Chapter 20 Quiz: The WKB Approximation

Instructions: This quiz covers the core concepts from Chapter 20. For multiple choice, select the single best answer. For true/false, provide a brief justification (1-2 sentences). For short answer, aim for 3-5 sentences. For applied scenarios, show your work.

Multiple Choice (10 questions)

Q1. The WKB approximation is valid when:

(a) The potential energy is much smaller than the kinetic energy (b) The de Broglie wavelength changes slowly over one wavelength (c) The particle is in the classically forbidden region (d) The quantum number $n$ is exactly zero

Q2. The WKB wavefunction in the classically allowed region has amplitude proportional to:

(a) $\sqrt{p(x)}$ (b) $1/\sqrt{p(x)}$ (c) $p(x)$ (d) $1/p(x)$

Q3. The WKB approximation breaks down at:

(a) The center of the classically allowed region (b) Points of maximum kinetic energy (c) Classical turning points where $E = V(x)$ (d) Points where the potential has a local minimum

Q4. Each classical turning point contributes a phase shift of:

(a) $0$ (b) $\pi/4$ (c) $\pi/2$ (d) $\pi$

Q5. The WKB quantization condition $\oint p\,dx = (n + \frac{1}{2})h$ gives exact energy levels for:

(a) The infinite square well (b) The hydrogen atom only (c) The quantum harmonic oscillator (d) Any potential with two turning points

Q6. The Gamow exponent for tunneling through a barrier is defined as $\gamma = \hbar^{-1}\int_{x_1}^{x_2}\sqrt{2m[V(x) - E]}\,dx$. Doubling the mass of the tunneling particle while keeping everything else the same will:

(a) Double $\gamma$ (b) Multiply $\gamma$ by $\sqrt{2}$ (c) Halve $\gamma$ (d) Have no effect on $\gamma$

Q7. In Gamow's alpha decay model, the extreme sensitivity of the half-life to the alpha particle energy arises because:

(a) The nuclear force strength depends sensitively on energy (b) The alpha particle speed changes dramatically with energy (c) The Gamow exponent appears in the argument of an exponential (d) The nuclear radius depends strongly on the alpha energy

Q8. The WKB tunneling probability $T \approx e^{-2\gamma}$ for a given barrier is largest when:

(a) The particle has the highest possible mass (b) The barrier is at its widest (c) The particle energy is closest to the barrier height (d) The potential varies most rapidly

Q9. The Airy functions $\text{Ai}(z)$ and $\text{Bi}(z)$ are solutions to:

(a) $d^2\psi/dz^2 = z^2\psi$ (b) $d^2\psi/dz^2 = z\psi$ (c) $d^2\psi/dz^2 = -z\psi$ (d) $d^2\psi/dz^2 + z^2\psi = 0$

Q10. The Geiger-Nuttall law relates alpha decay half-life $t_{1/2}$ to alpha energy $E_\alpha$ as:

(a) $\log t_{1/2} \propto E_\alpha$ (b) $\log t_{1/2} \propto 1/E_\alpha$ (c) $\log t_{1/2} \propto 1/\sqrt{E_\alpha}$ (d) $\log t_{1/2} \propto \sqrt{E_\alpha}$

True/False (4 questions)

Q11. True or False: The WKB approximation requires that the potential energy be much smaller than the total energy.

Justify your answer in 1-2 sentences.

Q12. True or False: The WKB tunneling formula $T \approx e^{-2\gamma}$ correctly captures both the exponential dependence and the algebraic prefactor of the exact transmission coefficient.

Justify your answer in 1-2 sentences.

Q13. True or False: The Bohr-Sommerfeld quantization condition (without the $+\frac{1}{2}$ correction) predicts zero-point energy for the harmonic oscillator.

Justify your answer in 1-2 sentences.

Q14. True or False: Gamow's WKB model of alpha decay correctly predicts half-lives across more than 20 orders of magnitude.

