}$
    • (c) 5.29 $\times 10^{6}$
    • (d) 5.30 $\times 10^{6}$

13. A particle in a box with $E_2 = 0.8 , \text{eV}$. Calculate $E_1$ in eV.

    • (a) 0.19 eV
    • (b) 0.20 eV
    • (c) 0.21 eV
    • (d) 0.22 eV

14. A wave function $\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{3\pi x}{a}\right)$ for $0 \leq x \leq a$. Calculate the probability between $x = \frac{a}{3}$ and $x = \frac{2a}{3}$.

    • (a) 0.33
    • (b) 0.34
    • (c) 0.35
    • (d) 0.36

15. A proton in a box ($L = 1 , \text{nm}$). Calculate $E_1$ in eV.

    • (a) 0.000204 eV
    • (b) 0.000205 eV
    • (c) 0.000206 eV
    • (d) 0.000207 eV

16. An electron with $\Delta E = 2 \times 10^{-19} , \text{J}$. Calculate the minimum $\Delta t$ in seconds.

    • (a) 2.63 $\times 10^{-16}$
    • (b) 2.64 $\times 10^{-16}$
    • (c) 2.65 $\times 10^{-16}$
    • (d) 2.66 $\times 10^{-16}$

17. An electron tunnels with $V_0 - E = 1.5 , \text{eV}$, $a = 0.2 , \text{nm}$. Calculate $T$.

    • (a) 0.168
    • (b) 0.169
    • (c) 0.170
    • (d) 0.171

18. A wave function $\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{\pi x}{a}\right)$ for $0 \leq x \leq a$. Calculate $\langle x \rangle$ in terms of $a$.

    • (a) $0.49a$
    • (b) $0.50a$
    • (c) $0.51a$
    • (d) $0.52a$

19. A particle in a box ($L = 0.1 , \text{nm}$, $m = 9.11 \times 10^{-31} , \text{kg}$). Calculate $E_1$ in eV.

    • (a) 37.5 eV
    • (b) 37.6 eV
    • (c) 37.7 eV
    • (d) 37.8 eV

20. A proton with $\Delta p = 3 \times 10^{-20} , \text{kg·m/s}$. Calculate the minimum $\Delta x$ in meters.

    • (a) 1.75 $\times 10^{-15}$
    • (b) 1.76 $\times 10^{-15}$
    • (c) 1.77 $\times 10^{-15}$
    • (d) 1.78 $\times 10^{-15}$

21. An electron in a box ($L = 1.5 , \text{nm}$). Calculate $E_3$ in eV.

    • (a) 0.149 eV
    • (b) 0.150 eV
    • (c) 0.151 eV
    • (d) 0.152 eV

22. A wave function $\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{2\pi x}{a}\right)$ for $0 \leq x \leq a$. Calculate the probability between $x = \frac{a}{4}$ and $x = \frac{3a}{4}$.

    • (a) 0.49
    • (b) 0.50
    • (c) 0.51
    • (d) 0.52

23. An electron tunnels with $V_0 - E = 0.5 , \text{eV}$, $a = 0.4 , \text{nm}$. Calculate $T$.

    • (a) 0.055
    • (b) 0.056
    • (c) 0.057
    • (d) 0.058

24. A particle with $\Delta x = 0.1 , \text{nm}$, $m = 1.67 \times 10^{-27} , \text{kg}$. Calculate the minimum $\Delta v$ in m/s.

    • (a) 3.15 $\times 10^{2}$
    • (b) 3.16 $\times 10^{2}$
    • (c) 3.17 $\times 10^{2}$
    • (d) 3.18 $\times 10^{2}$

25. A particle in a box with $E_1 = 0.05 , \text{eV}$, $n=4$. Calculate $E_4$ in eV.

    • (a) 0.79 eV
    • (b) 0.80 eV
    • (c) 0.81 eV
    • (d) 0.82 eV

26. A wave function $\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{3\pi x}{a}\right)$ for $0 \leq x \leq a$. Calculate $\langle x \rangle$ in terms of $a$.

