Chapter 20: The WKB Approximation: Semiclassical Quantum Mechanics

"The semiclassical approximation is the oldest, the most frequently used, and arguably the most physically illuminating approximation method in quantum mechanics." — Laurence Littenberg and Barry Holstein

"It is amusing that the WKB method was invented three times, independently, in the same year." — after Griffiths, Introduction to Quantum Mechanics

Every approximation method we have studied so far — perturbation theory (Chapter 17), degenerate perturbation theory (Chapter 18), and the variational principle (Chapter 19) — relies on a known, exactly solvable reference system. Perturbation theory starts from an unperturbed Hamiltonian $\hat{H}_0$ and systematically corrects it. The variational method starts from a trial wavefunction and optimizes it. Both are immensely powerful, but both share a limitation: they require you to already know something close to the answer.

The WKB approximation is different. It does not require a solvable reference system. Instead, it exploits the fact that classical mechanics is, in a precise sense, the $\hbar \to 0$ limit of quantum mechanics. When the potential varies slowly on the scale of the de Broglie wavelength — when the particle's wavelength is short compared to the scale over which the potential changes — the wavefunction can be written as a rapidly oscillating (or exponentially decaying) envelope modulated by a slowly varying amplitude. The result is an approximate wavefunction that bridges the gap between quantum and classical mechanics, illuminating the correspondence principle from the inside.

Named after Gregor Wentzel, Hendrik Kramers, and Léon Brillouin, who all published the method independently in 1926, the WKB approximation gives us tunneling rates through arbitrary barriers, a quantization condition for bound states that recovers and generalizes the old Bohr-Sommerfeld rule, and — in one of its most spectacular applications — a derivation of the lifetimes of alpha-decaying nuclei that explained a factor-of-10^{24} variation in half-lives from a simple integral.

🏃 Fast Track: If you are familiar with the physical reasoning behind the semiclassical limit and want to get to applications, read Section 20.1 for context, skim Section 20.2 for the form of WKB wavefunctions, then jump to Section 20.4 (connection formulas), Section 20.5 (Bohr-Sommerfeld), and Section 20.7 (alpha decay). Those sections contain the results that are used most often in practice.

🔗 Connection: The WKB method complements the other approximation methods in Part IV. Perturbation theory (Chapter 17) needs a solvable $\hat{H}_0$; the variational method (Chapter 19) needs a good trial wavefunction; WKB needs neither. Instead, it needs the potential to vary slowly — a completely different regime. Together, these three methods cover nearly every situation a physicist encounters. The table in Section 20.8.2 summarizes when to use each.

20.1 The Semiclassical Limit: When $\hbar$ Is "Small"

20.1.1 What Does It Mean for $\hbar$ to Be Small?

Planck's constant is a fixed number: $\hbar = 1.054571817 \times 10^{-34}$ J$\cdot$s. It is not adjustable. So what could it possibly mean to say "$\hbar$ is small"?

The answer is that $\hbar$ is small relative to the classical action of the system. The classical action $S$ of a particle — the integral of the Lagrangian over the trajectory — has dimensions of energy $\times$ time, the same as $\hbar$. When $S \gg \hbar$, the system behaves classically. When $S \sim \hbar$, quantum effects dominate. The relevant dimensionless quantity is $S/\hbar$.

Consider a particle of mass $m$ moving with momentum $p$ through a region where the potential $V(x)$ changes significantly over a length scale $L$. The de Broglie wavelength is:

$$\lambda_{\text{dB}} = \frac{2\pi\hbar}{p}$$

The condition for the semiclassical limit is:

$$\boxed{\lambda_{\text{dB}} \ll L \quad \Leftrightarrow \quad p \gg \frac{2\pi\hbar}{L}}$$

In words: the WKB approximation is valid when the de Broglie wavelength is much shorter than the length scale over which the potential varies. This is precisely the condition under which a particle "looks classical" — its wave nature is negligible compared to the scale of its environment.

💡 Key Insight: The semiclassical limit is not "classical mechanics." It is quantum mechanics in the regime where classical intuition is approximately correct. The WKB method captures the leading quantum corrections to classical behavior — including tunneling, which has no classical analogue at all.

20.1.2 The Classical Momentum

For a particle of mass $m$ and total energy $E$ in a potential $V(x)$, classical mechanics defines the momentum as:

$$p(x) = \sqrt{2m[E - V(x)]}$$

This is real when $E > V(x)$ (classically allowed region) and imaginary when $E < V(x)$ (classically forbidden region). The boundary between these regions — the points where $E = V(x)$ — are the classical turning points. At these points, $p(x) = 0$ and the classical particle momentarily stops before reversing direction.

