Chapter 3 Quiz

Instructions: Choose the best answer for each question. Each question has exactly one correct answer.

Q1. For the infinite square well of width $a$, the ground state energy is $E_1 = \pi^2\hbar^2/(2ma^2)$. If the width is doubled to $2a$, the new ground state energy is:

(A) $4E_1$ (B) $2E_1$ (C) $E_1$ (D) $E_1/2$ (E) $E_1/4$

Q2. In the infinite square well, the energy of the $n$-th level is proportional to:

(A) $n$ (B) $n^2$ (C) $n^3$ (D) $\sqrt{n}$ (E) $1/n^2$

Q3. The wavefunctions of the infinite square well form a complete orthonormal set. "Completeness" means:

(A) Each wavefunction satisfies the Schrödinger equation. (B) The wavefunctions are normalized to unity. (C) Any function satisfying the boundary conditions can be expanded as a sum of these wavefunctions. (D) There are infinitely many wavefunctions. (E) The wavefunctions have no nodes inside the well.

Q4. For the ground state of the infinite square well, $\langle x \rangle = a/2$. This means:

(A) The particle is located at $x = a/2$. (B) The probability density has its maximum at $x = a/2$. (C) The average over many measurements of position will give $a/2$. (D) Both (B) and (C) are correct. (E) All of (A), (B), and (C) are correct.

Q5. A free particle has a continuous (not discrete) energy spectrum because:

(A) The free particle has no potential energy. (B) The free particle is not confined, so no boundary conditions restrict the allowed wavelengths. (C) The Schrödinger equation has different mathematical structure for free particles. (D) Free particles are always relativistic. (E) The plane wave solutions are not normalizable.

Q6. A plane wave $\psi = Ae^{ikx}$ cannot represent a physical particle because:

(A) It has zero kinetic energy. (B) It is not a solution to the Schrödinger equation. (C) It is not normalizable — $\int_{-\infty}^{\infty}|\psi|^2\,dx$ diverges. (D) It violates the uncertainty principle. (E) It is complex-valued.

Q7. For a free non-relativistic particle, the group velocity of a wave packet is:

(A) Half the phase velocity. (B) Equal to the phase velocity. (C) Twice the phase velocity. (D) Equal to the speed of light. (E) Zero.

Q8. A Gaussian wave packet for a free particle spreads over time. This spreading occurs because:

(A) The particle loses energy. (B) Different momentum components travel at different speeds (dispersion). (C) The uncertainty principle forces the position uncertainty to grow. (D) External forces act on the particle. (E) The normalization of the wavefunction changes over time.

Q9. The finite square well always has at least one bound state (in one dimension). For the symmetric well of depth $V_0$ and half-width $a$, this is because:

(A) The ground state energy is always negative. (B) The even-parity transcendental equation always has at least one solution. (C) A deeper well has more bound states. (D) The wavefunctions must be continuous. (E) The infinite well limit guarantees it.

Q10. Compared to the infinite square well, the bound state energies of a finite square well of the same width are:

(A) Higher, because the finite well is less confining. (B) Lower, because the wavefunctions extend into the classically forbidden region. (C) The same, because both wells have the same width. (D) Higher for even states, lower for odd states. (E) Impossible to determine without specific parameters.

Q11. In the finite square well, the wavefunctions "leak" into the classically forbidden region. This means:

(A) The particle has negative kinetic energy in that region. (B) There is a nonzero probability of finding the particle outside the well. (C) The boundary conditions at the well edges are violated. (D) Both (A) and (B). (E) The wavefunction is not normalizable.

Q12. For a step potential with $E > V_0$, classical mechanics predicts $R = 0$ (no reflection). Quantum mechanics predicts:

(A) $R = 0$ as well. (B) $R = 1$ (total reflection). (C) $0 < R < 1$ (partial reflection). (D) $R > 1$ (amplified reflection). (E) $R$ is undefined for $E > V_0$.

Q13. In the expression for the transmission coefficient $T$ at a step potential, the factor $k_2/k_1$ appears because:

(A) The amplitudes must be equal on both sides. (B) The probability current depends on both the amplitude and the velocity. (C) The wavefunctions are not normalized. (D) The potential is discontinuous at the step. (E) This is a relativistic correction.