    • (a) $0.49a$
    • (b) $0.50a$
    • (c) $0.51a$
    • (d) $0.52a$

27. An electron with $\Delta E = 1 \times 10^{-18} , \text{J}$. Calculate the minimum $\Delta t$ in seconds.

    • (a) 5.27 $\times 10^{-17}$
    • (b) 5.28 $\times 10^{-17}$
    • (c) 5.29 $\times 10^{-17}$
    • (d) 5.30 $\times 10^{-17}$

28. A proton in a box ($L = 0.2 , \text{nm}$). Calculate $E_1$ in eV.

    • (a) 0.00510 eV
    • (b) 0.00511 eV
    • (c) 0.00512 eV
    • (d) 0.00513 eV

29. An electron tunnels with $V_0 - E = 2.0 , \text{eV}$, $a = 0.1 , \text{nm}$. Calculate $T$.

    • (a) 0.364
    • (b) 0.365
    • (c) 0.366
    • (d) 0.367

30. A wave function $\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{\pi x}{a}\right)$ for $0 \leq x \leq a$. Calculate the probability between $x = 0$ and $x = \frac{a}{4}$.

    • (a) 0.09
    • (b) 0.10
    • (c) 0.11
    • (d) 0.12

31. A spacecraft quantum system confines an electron in a box ($L = 0.5 , \text{nm}$). Calculate $E_1$ in eV.

    • (a) 1.49 eV
    • (b) 1.50 eV
    • (c) 1.51 eV
    • (d) 1.52 eV

32. A particle with $\Delta p = 2 \times 10^{-21} , \text{kg·m/s}$. Calculate the minimum $\Delta x$ in meters.

    • (a) 2.63 $\times 10^{-14}$
    • (b) 2.64 $\times 10^{-14}$
    • (c) 2.65 $\times 10^{-14}$
    • (d) 2.66 $\times 10^{-14}$

33. A particle in a box ($L = 1 , \text{nm}$, $m = 1.67 \times 10^{-27} , \text{kg}$). Calculate $E_2$ in eV.

    • (a) 0.000816 eV
    • (b) 0.000817 eV
    • (c) 0.000818 eV
    • (d) 0.000819 eV

34. A wave function $\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{2\pi x}{a}\right)$ for $0 \leq x \leq a$. Calculate $\langle x \rangle$ in terms of $a$.

    • (a) $0.49a$
    • (b) $0.50a$
    • (c) $0.51a$
    • (d) $0.52a$

35. An electron tunnels with $V_0 - E = 0.2 , \text{eV}$, $a = 0.6 , \text{nm}$. Calculate $T$.

    • (a) 0.042
    • (b) 0.043
    • (c) 0.044
    • (d) 0.045

Conceptual Problems

36. What does the wave function $\psi(x)$ represent?

    • (a) Probability of finding a particle
    • (b) Quantum state of a particle
    • (c) Energy of a particle
    • (d) Momentum of a particle

37. What does $|\psi(x)|^2$ represent?

    • (a) Wave function
    • (b) Probability density
    • (c) Energy density
    • (d) Momentum density

38. What is the unit of the wave function $\psi(x)$ in one dimension?

    • (a) m
    • (b) m$^{-1/2}$
    • (c) kg·m/s
    • (d) Joule

39. What happens to the energy levels in a particle in a box as $n$ increases?

    • (a) Decrease
    • (b) Increase as $n^2$
    • (c) Remain the same
    • (d) Become zero

40. What does quantum tunneling allow a particle to do?

    • (a) Increase its energy
    • (b) Pass through a potential barrier
    • (c) Reflect completely
    • (d) Absorb energy

41. What is the unit of tunneling probability $T$?

    • (a) Dimensionless
    • (b) Meter
    • (c) Joule
    • (d) kg·m/s

42. What does a smaller $\Delta x$ in the uncertainty principle indicate?

    • (a) Smaller $\Delta p$
    • (b) Larger $\Delta p$
    • (c) No change in $\Delta p$
    • (d) Zero $\Delta p$

43. What happens to the tunneling probability as the barrier width $a$ increases?

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

44. What does the particle in a box model demonstrate?

    • (a) Continuous energy levels
    • (b) Quantized energy levels
    • (c) Tunneling
    • (d) Uncertainty principle

45. What is the dimension of $E_n$ in the particle in a box?

    • (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 normalization condition $\int |\psi|^2 dx = 1$ ensure?

