Chapter 20 Exercises: The WKB Approximation
Part A: Conceptual Questions (7 problems)
These questions test your understanding of the core ideas. No calculations required unless stated.
A.1 Explain in your own words why the WKB approximation breaks down at classical turning points. What is physically special about a turning point, and why does the mathematical form of the WKB wavefunction ($\propto 1/\sqrt{p}$) become problematic there?
A.2 The WKB quantization condition is $\oint p\,dx = (n + \frac{1}{2})h$. The original Bohr quantization rule was $\oint p\,dx = nh$. Where does the extra $1/2$ come from physically, and what does it predict that Bohr's rule does not?
A.3 A friend claims: "The WKB approximation says that quantum mechanics reduces to classical mechanics when $\hbar$ is small." Is this correct? If not, explain what the WKB approximation actually says about the relationship between quantum and classical mechanics. Give an example of a purely quantum effect that the WKB method captures.
A.4 The WKB tunneling probability is $T \approx e^{-2\gamma}$. Explain qualitatively (without detailed math) why the tunneling probability depends on: (a) The width of the barrier (b) The height of the barrier above the particle energy (c) The mass of the particle For each, state whether increasing the quantity increases or decreases $T$ and explain why.
A.5 Why is the WKB approximation exact for the harmonic oscillator but poor for the ground state of the infinite square well? What property of the potential makes the difference?
A.6 In Gamow's alpha decay model, the half-life of $^{238}$U is approximately $4.5 \times 10^9$ years, while that of $^{212}$Po is approximately $0.3 \times 10^{-6}$ seconds — a ratio of $\sim 10^{23}$. Yet the alpha particle energies differ by only about a factor of 2 (4.27 MeV vs. 8.95 MeV). Explain qualitatively how such a small change in energy produces such an enormous change in lifetime.
A.7 The connection formulas contain a phase shift of $\pi/4$ at each turning point. What would happen to the WKB quantization condition if this phase shift were $\pi/2$ instead? What if it were zero? Would either of these give correct results for the harmonic oscillator?
Part B: Applied Problems (12 problems)
These problems require direct application of the chapter's key equations and techniques.
B.1: WKB Validity Check
For each of the following potentials, determine the regions where the WKB approximation is valid for a particle of energy $E$:
(a) $V(x) = V_0 e^{-x^2/a^2}$ (Gaussian barrier), $E < V_0$
(b) $V(x) = \frac{1}{2}m\omega^2 x^2$ (harmonic oscillator)
(c) $V(x) = -V_0/\cosh^2(x/a)$ (Pöschl-Teller potential), for a bound state
In each case, identify the classical turning points and mark the region(s) where $|d\lambda/dx| \ll 2\pi$ is violated.
B.2: WKB Wavefunctions for Linear Potential
A particle of mass $m$ moves in a linear potential $V(x) = mgx$ for $x > 0$, with an infinite wall at $x = 0$.
(a) Find the classical turning point $x_0(E)$ as a function of energy $E$.
(b) Write down the WKB wavefunction in the classically allowed region $0 < x < x_0$ and in the forbidden region $x > x_0$.
(c) Apply the boundary condition $\psi(0) = 0$ (infinite wall) and the connection formula at $x_0$ to derive the WKB quantization condition for this system.
(d) Show that the energy levels are approximately $E_n \approx \left(\frac{3\pi mg\hbar}{2}\right)^{2/3}\left(n + \frac{3}{4}\right)^{2/3}$, $n = 0, 1, 2, \ldots$
(e) Compare your WKB energy levels for $n = 0, 1, 2$ with the exact results from the zeros of the Airy function. (Hint: the exact condition is $\text{Ai}(-z_n) = 0$, with $z_n = E_n/(mg\hbar^2/(2m))^{1/3}$. The first three zeros of Ai are at $z_0 = 2.338$, $z_1 = 4.088$, $z_2 = 5.521$.)