In the classically allowed region, $p(x)$ is the local momentum of the particle. In the classically forbidden region, the local "momentum" is purely imaginary:

$$\kappa(x) = \frac{1}{\hbar}\sqrt{2m[V(x) - E]}$$

where we define $\kappa(x)$ as a real, positive quantity for convenience. The wavefunction oscillates in the allowed region and decays (or grows) exponentially in the forbidden region — exactly as we saw in the finite barrier problem (Chapter 3). The WKB approximation provides the precise form of this behavior for arbitrary potentials.

20.1.3 A Dimensional Analysis Argument

Before diving into the derivation, let us understand why the WKB form is natural from a dimensional analysis perspective.

In a constant potential, the wavefunction is $\psi(x) = A e^{\pm ikx}$ with $k = p/\hbar$. The phase accumulates linearly. For a slowly varying potential, we expect the same form but with $k$ becoming position-dependent:

$$\psi(x) \sim A(x) \, e^{i\phi(x)}$$

where $\phi(x)$ is a phase that accumulates as the particle moves and $A(x)$ is a slowly varying amplitude. The natural guess for the phase is:

$$\phi(x) = \frac{1}{\hbar}\int^x p(x')\, dx'$$

This is the eikonal approximation, borrowed from optics where it describes the phase of a light wave in a medium with varying index of refraction. The WKB method makes this precise and determines the amplitude $A(x)$.

🔵 Historical Note: The eikonal approximation predates quantum mechanics. It was developed by Sommerfeld and Runge in the early 1900s for optics — the propagation of light in media with slowly varying refractive index. Wentzel, Kramers, and Brillouin recognized that the Schrödinger equation has the same mathematical structure as the wave equation in optics, with $p(x)/\hbar$ playing the role of the wave number and $\hbar$ playing the role of the wavelength. The French call this the "BKW" method; the English-speaking world, "WKB." In the Soviet tradition, it is the "quasi-classical approximation," and Jeffreys had actually discovered it before all three in 1924 (the "JWKB" method). Physics has a naming problem.

20.1.4 The Optical-Mechanical Analogy

The connection between optics and mechanics is ancient — Hamilton himself formulated classical mechanics using ideas borrowed from optics. In modern terms, the analogy is:

The WKB approximation is, in essence, the ray-optics approximation applied to the Schrödinger equation. Just as geometric optics breaks down at caustics (where rays focus and the intensity formally diverges), WKB breaks down at turning points (where the amplitude formally diverges). And just as the wave theory of light resolves the caustic singularity through diffraction, the Airy function solution resolves the turning-point singularity in quantum mechanics.

This analogy is not merely pedagogical — it is a precise mathematical correspondence. The Helmholtz equation in optics and the time-independent Schrödinger equation are both second-order linear ODEs of the same form. Every theorem about WKB in quantum mechanics has an optical counterpart, and vice versa. Physicists have exploited this duality in both directions for over a century.

20.1.5 Worked Example: When Is WKB Valid?

Consider an electron ($m = 9.11 \times 10^{-31}$ kg) with kinetic energy $K = 10$ eV traveling through a region where the potential changes by $\Delta V = 1$ eV over a distance $L$.

De Broglie wavelength:

$$\lambda_{\text{dB}} = \frac{2\pi\hbar}{p} = \frac{2\pi\hbar}{\sqrt{2mK}} = \frac{2\pi(1.055 \times 10^{-34})}{\sqrt{2(9.11 \times 10^{-31})(10 \times 1.60 \times 10^{-19})}} \approx 3.9 \times 10^{-10}\,\text{m} = 3.9\,\text{\AA}$$

Case 1: $L = 100$ nm (slowly varying potential). $\lambda_{\text{dB}}/L = 3.9 \times 10^{-10}/10^{-7} = 0.004 \ll 1$. WKB is excellent.

Case 2: $L = 1$ nm (moderately varying). $\lambda_{\text{dB}}/L = 0.39$. WKB is marginal — expect $\sim 10$% errors.

Case 3: $L = 2$ \AA (atomic scale). $\lambda_{\text{dB}}/L = 2.0 \gg 1$. WKB fails completely. You need to solve the Schrödinger equation exactly or numerically.