Justify your answer in 1-2 sentences.

Short Answer (4 questions)

Q15. Explain the physical interpretation of the WKB amplitude factor $|\psi(x)|^2 \propto 1/p(x)$. How does this relate to classical mechanics?

Q16. Why does the WKB connection formula involve Airy functions rather than simpler functions? What approximation of the potential near the turning point leads to the Airy equation?

Q17. Describe two physical systems (other than alpha decay) where WKB tunneling plays an important role. For each, identify the tunneling particle, the nature of the barrier, and the observable consequence of tunneling.

Q18. Explain why the WKB approximation improves with increasing quantum number $n$. What is the physical reason, and how does this connect to the correspondence principle?

Applied Scenarios (2 questions)

Q19. A proton ($m = 1.67 \times 10^{-27}$ kg) with kinetic energy $E = 1.0$ MeV approaches a Coulomb barrier of a light nucleus with $Z = 6$ (carbon). The nuclear radius is $R = 3.0$ fm.

(a) Calculate the Coulomb barrier height at the nuclear surface. Use $ke^2 = 1.44$ MeV$\cdot$fm.

(b) Find the outer turning point $r_2$.

(c) Estimate the Gamow exponent $\gamma$ and the tunneling probability. (You may approximate the integral or evaluate it numerically.)

(d) If the proton bounces inside the nucleus with frequency $\sim 10^{22}$ Hz, estimate how many attempts per second it makes and the overall reaction rate per proton.

Q20. You have a double-well potential with two identical wells separated by a barrier. A particle in the left well has approximately the energy of the ground state of a single well, $E_0$.

(a) If the WKB tunneling integral through the barrier gives $\gamma = 5$, estimate the energy splitting $\Delta E$ between the symmetric and antisymmetric states. Express your answer in terms of $\hbar\omega$ (the oscillation frequency in one well).

(b) What is the tunneling period — the time for the particle to oscillate from the left well to the right well and back?

(c) If the barrier height is doubled (keeping width constant), how does the tunneling period change? Give a quantitative estimate of the ratio of new to old periods.

Answer Key

Q1: (b) — The WKB condition is $|d\lambda/dx| \ll 2\pi$, meaning the de Broglie wavelength changes slowly.

Q2: (b) — The WKB amplitude is $1/\sqrt{p(x)}$, reflecting the classical dwell-time distribution.

Q3: (c) — At turning points, $p \to 0$, so $1/\sqrt{p}$ diverges and the validity condition fails.

Q4: (b) — The Airy function matching gives a $\pi/4$ phase shift at each turning point.

Q5: (c) — The harmonic oscillator is the unique potential for which WKB is exact at all $n$.

Q6: (b) — $\gamma \propto \sqrt{m}$, so doubling $m$ multiplies $\gamma$ by $\sqrt{2}$.

Q7: (c) — The energy appears in the exponent $\gamma \propto 1/\sqrt{E}$, and $T = e^{-2\gamma}$ amplifies small changes exponentially.

Q8: (c) — When $E$ is close to the barrier height, $V(x) - E$ is small throughout the forbidden region, minimizing $\gamma$.

Q9: (b) — The Airy equation is $d^2\psi/dz^2 = z\psi$, arising from linearizing the potential near a turning point.

Q10: (c) — The Gamow exponent is $\gamma \propto Z/\sqrt{E_\alpha}$, giving $\log t_{1/2} \propto 1/\sqrt{E_\alpha}$.

Q11: False. The WKB approximation requires that the potential vary slowly on the scale of the de Broglie wavelength. The potential can be large — what matters is that it changes gradually.

Q12: False. The WKB formula $T \approx e^{-2\gamma}$ captures only the exponential (dominant) factor. The algebraic prefactor, which depends on the precise shape of the barrier near the turning points, is not captured.

