PHYS 165 MODULE 8 EXAMINATION 2026/2027
– QUANTUM MECHANICS, ATOMIC PHYSICS,
NUCLEAR PHYSICS & PARTICLE PHYSICS,
INTERMEDIATE TO ADVANCED LEVEL
SECTION A: QUANTUM MECHANICS FUNDAMENTALS
Multiple Choice Questions (1-50)
1. The wavefunction of a particle in an infinite potential well of width L is given by ψ(x) =
√(2/L) sin(nπx/L). What is the probability of finding the particle in the region L/4 to L/2 for
the ground state?
A) 0.25
B) 0.5
C) 0.091
D) 0.409
Answer: C
The probability is ∫(L/4 to L/2) |ψ|²dx = (2/L)∫(L/4 to L/2) sin²(πx/L)dx = (1/π)[π/4 - sin(π/2)/2]
= 0.091
,2. The Heisenberg uncertainty principle for position and momentum can be expressed as:
A) ΔxΔp ≥ ℏ/2
B) ΔxΔp ≥ ℏ
C) ΔxΔp ≥ 2ℏ
D) ΔxΔp ≥ ℏ/4
Answer: A
The standard Heisenberg uncertainty relation is ΔxΔp ≥ ℏ/2, where ℏ is the reduced
Planck's constant.
3. A quantum harmonic oscillator has energy levels given by E_n = (n + 1/2)ℏω. What is the
zero-point energy?
A) 0
B) ℏω
C) ℏω/2
D) 2ℏω
Answer: C
The zero-point energy corresponds to n=0, giving E₀ = ℏω/2, which is a purely quantum
mechanical effect.
4. The Pauli exclusion principle states that:
A) Two fermions can occupy the same quantum state
B) No two fermions can occupy the same quantum state
C) No two bosons can occupy the same quantum state
D) Electrons can share quantum numbers
Answer: B
The Pauli exclusion principle states that no two identical fermions can occupy the same
quantum state simultaneously.
5. Which of the following is NOT a valid quantum number for an electron in a hydrogen
atom?
A) n = 1
B) l = 0
C) m_l = 2
D) m_s = 1/2
Answer: C
For n=1, l can only be 0, and m_l can range from -l to +l, so m_l=2 is invalid.
,6. The de Broglie wavelength of a particle with momentum p is:
A) h/p
B) h/p²
C) p/h
D) h²/p
Answer: A
The de Broglie wavelength λ = h/p, where h is Planck's constant and p is the momentum.
7. In the quantum mechanical description, the expectation value of an operator  is given
by:
A) ⟨ψ|Â|ψ⟩
B) ⟨Âψ|ψ⟩
C) |⟨ψ|Â|ψ⟩|²
D) ∫ψÂψ*dτ
Answer: A
The expectation value is ⟨Â⟩ = ⟨ψ|Â|ψ⟩ = ∫ψ*Âψ dτ for normalized wavefunctions.
8. The normalization condition for a wavefunction ψ(x) in one dimension is:
A) ∫|ψ(x)|²dx = 1
B) ∫ψ(x)dx = 1
C) ∫|ψ(x)|dx = 1
D) ψ(x) = 1
Answer: A
The total probability of finding the particle somewhere in space must be unity: ∫|ψ(x)|²dx =
1.
9. Which operator represents momentum in quantum mechanics?
A) -iℏ∂/∂x
B) iℏ∂/∂x
C) -ℏ²∂²/∂x²
D) iℏ∂/∂t
Answer: A
The momentum operator is p
̂ = -iℏ∂/∂x in the position representation.
10. The commutator [x, p] equals:
A) 0
B) iℏ
, C) -iℏ
D) ℏ²
Answer: B
[x, p] = xp - px = iℏ, which is the fundamental commutation relation of quantum mechanics.
11. A particle in a box of length L has wavefunction ψ(x) = √(2/L)sin(nπx/L). The energy
eigenvalues are:
A) E_n = n²h²/8mL²
B) E_n = n²h²/2mL²
C) E_n = n²ℏ²π²/2mL²
D) Both A and C
Answer: D
The energy is E_n = n²h²/8mL² = n²ℏ²π²/2mL², as these are equivalent (h=2πℏ).
12. The spin quantum number for an electron is:
A) 1
B) 1/2
C) 0
D) -1/2
Answer: B
Electrons have spin s = 1/2, making them fermions.
13. The total number of electrons that can occupy a shell with principal quantum number n
is:
A) 2n
B) 2n²
C) n²
D) 2n+2
Answer: B
Each shell can hold 2n² electrons, accounting for all possible l, m_l, and m_s values.
