Chapter 20 Key Takeaways: The WKB Approximation
Core Message
The WKB approximation is the bridge between quantum and classical mechanics. By expanding the wavefunction in powers of $\hbar$, it reveals that the phase of a quantum wavefunction is the classical action and the amplitude reflects the classical dwell time. Unlike perturbation theory (which requires a solvable reference system), WKB works for any slowly varying potential and provides tunneling rates, bound-state energies, and semiclassical wavefunctions from a single, elegant framework.
Key Concepts
1. The Semiclassical Limit
The WKB approximation is valid when the de Broglie wavelength $\lambda_{\text{dB}} = 2\pi\hbar/p$ is much shorter than the length scale $L$ over which the potential varies: $\lambda_{\text{dB}} \ll L$. Equivalently, $|d\lambda/dx| \ll 2\pi$. This is the regime where classical intuition is approximately correct, but quantum corrections (including tunneling) are captured.
2. WKB Wavefunctions
In classically allowed regions: $\psi \propto p(x)^{-1/2}\exp(\pm i\int p\,dx/\hbar)$ — oscillatory with position-dependent wavelength and amplitude inversely proportional to $\sqrt{p}$. In forbidden regions: $\psi \propto \kappa(x)^{-1/2}\exp(\pm\int\kappa\,dx)$ — exponentially growing or decaying.
3. Connection Formulas and the $\pi/4$ Phase Shift
The WKB solutions on either side of a classical turning point are connected through Airy function matching. Each turning point introduces a phase shift of $\pi/4$. This seemingly small correction is essential — it produces the $+\frac{1}{2}$ in the Bohr-Sommerfeld rule and is responsible for zero-point energy.
4. WKB Tunneling
The transmission probability through an arbitrary barrier is $T \approx e^{-2\gamma}$, where $\gamma = \hbar^{-1}\int_{x_1}^{x_2}\sqrt{2m(V-E)}\,dx$. This single formula governs phenomena from alpha decay to enzyme catalysis to the scanning tunneling microscope.
5. Gamow's Alpha Decay Theory
Applying WKB tunneling to the nuclear Coulomb barrier explains alpha-decay half-lives spanning 24 orders of magnitude (from microseconds to billions of years) through a simple integral. The Geiger-Nuttall law $\log t_{1/2} \propto 1/\sqrt{E_\alpha}$ follows directly from the $1/r$ Coulomb potential.
Key Equations
| Equation | Name | Meaning |
|---|---|---|
| $\psi = \dfrac{C}{\sqrt{p}}\exp\!\left(\pm\dfrac{i}{\hbar}\int p\,dx\right)$ | WKB wavefunction (allowed) | Oscillatory solution with classical amplitude |
| $\psi = \dfrac{D}{\sqrt{\kappa}}\exp\!\left(\pm\int\kappa\,dx\right)$ | WKB wavefunction (forbidden) | Exponentially decaying/growing solution |
| $\left\vert\dfrac{d\lambda}{dx}\right\vert \ll 2\pi$ | Validity condition | WKB requires slowly varying potential |
| $\dfrac{D}{\sqrt{\kappa}}e^{-\Gamma} \longleftrightarrow \dfrac{2D}{\sqrt{p}}\sin(\Phi + \pi/4)$ | Connection formula | Links solutions across turning points |
| $\oint p\,dx = \left(n + \dfrac{1}{2}\right)h$ | Bohr-Sommerfeld (WKB) | Semiclassical quantization rule |
| $T \approx e^{-2\gamma},\quad \gamma = \dfrac{1}{\hbar}\displaystyle\int_{x_1}^{x_2}\!\sqrt{2m(V\!-\!E)}\,dx$ | WKB tunneling | Transmission through arbitrary barrier |
| $t_{1/2} = \dfrac{2R\ln 2}{v}\,e^{2\gamma}$ | Gamow alpha-decay lifetime | Half-life from nuclear Coulomb tunneling |
Key Physical Insights
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$|\psi_{\text{WKB}}|^2 \propto 1/p(x)$ is the classical probability distribution. The quantum particle spends the most time where it moves slowest — exactly as a classical particle does. This is the correspondence principle made quantitative.