Q13: False. The original Bohr rule $\oint p\,dx = nh$ gives $E_n = n\hbar\omega$ for the harmonic oscillator, so $E_0 = 0$ (no zero-point energy). The WKB correction $n \to n + 1/2$ is needed to obtain the correct zero-point energy $\frac{1}{2}\hbar\omega$.

Q14: True. The Gamow model predicts half-lives from microseconds ($^{212}$Po) to billions of years ($^{238}$U) with agreement to within a factor of about 2 — across more than 23 orders of magnitude.

Q15: The probability density $|\psi|^2 \propto 1/p(x)$ means the particle is most likely found where it moves slowly (low momentum) and least likely where it moves fast. This matches the classical probability distribution: a classical particle spends time $dt = dx/v$ at position $x$, so the probability of finding it near $x$ is proportional to $1/v \propto 1/p$.

Q16: Near a turning point, the potential is approximately linear: $V(x) \approx E + V'(x_0)(x - x_0)$. Substituting this linear approximation into the Schrödinger equation and rescaling gives the Airy equation $d^2\psi/dz^2 = z\psi$, whose solutions (Airy functions) naturally interpolate between oscillatory and exponential behavior. Simpler functions like sines and exponentials cannot smoothly transition between the two regimes.

Q17: (1) Scanning tunneling microscope: electrons tunnel through a vacuum gap ($\sim$1 nm) between a metallic tip and a surface, producing a current exponentially sensitive to distance. (2) Nuclear fusion in stars: protons tunnel through the Coulomb barrier ($\sim$1 MeV) at thermal energies ($\sim$1 keV), enabling hydrogen burning. Other valid examples: Josephson junctions, field emission, proton transfer in biological systems.

Q18: As $n$ increases, the de Broglie wavelength $\lambda = h/p$ decreases (because the energy and thus $p$ increase). For large $n$, $\lambda$ becomes much shorter than the scale over which $V(x)$ varies, satisfying the WKB validity condition $|d\lambda/dx| \ll 2\pi$. This is the correspondence principle: at high quantum numbers, quantum mechanics approaches classical behavior, which is precisely what WKB captures.

Q19: (a) $V(R) = kZe^2/R = (1.44)(6)/(3.0) = 2.88$ MeV. (Note: the proton has charge $e$, so the Coulomb potential is $kZe^2/r$, not $2kZe^2/r$ as for alpha decay.) (b) $r_2 = kZe^2/E = (1.44)(6)/(1.0) = 8.64$ fm. (c) $\gamma \approx (\sqrt{2m_p}/\hbar)\int_R^{r_2}\sqrt{kZe^2/r - E}\,dr$. Numerical evaluation gives $\gamma \approx 4.9$, so $T \approx e^{-9.8} \approx 5.5 \times 10^{-5}$. (d) Rate $\approx 10^{22} \times 5.5 \times 10^{-5} \approx 5.5 \times 10^{17}$ s$^{-1}$. (This is the rate per proton already at the nuclear surface; astrophysical rates also include the thermal distribution factor.)

Q20: (a) $\Delta E \approx (\hbar\omega/\pi)e^{-\gamma} = (\hbar\omega/\pi)e^{-5} \approx 2.1 \times 10^{-3}\hbar\omega$. (b) Tunneling period $\tau = h/\Delta E = 2\pi\hbar/\Delta E \approx \pi^2/(\omega \cdot e^{-5}/\pi) \approx (2\pi^2/\omega)e^5 \approx 2930/\omega$. (c) Doubling barrier height increases $\gamma$ by $\sqrt{2}$ (since $\gamma \propto \sqrt{V_0 - E}$ for a rectangular barrier). New $\gamma \approx 5\sqrt{2} \approx 7.07$. Ratio of periods: $e^{\gamma_{\text{new}}}/e^{\gamma_{\text{old}}} = e^{7.07 - 5.0} = e^{2.07} \approx 7.9$. The tunneling period increases by about a factor of 8.