Q14. For a rectangular barrier with $E < V_0$, the tunneling probability in the thick-barrier limit is approximately:

(A) $T \propto e^{-\kappa d}$ (B) $T \propto e^{-2\kappa d}$ (C) $T \propto e^{-\kappa^2 d}$ (D) $T \propto 1/d$ (E) $T \propto 1/d^2$

Q15. Which of the following most strongly affects the tunneling probability through a rectangular barrier?

(A) The mass of the particle. (B) The exact shape of the barrier (rectangular vs. triangular). (C) The de Broglie wavelength of the incident wave. (D) The phase of the incident wave. (E) The temperature of the barrier.

Q16. The exponential sensitivity of tunneling to particle mass explains why:

(A) Electrons tunnel much more readily than protons. (B) All particles tunnel equally well. (C) Heavier particles tunnel faster. (D) Tunneling only occurs for photons. (E) Tunneling depends only on barrier width, not mass.

Q17. Quantum tunneling is responsible for all of the following EXCEPT:

(A) Alpha decay of radioactive nuclei. (B) The operation of the scanning tunneling microscope. (C) Data storage in flash memory. (D) The photoelectric effect. (E) Nuclear fusion in the Sun's core.

Q18. In the finite difference method for solving the Schrödinger equation, the second derivative $d^2\psi/dx^2$ is approximated by:

(A) $(\psi_{i+1} - \psi_{i-1})/(2\Delta x)$ (B) $(\psi_{i+1} - 2\psi_i + \psi_{i-1})/(\Delta x)^2$ (C) $(\psi_{i+1} - \psi_i)/(\Delta x)^2$ (D) $(\psi_{i+1} + \psi_{i-1})/(\Delta x)^2$ (E) $(\psi_{i+1} - \psi_{i-1})/(\Delta x)^2$

Q19. The finite difference method converts the TISE into:

(A) A set of coupled first-order ODEs. (B) A matrix eigenvalue problem. (C) An integral equation. (D) A partial differential equation. (E) A single algebraic equation.

Q20. A particle confined to a 1D box has zero-point energy $E_1 > 0$. This is a consequence of:

(A) The particle having thermal energy. (B) The uncertainty principle — confinement in position requires nonzero momentum spread. (C) Relativistic effects. (D) The particle interacting with the walls. (E) A computational artifact with no physical meaning.

Answer Key

Q Answer Explanation
1 (E) $E_1 \propto 1/a^2$. Doubling $a$ gives $E_1/4$.
2 (B) $E_n = n^2\pi^2\hbar^2/(2ma^2)$.
3 (C) Completeness = any function in the space can be expanded in the basis.
4 (D) $\langle x \rangle$ is the average of many measurements; for $n=1$, the density $\sin^2(\pi x/a)$ peaks at $a/2$.
5 (B) No confinement → no boundary conditions → no quantization.
6 (C) $\int|e^{ikx}|^2\,dx = \int 1\,dx = \infty$.
7 (C) $v_g = d\omega/dk = \hbar k/m = 2(\hbar k/2m) = 2v_\phi$.
8 (B) Different $k$-components have different $\omega(k)$, causing dispersion.
9 (B) The graphical solution always has one intersection for even-parity states.
10 (B) Wavefunction leakage reduces effective confinement, lowering energies.
11 (D) The particle has a classically impossible negative KE there, and the probability is nonzero.
12 (C) Quantum partial reflection occurs for any step, even when $E > V_0$.
13 (B) Probability current $j = (\hbar k/m)|A|^2$ — different $k$ means different current for same amplitude.
14 (B) $T \approx \text{prefactor}\times e^{-2\kappa d}$. The exponent is $-2\kappa d$.
15 (A) $\kappa = \sqrt{2m(V_0-E)}/\hbar$; mass enters the exponent, giving exponential sensitivity.
16 (A) $\kappa \propto \sqrt{m}$, so heavier particles have much smaller tunneling probability.
17 (D) The photoelectric effect involves photon absorption, not tunneling.
18 (B) Standard central difference formula for the second derivative.
19 (B) The discretized TISE becomes $\mathbf{H}\boldsymbol{\psi} = E\boldsymbol{\psi}$.
20 (B) Confinement ($\Delta x \lesssim a$) → $\Delta p \ge \hbar/2a$ → minimum KE.