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The $\pi/4$ phase shift has real physical consequences. Without it, the Bohr-Sommerfeld rule gives $\oint p\,dx = nh$ instead of $(n+1/2)h$, missing zero-point energy entirely. The $1/2$ is not an ad hoc addition — it emerges inevitably from the turning-point structure.
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Tunneling rates are exponentially sensitive to the barrier integral. For alpha decay, a factor-of-2 change in $E_\alpha$ changes $2\gamma$ by $\sim 50$, producing a $10^{24}$-fold change in half-life. This exponential sensitivity is the physical essence of quantum tunneling.
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WKB is exact for the harmonic oscillator and the Coulomb potential. This is not a coincidence — both potentials have special algebraic structure (ladder operators, $SO(4)$ symmetry) that causes higher-order WKB corrections to vanish.
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WKB gets better with increasing quantum number. For $n \gg 1$, the de Broglie wavelength is short, the potential is "slowly varying" on the scale of $\lambda$, and WKB becomes increasingly accurate. This is the correspondence principle at work.
Common Mistakes to Avoid
| Mistake | Correction |
|---|---|
| Applying WKB at a turning point | WKB breaks down where $p(x) = 0$; use Airy function matching instead |
| Forgetting the $\pi/4$ phase shift | Each turning point contributes $\pi/4$; two turning points give the $+1/2$ in Bohr-Sommerfeld |
| Trusting WKB for the ground state of hard-wall potentials | Hard walls violate the slowly varying condition; WKB is poor for $n = 0$ in the infinite square well |
| Interpreting $T \approx e^{-2\gamma}$ as exact | This formula captures the dominant exponential behavior but misses the algebraic prefactor |
| Assuming WKB requires a constant potential | WKB requires a slowly varying potential (relative to $\lambda_{\text{dB}}$), not a constant one |
| Using the connection formulas "backwards" | The connection from decaying exponential to sine is robust; the reverse is exponentially ill-conditioned |
Connections to Other Chapters
| Chapter | Connection |
|---|---|
| Ch 3 (Tunneling) | WKB generalizes the rectangular barrier result $T = e^{-2\kappa a}$ to arbitrary barrier shapes |
| Ch 4 (QHO) | WKB quantization is exact for the harmonic oscillator; the envelope of high-$n$ wavefunctions is the WKB amplitude |
| Ch 5 (Hydrogen) | WKB quantization of the radial equation (with Langer modification) gives exact hydrogen energy levels |
| Ch 8 (Dirac notation) | WKB wavefunctions in position representation; semiclassical limit of $\langle x|\psi\rangle$ |
| Ch 17 (Perturbation theory) | WKB complements perturbation theory: perturbation needs a solvable $\hat{H}_0$; WKB needs a slowly varying $V(x)$ |
| Ch 19 (Variational) | Variational gives upper bounds on $E_0$; WKB gives all energy levels but is less accurate for low $n$ |
| Ch 22 (Scattering) | WKB phase shifts for partial waves in the semiclassical limit |
| Ch 31 (Path integrals) | WKB is the stationary-phase approximation to the Feynman path integral |
What You Should Be Able to Do After This Chapter
- Write down the WKB wavefunction in both allowed and forbidden regions for any given potential and energy.
- Apply the connection formulas at turning points to match WKB solutions.
- Use the Bohr-Sommerfeld quantization condition to estimate energy levels, and know when it is accurate (high $n$, smooth potentials) and when it is not (low $n$, hard walls).
- Calculate tunneling rates through arbitrary barriers using $T \approx e^{-2\gamma}$.
- Apply the Gamow model to alpha decay, computing the Gamow exponent and estimating half-lives.
- Explain the physical content of the WKB approximation: why $|\psi|^2 \propto 1/p$, why each turning point gives a $\pi/4$ phase shift, and why tunneling rates are exponentially sensitive to barrier parameters.