QUANTUM MECHANICS SOLUTION EXAM QUESTIONS AND ANSWERS
2025/2026
1. A normalized wavefunction ψ(x)\psi(x)ψ(x) must satisfy:
A) ∫−∞∞∣ψ∣2dx=1\int_{-\infty}^{\infty}|\psi|^2 dx =
1∫−∞∞∣ψ∣2dx=1
B) ∫−∞∞ψ dx=1\int_{-\infty}^{\infty}\psi \, dx = 1∫−∞∞
ψdx=1
C) ψ(0)=1\psi(0)=1ψ(0)=1
D) ∣ψ(x)∣≤1|\psi(x)|\le 1∣ψ(x)∣≤1
Answer: A. Probability density integrates to 1 for
normalization.
2. The time-dependent Schrödinger equation (TDSE) for a
particle of mass mmm in potential VVV is:
A) iℏ∂tψ=−ℏ22m∂x2ψ+Vψi\hbar \partial_t\psi = -\frac{\
hbar^2}{2m}\partial_x^2\psi + V\psiiℏ∂tψ=−2mℏ2∂x2ψ+Vψ
B) −ℏ22m∂x2ψ=Eψ-\frac{\hbar^2}{2m}\partial_x^2\psi =
E\psi−2mℏ2∂x2ψ=Eψ
,QUANTUM MECHANICS SOLUTION EXAM QUESTIONS AND ANSWERS
2025/2026
C) ∂tψ=0\partial_t\psi = 0∂tψ=0
D) Hψ=0H\psi = 0Hψ=0
Answer: A. TDSE: iℏ∂∂tψ=H^ψi\hbar\frac{\partial}{\partial
t}\psi = \hat H\psiiℏ∂t∂ψ=H^ψ.
3. Stationary states have time dependence:
A) e−iEt/ℏe^{-iEt/\hbar}e−iEt/ℏ
B) cos(Et/ℏ)\cos(Et/\hbar)cos(Et/ℏ)
C) e+iEt/ℏe^{+iEt/\hbar}e+iEt/ℏ
D) No time dependence at all
Answer: A. Energy eigenstates pick up a phase e−iEt/ℏe^{-
iEt/\hbar}e−iEt/ℏ.
4. The momentum operator in position representation is:
A) −iℏddx-i\hbar \frac{d}{dx}−iℏdxd
B) iℏddxi\hbar \frac{d}{dx}iℏdxd
C) −ℏddx-\hbar \frac{d}{dx}−ℏdxd
D) ℏx\hbar xℏx
,QUANTUM MECHANICS SOLUTION EXAM QUESTIONS AND ANSWERS
2025/2026
Answer: A. p^=−iℏ∂x\hat p = -i\hbar \partial_xp^=−iℏ∂x
(in 1D).
5. Expectation value ⟨A^⟩\langle \hat A\rangle⟨A^⟩ equals:
A) ∫ψ∗A^ψ dx\int \psi^* \hat A \psi\, dx∫ψ∗A^ψdx
B) ∫A^ψ dx\int \hat A \psi\, dx∫A^ψdx
C) A^ψ\hat A \psiA^ψ
D) ∫∣ψ∣2dx\int |\psi|^2 dx∫∣ψ∣2dx
Answer: A. Definition of expectation value in wave
mechanics.
6. Two operators that commute [A^,B^]=0[\hat A,\hat
B]=0[A^,B^]=0 imply:
A) They share a common set of eigenfunctions (if non-
degenerate)
B) They cannot be measured simultaneously
C) They are identical
D) Their eigenvalues multiply to zero
, QUANTUM MECHANICS SOLUTION EXAM QUESTIONS AND ANSWERS
2025/2026
Answer: A. Commuting Hermitian operators have
simultaneous eigenstates (generically).
7. The Heisenberg uncertainty principle between xxx and ppp
is:
A) Δx Δp≥ℏ/2\Delta x\,\Delta p \ge \hbar/2ΔxΔp≥ℏ/2
B) Δx Δp=0\Delta x\,\Delta p = 0ΔxΔp=0
C) Δx+Δp≥ℏ\Delta x+\Delta p \ge \hbarΔx+Δp≥ℏ
D) Δx Δp≤ℏ/2\Delta x\,\Delta p \le \hbar/2ΔxΔp≤ℏ/2
Answer: A. Standard uncertainty relation.