    • (a) Energy conservation
    • (b) Total probability is 1
    • (c) Momentum conservation
    • (d) Wave function is real

47. What is the significance of $\frac{n^2 \pi^2 \hbar^2}{2 m L^2}$?

    • (a) Wave function
    • (b) Energy levels in a box
    • (c) Tunneling probability
    • (d) Uncertainty product

48. What happens to the energy levels in a box as $L$ increases?

    • (a) Increase
    • (b) Decrease
    • (c) Remain the same
    • (d) Become zero

49. What does the uncertainty principle apply to?

    • (a) Classical particles only
    • (b) Quantum particles only
    • (c) Both classical and quantum particles
    • (d) Photons only

50. How does quantum tunneling assist in spacecraft nanotechnology?

    • (a) Increases energy
    • (b) Enables quantum switching in devices
    • (c) Reduces momentum
    • (d) Increases wavelength

Derivation Problems

51. Derive the probability density $|\psi(x)|^2$ for a particle in a box with $\psi(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi x}{L}\right)$.

52. Derive the energy levels $E_n = \frac{n^2 \pi^2 \hbar^2}{2 m L^2}$ for a particle in a box.

53. Derive the tunneling probability $T \approx e^{-2 \kappa a}$ for a rectangular barrier.

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

55. Derive the wave function $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi x}{L}\right)$ for a particle in a box.

56. Derive the expectation value $\langle x \rangle$ for $\psi(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right)$.

57. Derive the probability of finding a particle in a box between $x = 0$ and $x = \frac{L}{2}$ for $n=1$.

58. Derive the energy $E_1$ for an electron in a box of length $L$.

59. Derive the tunneling probability $T$ for an electron with given $V_0 - E$ and $a$.

60. Derive the minimum $\Delta p$ for a given $\Delta x$ using the uncertainty principle.

61. Derive the number of nodes in a particle in a box wave function for a given $n$.

62. Derive the minimum $\Delta x$ for a given $\Delta p$ using the uncertainty principle.

63. Derive the energy difference $E_2 - E_1$ for a particle in a box.

64. Derive the probability density for a particle in a box with $n=2$.

65. Derive the minimum $\Delta t$ for a given $\Delta E$ using the uncertainty principle.

NEET-style Conceptual Problems

66. What is the unit of energy $E_n$ in the particle in a box model?

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

67. What does a larger $n$ in the particle in a box indicate?

    • (a) Lower energy
    • (b) Higher energy
    • (c) No energy change
    • (d) Zero energy

68. What is the relationship between $E_n$ and $L$ in the particle in a box?

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

69. What happens to the tunneling probability if $V_0 - E$ increases?

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

70. 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) $[\text{M} \text{L}^2 \text{T}^{-2}]$

71. What does the boundary condition $\psi(0) = 0$ in a box represent?

    • (a) Infinite potential at the walls
    • (b) Zero probability at the walls
    • (c) Maximum probability at the walls
    • (d) Energy at the walls

72. What is the role of the uncertainty principle in electron microscopy?

    • (a) Increases energy
    • (b) Limits resolution due to position-momentum uncertainty
    • (c) Reduces momentum
    • (d) Increases wavelength

73. What happens to the number of nodes in a particle in a box as $n$ increases?

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

74. Why does quantum tunneling occur in quantum mechanics?

    • (a) Due to classical mechanics
    • (b) Due to the wave nature of particles
    • (c) Due to high energy
    • (d) Due to low energy

75. What is the unit of probability density $|\psi(x)|^2$ in one dimension?

    • (a) m$^{-1}$
    • (b) m
    • (c) Joule
    • (d) kg·m/s

76. What does a larger $\Delta E$ in the uncertainty principle indicate?

    • (a) Larger $\Delta t$
    • (b) Smaller $\Delta t$
    • (c) No change in $\Delta t$
    • (d) Zero $\Delta t$

77. Which quantum number determines the energy in a particle in a box?

    • (a) $l$
    • (b) $n$
    • (c) $m$
    • (d) $s$

78. What is the effect of infinite potential walls in a box?

    • (a) Continuous energy levels
    • (b) Quantized energy levels
    • (c) No energy levels
    • (d) Random energy levels