The lesson: for a 10-eV electron, WKB works beautifully for macroscopic and mesoscopic potentials but fails at the atomic scale. For a proton or alpha particle with the same kinetic energy, $\lambda_{\text{dB}}$ is $\sim 40$ times shorter, and WKB would work even at the 1-nm scale. This is why WKB is so effective in nuclear physics — nuclear energies are in MeV and nuclear masses are thousands of MeV/$c^2$, making $\lambda_{\text{dB}}$ very small compared to nuclear length scales.

20.2 WKB Wavefunctions in Classically Allowed Regions

20.2.1 The Formal Derivation

We start from the time-independent Schrödinger equation in one dimension:

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi$$

Rearranging:

$$\frac{d^2\psi}{dx^2} = -\frac{p(x)^2}{\hbar^2}\psi$$

where $p(x) = \sqrt{2m[E - V(x)]}$ is the local classical momentum. For a constant potential, this has solutions $\psi = e^{\pm ipx/\hbar}$. For a varying potential, we try the ansatz:

$$\psi(x) = A(x)\, e^{i\phi(x)/\hbar}$$

where $A(x)$ and $\phi(x)$ are real-valued functions. (Taking $\phi$ to be real with the $\hbar$ written explicitly will help us track orders of approximation.) Substituting into the Schrödinger equation and separating real and imaginary parts, we find two coupled equations.

Substitution: Computing $d^2\psi/dx^2$ from the ansatz:

$$\frac{d\psi}{dx} = \left(A' + \frac{i}{\hbar}A\phi'\right) e^{i\phi/\hbar}$$

$$\frac{d^2\psi}{dx^2} = \left[A'' + \frac{2i}{\hbar}A'\phi' + \frac{i}{\hbar}A\phi'' - \frac{1}{\hbar^2}A(\phi')^2\right] e^{i\phi/\hbar}$$

Setting this equal to $-\frac{p^2}{\hbar^2}A\,e^{i\phi/\hbar}$ and separating real and imaginary parts:

Real part:

$$A'' - \frac{A(\phi')^2}{\hbar^2} = -\frac{p^2}{\hbar^2}A$$

$$\Rightarrow \quad (\phi')^2 = p(x)^2 + \hbar^2\frac{A''}{A}$$

Imaginary part:

$$2A'\phi' + A\phi'' = 0$$

$$\Rightarrow \quad \frac{d}{dx}\!\left(A^2 \phi'\right) = 0$$

20.2.2 The WKB Expansion

The imaginary part gives us a conservation law immediately. Define $j = A^2\phi'$; then $j$ is constant. This is a statement of probability current conservation — exactly what we expect.

The real part is exact but involves the unknown $A(x)$. Here is where the approximation enters. We expand $\phi$ in powers of $\hbar$:

$$\phi(x) = \phi_0(x) + \hbar\,\phi_1(x) + \hbar^2\,\phi_2(x) + \cdots$$

At zeroth order in $\hbar$ (i.e., dropping the $\hbar^2 A''/A$ term):

$$(\phi_0')^2 = p(x)^2 \quad \Rightarrow \quad \phi_0'(x) = \pm p(x)$$

$$\boxed{\phi_0(x) = \pm \int^x p(x')\, dx'}$$

This is the Hamilton-Jacobi equation of classical mechanics! The phase of the WKB wavefunction is the classical action. At the semiclassical level, quantum mechanics remembers its classical heritage.

From the imaginary part, with $\phi' \approx \pm p(x)$:

$$A^2 p(x) = \text{constant} \quad \Rightarrow \quad A(x) = \frac{C}{\sqrt{p(x)}}$$

💡 Key Insight: The amplitude $A(x) \propto 1/\sqrt{p(x)}$ has a beautiful classical interpretation. In classical mechanics, a particle spends more time where it moves slowly (low momentum) and less time where it moves fast (high momentum). The probability of finding a classical particle near position $x$ is proportional to $dt/dx = 1/v(x) \propto 1/p(x)$. Therefore $|\psi|^2 \propto 1/p(x)$, which means $A \propto 1/\sqrt{p(x)}$. The WKB amplitude is the quantum expression of the classical dwell-time distribution.