8. A free particle plane wave ψ∝eikx\psi\propto e^{ikx}ψ∝eikx
has:
A) Well-defined momentum, ill-defined position
B) Well-defined position, ill-defined momentum
C) Both well-defined
D) Neither well-defined
2025/2026
1. A normalized wavefunction ψ(x)\psi(x)ψ(x) must satisfy:
A) ∫−∞∞∣ψ∣2dx=1\int_{-\infty}^{\infty}|\psi|^2 dx =
1∫−∞∞∣ψ∣2dx=1
B) ∫−∞∞ψ dx=1\int_{-\infty}^{\infty}\psi \, dx = 1∫−∞∞
ψdx=1
C) ψ(0)=1\psi(0)=1ψ(0)=1
D) ∣ψ(x)∣≤1|\psi(x)|\le 1∣ψ(x)∣≤1
Answer: A. Probability density integrates to 1 for
normalization.
2. The time-dependent Schrödinger equation (TDSE) for a
particle of mass mmm in potential VVV is:
A) iℏ∂tψ=−ℏ22m∂x2ψ+Vψi\hbar \partial_t\psi = -\frac{\
hbar^2}{2m}\partial_x^2\psi + V\psiiℏ∂tψ=−2mℏ2∂x2ψ+Vψ
B) −ℏ22m∂x2ψ=Eψ-\frac{\hbar^2}{2m}\partial_x^2\psi =
E\psi−2mℏ2∂x2ψ=Eψ
,QUANTUM MECHANICS SOLUTION EXAM QUESTIONS AND ANSWERS
2025/2026
C) ∂tψ=0\partial_t\psi = 0∂tψ=0
D) Hψ=0H\psi = 0Hψ=0
Answer: A. TDSE: iℏ∂∂tψ=H^ψi\hbar\frac{\partial}{\partial
t}\psi = \hat H\psiiℏ∂t∂ψ=H^ψ.
3. Stationary states have time dependence:
A) e−iEt/ℏe^{-iEt/\hbar}e−iEt/ℏ
B) cos(Et/ℏ)\cos(Et/\hbar)cos(Et/ℏ)
C) e+iEt/ℏe^{+iEt/\hbar}e+iEt/ℏ
D) No time dependence at all
Answer: A. Energy eigenstates pick up a phase e−iEt/ℏe^{-
iEt/\hbar}e−iEt/ℏ.
4. The momentum operator in position representation is:
A) −iℏddx-i\hbar \frac{d}{dx}−iℏdxd
B) iℏddxi\hbar \frac{d}{dx}iℏdxd
C) −ℏddx-\hbar \frac{d}{dx}−ℏdxd
D) ℏx\hbar xℏx
,QUANTUM MECHANICS SOLUTION EXAM QUESTIONS AND ANSWERS
2025/2026
Answer: A. p^=−iℏ∂x\hat p = -i\hbar \partial_xp^=−iℏ∂x
(in 1D).
5. Expectation value ⟨A^⟩\langle \hat A\rangle⟨A^⟩ equals:
A) ∫ψ∗A^ψ dx\int \psi^* \hat A \psi\, dx∫ψ∗A^ψdx
B) ∫A^ψ dx\int \hat A \psi\, dx∫A^ψdx
C) A^ψ\hat A \psiA^ψ
D) ∫∣ψ∣2dx\int |\psi|^2 dx∫∣ψ∣2dx
Answer: A. Definition of expectation value in wave
mechanics.
6. Two operators that commute [A^,B^]=0[\hat A,\hat
B]=0[A^,B^]=0 imply:
A) They share a common set of eigenfunctions (if non-
degenerate)
B) They cannot be measured simultaneously
C) They are identical
D) Their eigenvalues multiply to zero
, QUANTUM MECHANICS SOLUTION EXAM QUESTIONS AND ANSWERS
2025/2026
Answer: A. Commuting Hermitian operators have
simultaneous eigenstates (generically).
7. The Heisenberg uncertainty principle between xxx and ppp
is:
A) Δx Δp≥ℏ/2\Delta x\,\Delta p \ge \hbar/2ΔxΔp≥ℏ/2
B) Δx Δp=0\Delta x\,\Delta p = 0ΔxΔp=0
C) Δx+Δp≥ℏ\Delta x+\Delta p \ge \hbarΔx+Δp≥ℏ
D) Δx Δp≤ℏ/2\Delta x\,\Delta p \le \hbar/2ΔxΔp≤ℏ/2
Answer: A. Standard uncertainty relation.
8. A free particle plane wave ψ∝eikx\psi\propto e^{ikx}ψ∝eikx
has:
A) Well-defined momentum, ill-defined position
B) Well-defined position, ill-defined momentum
C) Both well-defined
D) Neither well-defined