14. The Schrödinger equation in time-independent form is:
A) Ĥψ = Eψ
B) iℏ∂ψ/∂t = Ĥψ
C) Ĥψ = 0
D) ∂ψ/∂t = Ĥψ
– QUANTUM MECHANICS, ATOMIC PHYSICS,
NUCLEAR PHYSICS & PARTICLE PHYSICS,
INTERMEDIATE TO ADVANCED LEVEL
SECTION A: QUANTUM MECHANICS FUNDAMENTALS
Multiple Choice Questions (1-50)
1. The wavefunction of a particle in an infinite potential well of width L is given by ψ(x) =
√(2/L) sin(nπx/L). What is the probability of finding the particle in the region L/4 to L/2 for
the ground state?
A) 0.25
B) 0.5
C) 0.091
D) 0.409
Answer: C
The probability is ∫(L/4 to L/2) |ψ|²dx = (2/L)∫(L/4 to L/2) sin²(πx/L)dx = (1/π)[π/4 - sin(π/2)/2]
= 0.091
,2. The Heisenberg uncertainty principle for position and momentum can be expressed as:
A) ΔxΔp ≥ ℏ/2
B) ΔxΔp ≥ ℏ
C) ΔxΔp ≥ 2ℏ
D) ΔxΔp ≥ ℏ/4
Answer: A
The standard Heisenberg uncertainty relation is ΔxΔp ≥ ℏ/2, where ℏ is the reduced
Planck's constant.
3. A quantum harmonic oscillator has energy levels given by E_n = (n + 1/2)ℏω. What is the
zero-point energy?
A) 0
B) ℏω
C) ℏω/2
D) 2ℏω
Answer: C
The zero-point energy corresponds to n=0, giving E₀ = ℏω/2, which is a purely quantum
mechanical effect.
4. The Pauli exclusion principle states that:
A) Two fermions can occupy the same quantum state
B) No two fermions can occupy the same quantum state
C) No two bosons can occupy the same quantum state
D) Electrons can share quantum numbers
Answer: B
The Pauli exclusion principle states that no two identical fermions can occupy the same
quantum state simultaneously.
5. Which of the following is NOT a valid quantum number for an electron in a hydrogen
atom?
A) n = 1
B) l = 0
C) m_l = 2
D) m_s = 1/2
Answer: C
For n=1, l can only be 0, and m_l can range from -l to +l, so m_l=2 is invalid.
,6. The de Broglie wavelength of a particle with momentum p is:
A) h/p
B) h/p²
C) p/h
D) h²/p
Answer: A
The de Broglie wavelength λ = h/p, where h is Planck's constant and p is the momentum.
7. In the quantum mechanical description, the expectation value of an operator  is given
by:
A) ⟨ψ|Â|ψ⟩
B) ⟨Âψ|ψ⟩
C) |⟨ψ|Â|ψ⟩|²
D) ∫ψÂψ*dτ
Answer: A
The expectation value is ⟨Â⟩ = ⟨ψ|Â|ψ⟩ = ∫ψ*Âψ dτ for normalized wavefunctions.
8. The normalization condition for a wavefunction ψ(x) in one dimension is:
A) ∫|ψ(x)|²dx = 1
B) ∫ψ(x)dx = 1
C) ∫|ψ(x)|dx = 1
D) ψ(x) = 1
Answer: A
The total probability of finding the particle somewhere in space must be unity: ∫|ψ(x)|²dx =
1.
9. Which operator represents momentum in quantum mechanics?
A) -iℏ∂/∂x
B) iℏ∂/∂x
C) -ℏ²∂²/∂x²
D) iℏ∂/∂t
Answer: A
The momentum operator is p
̂ = -iℏ∂/∂x in the position representation.
10. The commutator [x, p] equals:
A) 0
B) iℏ
, C) -iℏ
D) ℏ²
Answer: B
[x, p] = xp - px = iℏ, which is the fundamental commutation relation of quantum mechanics.
11. A particle in a box of length L has wavefunction ψ(x) = √(2/L)sin(nπx/L). The energy
eigenvalues are:
A) E_n = n²h²/8mL²
B) E_n = n²h²/2mL²
C) E_n = n²ℏ²π²/2mL²
D) Both A and C
Answer: D
The energy is E_n = n²h²/8mL² = n²ℏ²π²/2mL², as these are equivalent (h=2πℏ).
12. The spin quantum number for an electron is:
A) 1
B) 1/2
C) 0
D) -1/2
Answer: B
Electrons have spin s = 1/2, making them fermions.
13. The total number of electrons that can occupy a shell with principal quantum number n
is:
A) 2n
B) 2n²
C) n²
D) 2n+2
Answer: B
Each shell can hold 2n² electrons, accounting for all possible l, m_l, and m_s values.
14. The Schrödinger equation in time-independent form is:
A) Ĥψ = Eψ
B) iℏ∂ψ/∂t = Ĥψ
C) Ĥψ = 0
D) ∂ψ/∂t = Ĥψ