B.3: Bohr-Sommerfeld for the Morse Potential
The Morse potential models a diatomic molecule:
$$V(r) = D_e\left(1 - e^{-\beta(r - r_e)}\right)^2$$
where $D_e$ is the dissociation energy, $\beta$ is the range parameter, and $r_e$ is the equilibrium distance.
(a) Find the classical turning points for a bound state of energy $E < D_e$.
(b) The Bohr-Sommerfeld integral for the Morse potential can be evaluated analytically. Show that the WKB energy levels are:
$$E_n = \hbar\omega_e\left(n + \frac{1}{2}\right) - \frac{[\hbar\omega_e(n + \frac{1}{2})]^2}{4D_e}$$
where $\omega_e = \beta\sqrt{2D_e/m}$ is the harmonic frequency.
(c) Compare this with the exact Morse oscillator eigenvalues (same formula!). Why is WKB exact for the Morse potential?
B.4: Tunneling Through a Triangular Barrier
In field emission, electrons tunnel through a triangular barrier:
$$V(x) = \begin{cases} V_0 - eFx & 0 < x < V_0/(eF) \\ 0 & \text{otherwise} \end{cases}$$
where $F$ is the applied electric field strength and $V_0$ is the work function.
(a) Find the classical turning points for an electron at the Fermi level ($E = 0$).
(b) Compute the Gamow exponent $\gamma$ and show that the WKB tunneling probability is:
$$T \approx \exp\!\left(-\frac{4\sqrt{2m}}{3e\hbar F}V_0^{3/2}\right)$$
(c) Evaluate $T$ for aluminum ($V_0 = 4.3$ eV) in a field of $F = 5 \times 10^9$ V/m. Is this field experimentally achievable?
(d) Show that the Fowler-Nordheim law $J \propto F^2 \exp(-b/F)$ follows from this result, where $J$ is the emission current density.
B.5: WKB for the Quartic Oscillator
Consider the pure quartic potential $V(x) = \lambda x^4$.
(a) Find the classical turning points for a state of energy $E$.
(b) Evaluate the Bohr-Sommerfeld integral $\int_{-x_0}^{x_0}\sqrt{2m(E - \lambda x^4)}\,dx$. (Hint: substitute $u = x/x_0$ and use $\int_0^1\sqrt{1-u^4}\,du = \Gamma(5/4)\Gamma(3/2)/\Gamma(11/4) \approx 0.8740$.)
(c) Show that the WKB energy levels scale as $E_n \propto n^{4/3}$.
(d) Compute the first three energy levels numerically for $\lambda = 1$, $m = 1$, $\hbar = 1$ and compare with the exact numerical eigenvalues (obtainable from a finite-difference solver or your toolkit from Chapter 3): $E_0 = 0.6680$, $E_1 = 2.394$, $E_2 = 4.697$.
B.6: Alpha Decay of Polonium-212
$^{212}$Po emits an alpha particle with kinetic energy $E_\alpha = 8.95$ MeV, leaving $^{208}$Pb ($Z = 82$).
(a) Calculate the nuclear radius using $R = r_0 A^{1/3}$ with $r_0 = 1.2$ fm and $A = 212$.
(b) Find the Coulomb barrier height at the nuclear surface.
(c) Find the outer turning point $r_2$.
(d) Compute the Gamow exponent $2\gamma$. (Use $ke^2 = 1.44$ MeV$\cdot$fm.)
(e) Estimate the half-life and compare with the experimental value of $0.30\,\mu$s.
B.7: Tunneling in a Magnetic Field
A charged particle of mass $m$ and charge $q$ in a uniform magnetic field $B$ along $z$ has an effective 1D potential for motion in the $x$-direction:
$$V_{\text{eff}}(x) = \frac{1}{2}m\omega_c^2(x - x_0)^2$$
where $\omega_c = qB/m$ is the cyclotron frequency and $x_0$ depends on the canonical momentum.
(a) Using the WKB quantization condition, recover the Landau levels $E_n = (n + \frac{1}{2})\hbar\omega_c$.
(b) Is this result exact or approximate? Explain by connecting to the harmonic oscillator result from Section 20.5.3.