20.2.3 The WKB Wavefunction (Classically Allowed Region)

Combining the results, the general WKB wavefunction in a classically allowed region ($E > V(x)$) is:

$$\boxed{\psi_{\text{WKB}}(x) = \frac{C_1}{\sqrt{p(x)}}\exp\!\left(\frac{i}{\hbar}\int^x p(x')\,dx'\right) + \frac{C_2}{\sqrt{p(x)}}\exp\!\left(-\frac{i}{\hbar}\int^x p(x')\,dx'\right)}$$

This is a superposition of a "rightward-traveling" wave and a "leftward-traveling" wave, each with: - Amplitude $\propto 1/\sqrt{p(x)}$ — largest where the particle moves slowly - Local wave number $k(x) = p(x)/\hbar$ — shortest wavelength where the particle has the most kinetic energy

20.2.4 Validity Condition

When is the WKB approximation valid? The key assumption was that $\hbar^2 A''/A$ is negligible compared to $p^2$. Using $A = C/\sqrt{p}$, this requires:

$$\hbar\left|\frac{dp/dx}{p^2}\right| \ll 1$$

Equivalently, defining the local de Broglie wavelength $\lambda(x) = 2\pi\hbar/p(x)$:

$$\boxed{\left|\frac{d\lambda}{dx}\right| \ll 2\pi}$$

In words: the de Broglie wavelength must change by a small fraction of itself over one wavelength. This is the condition for a "slowly varying" potential.

⚠️ Common Misconception: The WKB approximation does NOT require a "nearly constant" potential. It requires a slowly varying potential relative to the local wavelength. A potential can change enormously over macroscopic distances and still satisfy the WKB condition, as long as the de Broglie wavelength is correspondingly short. Conversely, even a gently sloping potential violates WKB near a turning point where $p \to 0$ and $\lambda \to \infty$.

20.2.5 Worked Example: WKB Wavefunction for a Linear Potential

Consider a particle of mass $m$ in a linear potential $V(x) = mgx$ (gravity, or a uniform electric field), with energy $E$. The turning point is at $x_0 = E/(mg)$.

Classically allowed region ($x < x_0$): The local momentum is $p(x) = \sqrt{2m(E - mgx)} = \sqrt{2m^2g(x_0 - x)}$. The WKB wavefunction is:

$$\psi(x) = \frac{C}{\left[2m^2g(x_0 - x)\right]^{1/4}}\sin\!\left(\frac{1}{\hbar}\int_x^{x_0}\sqrt{2m^2g(x_0 - x')}\,dx' + \frac{\pi}{4}\right)$$

The phase integral evaluates to:

$$\frac{1}{\hbar}\int_x^{x_0}\sqrt{2m^2g(x_0 - x')}\,dx' = \frac{2}{3}\frac{\sqrt{2m^2g}}{\hbar}(x_0 - x)^{3/2}$$

Forbidden region ($x > x_0$): The wavefunction decays:

$$\psi(x) = \frac{D}{\left[2m^2g(x - x_0)\right]^{1/4}}\exp\!\left(-\frac{2}{3}\frac{\sqrt{2m^2g}}{\hbar}(x - x_0)^{3/2}\right)$$

Notice the characteristic $3/2$ power in the exponent — this comes from the linear potential and produces exactly the Airy function behavior, as it must.

✅ Checkpoint: Verify that the WKB wavefunction for the linear potential, when written in the dimensionless variable $z = \alpha(x - x_0)$ with $\alpha = (2m^2g/\hbar^2)^{1/3}$, reduces to $\text{Ai}(z)$ in the appropriate limits. This is a self-consistency check: the Airy function is the exact solution for a linear potential, so WKB should reproduce it (up to the usual prefactor corrections near $z = 0$).

20.2.6 The WKB Approximation and the Correspondence Principle

The WKB wavefunction makes the correspondence principle (Chapter 1) quantitatively precise. For large quantum numbers, the de Broglie wavelength becomes short compared to the size of the potential, the WKB approximation becomes excellent, and the quantum probability distribution $|\psi|^2 \propto 1/p(x)$ approaches the classical dwell-time distribution.

🔗 Connection: In Chapter 4, we noted that for high-$n$ harmonic oscillator states, the probability distribution develops many oscillations but its envelope matches the classical distribution $\rho_{\text{cl}}(x) \propto 1/\sqrt{A^2 - x^2}$, where $A$ is the classical amplitude. This is exactly the WKB result $|\psi|^2 \propto 1/p(x) = 1/\sqrt{2m(E - \frac{1}{2}m\omega^2 x^2)}$, which diverges at the turning points $x = \pm A$. The envelope match you saw in Chapter 4 was WKB in disguise.