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

    • (a) Affects perceived wave function
    • (b) Affects energy
    • (c) Creates tunneling
    • (d) Reduces momentum

80. What is the dimension of $\frac{\hbar^2}{2 m L^2}$?

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

81. What is the role of quantum tunneling in spacecraft quantum devices?

    • (a) Increases energy
    • (b) Enables nanoscale switching
    • (c) Reduces momentum
    • (d) Increases wavelength

82. What happens to the wave function at the boundaries of a box?

    • (a) Maximum value
    • (b) Zero value
    • (c) Constant value
    • (d) Infinite value

83. Why does the uncertainty principle limit precision in measurements?

    • (a) Due to $\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$
    • (b) Due to classical mechanics
    • (c) Due to high energy
    • (d) Due to low energy

84. What is the significance of $e^{-2 \kappa a}$?

    • (a) Wave function
    • (b) Energy level
    • (c) Tunneling probability
    • (d) Uncertainty product

85. What is the unit of $\kappa$ in tunneling probability?

    • (a) m$^{-1}$
    • (b) m
    • (c) Joule
    • (d) kg·m/s

86. What does a high tunneling probability indicate?

    • (a) High barrier width
    • (b) Low barrier width
    • (c) No barrier
    • (d) Constant barrier

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

    • (a) Energy level
    • (b) Minimum uncertainty product
    • (c) Wave function
    • (d) Tunneling probability

88. Why does the particle in a box have quantized energy levels?

    • (a) Due to infinite potential walls
    • (b) Due to continuous potential
    • (c) Due to tunneling
    • (d) Due to uncertainty

89. What is the dimension of $\sin\left(\frac{n \pi x}{L}\right)$?

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

90. How does the uncertainty principle affect quantum systems in spacecraft?

    • (a) Increases energy
    • (b) Limits precision in navigation sensors
    • (c) Reduces momentum
    • (d) Increases wavelength

91. What is the role of the quantum number $n$ in a box?

    • (a) Determines the wavelength
    • (b) Determines the energy level
    • (c) Determines the momentum
    • (d) Determines the position

92. What does a large probability density $|\psi(x)|^2$ indicate?

    • (a) Low probability of finding the particle
    • (b) High probability of finding the particle
    • (c) No probability
    • (d) Constant probability

93. What is the physical significance of $\frac{2}{L} \sin^2\left(\frac{n \pi x}{L}\right)$?

    • (a) Energy level
    • (b) Probability density in a box
    • (c) Tunneling probability
    • (d) Uncertainty product

94. What is the dimension of $\Delta E \cdot \Delta t$?

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

95. Why does tunneling probability depend on the particle’s mass?

    • (a) Due to $\kappa \propto \sqrt{m}$
    • (b) Due to energy levels
    • (c) Due to wave function
    • (d) Due to uncertainty

NEET-style Numerical Problems

96. A particle has $\psi(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{\pi x}{a}\right)$ for $0 \leq x \leq a$. Calculate the probability between $x = \frac{a}{4}$ and $x = \frac{3a}{4}$.

    • (a) 0.817
    • (b) 0.818
    • (c) 0.819
    • (d) 0.820

97. An electron in a box ($L = 1 , \text{nm}$). Calculate $E_1$ in eV.

    • (a) 0.375 eV
    • (b) 0.376 eV
    • (c) 0.377 eV
    • (d) 0.378 eV

98. An electron tunnels with $V_0 - E = 1 , \text{eV}$, $a = 0.2 , \text{nm}$. Calculate $T$.

    • (a) 0.364
    • (b) 0.365
    • (c) 0.366
    • (d) 0.367

99. A particle with $\Delta x = 0.5 , \text{nm}$. Calculate the minimum $\Delta p$ in kg·m/s.

    • (a) 1.05 $\times 10^{-25}$
    • (b) 1.06 $\times 10^{-25}$
    • (c) 1.07 $\times 10^{-25}$
    • (d) 1.08 $\times 10^{-25}$

100. A particle in a box with $E_1 = 0.3 , \text{eV}$, $n=2$. Calculate $E_2$ in eV.
     - (a) 1.1 eV
     - (b) 1.2 eV
     - (c) 1.3 eV
     - (d) 1.4 eV

Back to Chapter

Return to More About Matter Waves Chapter