B.8: Double-Well Tunneling and Energy Splitting
Consider a symmetric double-well potential with minima at $x = \pm a$ and a barrier of height $V_0$ between them. The WKB tunneling integral through the central barrier is $\gamma$.
(a) Argue that the two lowest energy levels form a doublet with splitting:
$$\Delta E = E_1 - E_0 \approx \frac{\hbar\omega}{\pi}\, e^{-\gamma}$$
where $\omega$ is the angular frequency of oscillation in one well.
(b) For the ammonia molecule (NH$_3$), the nitrogen atom tunnels through a barrier of height $\approx 0.25$ eV with an effective mass $\approx 2.4 \times 10^{-26}$ kg and barrier width $\approx 0.38$ \AA. Estimate $\gamma$ and the tunneling frequency $\Delta E/h$. Compare with the experimental microwave frequency of 23.87 GHz.
B.9: WKB Phase Shifts for Scattering
A particle of mass $m$ and energy $E$ scatters off a potential $V(r) = V_0 e^{-r/a}$ (Yukawa-type). For the $l = 0$ partial wave:
(a) Find the classical turning point $r_0$ (if $E < V_0$).
(b) Write the WKB expression for the $s$-wave phase shift:
$$\delta_0 \approx \int_{r_0}^{\infty}\left[\sqrt{k^2 - \frac{2mV(r)}{\hbar^2}} - k\right]dr - kr_0 + \frac{\pi}{4}$$
where $k = \sqrt{2mE}/\hbar$.
(c) Evaluate this integral numerically for $V_0 = 10E$, $a = 1/k$ and compare with the exact numerical phase shift.
B.10: Proton Tunneling in DNA
Proton transfer between base pairs in DNA involves tunneling through a barrier. Model the barrier as a symmetric double well with barrier height $V_0 = 0.4$ eV, barrier width $d = 0.7$ \AA, and proton mass $m_p = 1.67 \times 10^{-27}$ kg.
(a) Estimate the Gamow exponent for a rectangular barrier approximation.
(b) Compute the tunneling probability $T$.
(c) If the proton attempts to cross the barrier with frequency $\nu \approx 10^{13}$ Hz (typical O-H stretching frequency), what is the tunneling rate in events per second?
(d) Compare this rate with the rate of DNA replication ($\sim 1000$ bases/second). Does proton tunneling contribute significantly to spontaneous mutation rates?
B.11: WKB Quantization of the Coulomb Potential
Apply the Bohr-Sommerfeld quantization condition to the radial equation for hydrogen (with the substitution $u(r) = rR(r)$ and the effective potential $V_{\text{eff}}(r) = -ke^2/r + l(l+1)\hbar^2/(2mr^2)$).
(a) Show that the quantization condition gives:
$$E_n = -\frac{mk^2e^4}{2\hbar^2(n_r + l + 1)^2}$$
where $n_r = 0, 1, 2, \ldots$ is the radial quantum number. Define $n = n_r + l + 1$.
(b) Show this reproduces the exact hydrogen energy levels $E_n = -13.6\,\text{eV}/n^2$.
(c) The WKB result is exact for the Coulomb potential (as for the harmonic oscillator). What symmetry of the Coulomb problem ensures this?
B.12: Tunneling Time
The "tunneling time" problem remains controversial, but a simple WKB estimate can be obtained.
(a) For a rectangular barrier of width $a$ and height $V_0$, calculate the group delay (phase time):
$$\tau_{\text{phase}} = \hbar\frac{d\phi}{dE}$$
where $\phi$ is the transmission phase.
(b) In the thick-barrier (opaque) limit $\kappa a \gg 1$, show that $\tau_{\text{phase}}$ becomes independent of barrier width — the Hartman effect. Is this superluminal tunneling?
(c) Explain why the Hartman effect does not violate special relativity. (Hint: consider what happens to the amplitude of the transmitted wave in the thick-barrier limit.)
Part C: Advanced Problems (9 problems)
These problems require synthesis of multiple concepts and extended calculation.
