CHEM 332: Exam 1 (Version B) Questions And
Answers With Rationales/ Graded A+/2026
Update /100%Correct
Section 1: Multiple Choice (Questions 1-40)
1. Which of the following is a consequence of the Heisenberg Uncertainty Principle?
a) Energy is quantized
b) Electrons have a spin of 1/2
c) It is impossible to know both the position and momentum of a particle with arbitrary precision
simultaneously
d) The wave function must be normalized
Answer: c
Rationale: The Heisenberg Uncertainty Principle states ΔxΔp≥ℏ/2ΔxΔp≥ℏ/2. It is a fundamental limit on
the precision with which complementary variables can be known.
2. A wavefunction ψ(x)ψ(x) is said to be "normalized" if:
a) ∫ψ∗ψ dτ=1∫ψ∗ψdτ=1
b) ∫ψ∗ψ dτ=0∫ψ∗ψdτ=0
c) ψψ is continuous
d) ψψ is an eigenfunction of the Hamiltonian
Answer: a
Rationale: Normalization ensures the total probability of finding the particle somewhere in space is 1.
3. For the particle in a 1D box of length LL, the energy is proportional to:
a) nn
b) n2n2
,c) 1/n1/n
d) 1/n21/n2
Answer: b
Rationale: En=n2h28mL2En=8mL2n2h2. Therefore, energy increases quadratically with the quantum
number nn.
4. How many nodes (excluding boundaries) does the n=3n=3 wavefunction have for a particle in a 1D
box?
a) 0
b) 1
c) 2
d) 3
Answer: c
*Rationale: The number of nodes = n−1n−1. For n=3n=3, nodes = 2.*
5. The zero-point energy of the quantum harmonic oscillator is:
a) 0
b) hνhν
c) 12hν21hν
d) 32hν23hν
Answer: c
Rationale: Ev=(v+12)hνEv=(v+21)hν. At v=0v=0, E0=12hνE0=21hν. This is a consequence of the
uncertainty principle; the particle cannot be at rest at the bottom of the well.
6. The selection rule for a harmonic oscillator in infrared spectroscopy is:
a) Δv=±1Δv=±1
b) Δv=0Δv=0
c) Δv=±2Δv=±2
d) ΔJ=±1ΔJ=±1
Answer: a
Rationale: For a harmonic oscillator, the transition dipole moment requires Δv=±1Δv=±1. Anharmonicity
allows overtones (Δv=±2Δv=±2), but the fundamental harmonic rule is ±1±1.
, 7. The wavefunction for the hydrogen atom depends on which quantum numbers?
a) n,l,mln,l,ml
b) n,l,sn,l,s
c) n,ml,msn,ml,ms
d) l,ml,msl,ml,ms
Answer: a
Rationale: The spatial wavefunction (orbital) is defined by principal (nn), azimuthal (ll), and magnetic
(mlml) quantum numbers. Spin (msms) is added later for the full electron state.
8. Which of the following is a Hermitian operator?
a) ddxdxd
b) H^H^
c) x^x^
d) Both b and c
Answer: d
Rationale: Hermitian operators have real eigenvalues, corresponding to observable quantities. Both the
Hamiltonian (H^H^) and position (x^x^) are Hermitian. The momentum operator is also Hermitian,
but ddxdxd alone is anti-Hermitian unless multiplied by −iℏ−iℏ.
9. The expectation value ⟨x⟩⟨x⟩ for the n=1n=1 state of the particle in a box is:
a) 0
b) L/2L/2
c) LL
d) L/4L/4
Answer: b
Rationale: The probability distribution is symmetric about the center of the box. The average position is
the midpoint, L/2L/2.
10. What is the degeneracy of the n=3n=3 energy level for a particle in a 3D cubic box?
a) 1
b) 3
c) 6
d) 10
Answers With Rationales/ Graded A+/2026
Update /100%Correct
Section 1: Multiple Choice (Questions 1-40)
1. Which of the following is a consequence of the Heisenberg Uncertainty Principle?
a) Energy is quantized
b) Electrons have a spin of 1/2
c) It is impossible to know both the position and momentum of a particle with arbitrary precision
simultaneously
d) The wave function must be normalized
Answer: c
Rationale: The Heisenberg Uncertainty Principle states ΔxΔp≥ℏ/2ΔxΔp≥ℏ/2. It is a fundamental limit on
the precision with which complementary variables can be known.
2. A wavefunction ψ(x)ψ(x) is said to be "normalized" if:
a) ∫ψ∗ψ dτ=1∫ψ∗ψdτ=1
b) ∫ψ∗ψ dτ=0∫ψ∗ψdτ=0
c) ψψ is continuous
d) ψψ is an eigenfunction of the Hamiltonian
Answer: a
Rationale: Normalization ensures the total probability of finding the particle somewhere in space is 1.
3. For the particle in a 1D box of length LL, the energy is proportional to:
a) nn
b) n2n2
,c) 1/n1/n
d) 1/n21/n2
Answer: b
Rationale: En=n2h28mL2En=8mL2n2h2. Therefore, energy increases quadratically with the quantum
number nn.
4. How many nodes (excluding boundaries) does the n=3n=3 wavefunction have for a particle in a 1D
box?
a) 0
b) 1
c) 2
d) 3
Answer: c
*Rationale: The number of nodes = n−1n−1. For n=3n=3, nodes = 2.*
5. The zero-point energy of the quantum harmonic oscillator is:
a) 0
b) hνhν
c) 12hν21hν
d) 32hν23hν
Answer: c
Rationale: Ev=(v+12)hνEv=(v+21)hν. At v=0v=0, E0=12hνE0=21hν. This is a consequence of the
uncertainty principle; the particle cannot be at rest at the bottom of the well.
6. The selection rule for a harmonic oscillator in infrared spectroscopy is:
a) Δv=±1Δv=±1
b) Δv=0Δv=0
c) Δv=±2Δv=±2
d) ΔJ=±1ΔJ=±1
Answer: a
Rationale: For a harmonic oscillator, the transition dipole moment requires Δv=±1Δv=±1. Anharmonicity
allows overtones (Δv=±2Δv=±2), but the fundamental harmonic rule is ±1±1.
, 7. The wavefunction for the hydrogen atom depends on which quantum numbers?
a) n,l,mln,l,ml
b) n,l,sn,l,s
c) n,ml,msn,ml,ms
d) l,ml,msl,ml,ms
Answer: a
Rationale: The spatial wavefunction (orbital) is defined by principal (nn), azimuthal (ll), and magnetic
(mlml) quantum numbers. Spin (msms) is added later for the full electron state.
8. Which of the following is a Hermitian operator?
a) ddxdxd
b) H^H^
c) x^x^
d) Both b and c
Answer: d
Rationale: Hermitian operators have real eigenvalues, corresponding to observable quantities. Both the
Hamiltonian (H^H^) and position (x^x^) are Hermitian. The momentum operator is also Hermitian,
but ddxdxd alone is anti-Hermitian unless multiplied by −iℏ−iℏ.
9. The expectation value ⟨x⟩⟨x⟩ for the n=1n=1 state of the particle in a box is:
a) 0
b) L/2L/2
c) LL
d) L/4L/4
Answer: b
Rationale: The probability distribution is symmetric about the center of the box. The average position is
the midpoint, L/2L/2.
10. What is the degeneracy of the n=3n=3 energy level for a particle in a 3D cubic box?
a) 1
b) 3
c) 6
d) 10