C.1: Higher-Order WKB Corrections
The WKB expansion to the next order gives a correction to the quantization condition:
$$\oint p\,dx = \left(n + \frac{1}{2}\right)h - \frac{\hbar^2}{48}\oint\frac{1}{p}\frac{d^2V/dx^2}{p^2/(2m)}\,dx + \cdots$$
(a) Evaluate the second-order correction for the harmonic oscillator and show it vanishes.
(b) Evaluate the second-order correction for the infinite square well and show it partially corrects the ground-state error noted in Section 20.5.4.
(c) For the Morse potential, does the second-order correction vanish? Interpret your result.
C.2: WKB and the Path Integral
In Feynman's path integral formulation (previewed here, developed in Chapter 31), the propagator is:
$$K(x_b, t; x_a, 0) = \int \mathcal{D}[x(t)]\, \exp\!\left(\frac{i}{\hbar}S[x(t)]\right)$$
where $S$ is the classical action.
(a) Apply the stationary-phase approximation to the path integral (expand $S$ to second order about the classical path). Show that the result is the WKB wavefunction.
(b) What is the physical meaning of the stationary-phase condition $\delta S = 0$? How does this connect to the correspondence principle?
(c) Show that the WKB amplitude factor $1/\sqrt{p(x)}$ arises from the Gaussian integral over fluctuations about the classical path.
C.3: Gamow-Siegert Resonances
Metastable states (resonances) can be treated in WKB by allowing complex energies $E = E_R - i\Gamma/2$.
(a) Show that for a potential with a barrier (e.g., a well surrounded by a finite barrier), the WKB quantization condition with complex energy gives:
$$\oint p(x; E)\,dx = \left(n + \frac{1}{2}\right)h$$
where $p$ now involves the complex energy.
(b) To first order in $\Gamma$, show that the real part of this condition gives the resonance energy $E_R$ and the imaginary part gives the width:
$$\Gamma = \frac{\hbar\omega}{2\pi}e^{-2\gamma}$$
where $\omega$ is the oscillation frequency in the well and $\gamma$ is the barrier penetration integral.
(c) Apply this to estimate the width of the quasi-bound state in a potential $V(r) = 100\hbar^2/(2mr^2) - 50\hbar^2/(2mr)$ (in appropriate units).
C.4: WKB in Higher Dimensions — Radial Equation
For a central potential $V(r)$ in three dimensions, the radial Schrödinger equation for $u(r) = rR(r)$ is:
$$-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + V_{\text{eff}}(r)\, u = Eu, \quad V_{\text{eff}}(r) = V(r) + \frac{l(l+1)\hbar^2}{2mr^2}$$
(a) Apply the WKB approximation to this equation. What modification, if any, is needed compared to the one-dimensional case?
(b) The Langer modification replaces $l(l+1)$ with $(l + \frac{1}{2})^2$ in the centrifugal barrier. Show that this modification improves the WKB results for small $l$ and explain its origin from the singular behavior at $r = 0$.
(c) Apply the Langer-modified WKB to the hydrogen atom and show that you recover the exact energy levels.
C.5: Uniform WKB and Langer's Modification
The "uniform" WKB method avoids the connection formula entirely by writing the solution as:
$$\psi(x) = \alpha(x)\text{Ai}[\zeta(x)] + \beta(x)\text{Bi}[\zeta(x)]$$
where $\zeta(x)$ is a mapping function.
(a) Show that demanding $\psi$ satisfy the Schrödinger equation exactly (to leading order) gives:
$$\left(\frac{d\zeta}{dx}\right)^2 \zeta = \frac{p^2(x)}{\hbar^2}$$
and determine $\alpha(x)$.
(b) Verify that in the limits $|x - x_0| \gg$ (away from the turning point), this reduces to the standard WKB solutions with the correct connection formula.
(c) What advantage does the uniform method have over the standard WKB with patched connection formulas?
C.6: Fusion in the Sun — The Gamow Peak
The rate of nuclear fusion reactions in stars depends on the product of the Maxwell-Boltzmann energy distribution and the WKB tunneling probability:
$$R \propto \int_0^\infty e^{-E/(k_BT)}\, e^{-b/\sqrt{E}}\, dE$$
where $b = \pi kZ_1Z_2 e^2\sqrt{2\mu}/\hbar$ is the Gamow parameter.
(a) Show that the integrand has a sharp maximum (the Gamow peak) at energy:
$$E_0 = \left(\frac{bk_BT}{2}\right)^{2/3}$$
(b) Approximate the integrand as a Gaussian about $E_0$ and evaluate the integral in terms of $T$ and $b$.
(c) For the $p$-$p$ reaction ($Z_1 = Z_2 = 1$, $\mu = m_p/2$) at the solar core temperature $T = 1.5 \times 10^7$ K, calculate $E_0$ and the effective width $\Delta E$ of the Gamow peak. Express $E_0$ in keV and compare with $k_BT$.
(d) Show that the reaction rate scales as $R \propto T^\alpha$ where $\alpha = (E_0/k_BT - 2)/3$. For the $p$-$p$ reaction, what is $\alpha$? What does this tell you about the Sun's sensitivity to core temperature?
C.7: WKB for the Symmetric Double Well — Instantons
Consider a symmetric double-well potential $V(x) = \lambda(x^2 - a^2)^2$.
(a) Find the three turning points for a state at the bottom of the wells ($E = 0$).
(b) Compute the tunneling integral through the central barrier:
$$\gamma = \frac{1}{\hbar}\int_{-a}^{a}\sqrt{2m\lambda(x^2 - a^2)^2}\,dx$$
and evaluate it analytically.
(c) The energy splitting between the symmetric and antisymmetric ground states is $\Delta E \propto e^{-\gamma}$. Use the result from (b) to determine how $\Delta E$ depends on $a$, $\lambda$, and $m$.
(d) In the path integral formulation, the tunneling amplitude is dominated by the "instanton" — the classical trajectory in imaginary time. Show that the instanton solution for this potential is $x_{\text{inst}}(t) = a\tanh[\omega(t - t_0)/2]$ where $\omega = 2a\sqrt{2\lambda/m}$, and verify that its action $S_{\text{inst}} = \hbar\gamma$.
C.8: Three-Dimensional WKB and the EBK Quantization
The WKB method generalizes to integrable systems with multiple degrees of freedom via the Einstein-Brillouin-Keller (EBK) quantization:
$$\oint_{C_i} \mathbf{p}\cdot d\mathbf{q} = \left(n_i + \frac{\alpha_i}{4}\right)h, \quad i = 1, 2, \ldots, f$$
where the integrals are over irreducible cycles $C_i$ on the classical torus and $\alpha_i$ is the Maslov index (number of turning points on cycle $i$).
(a) For the 2D isotropic harmonic oscillator $V = \frac{1}{2}m\omega^2(x^2 + y^2)$, identify the two irreducible cycles in polar coordinates and compute the two quantization integrals.
(b) Show that the resulting energy levels are $E_{n_r, l} = (2n_r + |l| + 1)\hbar\omega$, consistent with the exact answer.
(c) What happens to EBK quantization for non-integrable (chaotic) systems? How does this connect to the field of "quantum chaos"?
C.9: WKB Beyond One Dimension — Maslov Index
The Maslov index $\mu$ counts the number of classical turning points (caustics) encountered along a closed classical orbit. For a standard 1D bound state, $\mu = 2$ (one at each turning point), giving the quantization rule $\oint p\,dx = (n + 1/2)h$ since $\mu/4 = 1/2$.
(a) For a bound state in a potential with a hard wall on one side and a smooth turning point on the other (like the bouncing-ball quantum), show that $\mu = 3$ and derive the quantization rule $\oint p\,dx = (n + 3/4)h$.
(b) For a particle on a ring (periodic boundary conditions, no turning points), show that $\mu = 0$ and the quantization rule reduces to $\oint p\,dx = nh$.
(c) For the radial hydrogen problem with the Langer correction, show that $\mu = 2$ regardless of $l$, and verify that this gives the correct energy levels.