20.3 WKB Wavefunctions in Classically Forbidden Regions

20.3.1 Exponential Behavior

In the classically forbidden region ($E < V(x)$), the classical momentum is imaginary: $p(x) = i\hbar\kappa(x)$ where $\kappa(x) = \sqrt{2m[V(x) - E]}/\hbar$ is real and positive. The Schrödinger equation becomes:

$$\frac{d^2\psi}{dx^2} = \kappa(x)^2\,\psi$$

and the WKB solutions are:

$$\boxed{\psi_{\text{WKB}}(x) = \frac{D_1}{\sqrt{\kappa(x)}}\exp\!\left(\int^x \kappa(x')\,dx'\right) + \frac{D_2}{\sqrt{\kappa(x)}}\exp\!\left(-\int^x \kappa(x')\,dx'\right)}$$

The first term grows exponentially; the second decays. For a particle incident from the left tunneling through a barrier, the physical solution deep inside the barrier is the decaying one.

💡 Key Insight: The forbidden-region wavefunction is not zero — it is exponentially small but nonzero. This is the essence of tunneling: the wavefunction leaks into the classically forbidden region, and if the barrier is thin enough, a measurable fraction of the probability amplitude reaches the other side. Classical mechanics says the probability is exactly zero; quantum mechanics says it is exponentially small but nonzero. The difference between "exactly zero" and "exponentially small" is the difference between "the Sun doesn't shine" and "the Sun has been burning hydrogen for 4.6 billion years."

20.3.2 Physical Interpretation of the Forbidden-Region Wavefunction

In the classically forbidden region, the particle's "kinetic energy" $E - V(x)$ is negative. What does this mean physically?

It is tempting to say that the particle has "negative kinetic energy" in the barrier, but this is misleading. A measurement of the particle's position that finds it inside the barrier would reveal a kinetic energy that is positive — because the act of measurement localizes the particle and, by the uncertainty principle, gives it enough momentum uncertainty to account for the energy discrepancy.

The correct interpretation is this: the WKB wavefunction in the forbidden region describes the amplitude for the particle to be found there, but it does not describe a classical trajectory. The particle does not "travel through" the barrier in any classical sense. The wavefunction is a mathematical object that encodes probabilities; the exponential decay in the forbidden region is a statement about how those probabilities diminish with distance.

⚖️ Interpretation: Different interpretations of quantum mechanics offer different pictures of tunneling. In the Copenhagen interpretation, nothing physical happens in the barrier — only the probability amplitude penetrates. In Bohmian mechanics, the particle does traverse the barrier, guided by the quantum potential, which exactly compensates for the classical barrier. In the many-worlds interpretation, the particle is in a superposition of being reflected and transmitted, and both outcomes are realized in different branches. The WKB formalism is interpretation-neutral: it gives correct probabilities regardless of which interpretation you prefer.

20.3.3 The Tunneling Amplitude

Consider a barrier extending from $x_1$ to $x_2$ (the two classical turning points where $E = V(x)$). The WKB tunneling amplitude — the ratio of transmitted to incident amplitudes — depends on the integral:

$$\gamma = \int_{x_1}^{x_2} \kappa(x)\,dx = \frac{1}{\hbar}\int_{x_1}^{x_2} \sqrt{2m[V(x) - E]}\,dx$$

This dimensionless quantity, sometimes called the Gamow exponent or the barrier action (in units of $\hbar$), controls the tunneling probability. Large $\gamma$ means exponentially suppressed tunneling.

📊 By the Numbers: For a typical nuclear physics barrier (alpha decay), $\gamma \sim 30$–$90$, giving tunneling probabilities $e^{-2\gamma}$ ranging from $\sim 10^{-26}$ to $\sim 10^{-78}$. This is how a single integral explains the enormous range of alpha-decay half-lives observed in nature — from microseconds to billions of years.

20.3.3 Connection to Rectangular Barrier Results

In Chapter 3, we solved the rectangular barrier exactly. For a barrier of height $V_0 > E$ and width $a$:

$$T_{\text{exact}} = \left[1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0 - E)}\right]^{-1}$$

In the thick-barrier limit ($\kappa a \gg 1$), this reduces to:

$$T \approx \frac{16E(V_0 - E)}{V_0^2}\, e^{-2\kappa a}$$

The WKB transmission coefficient gives:

$$T_{\text{WKB}} = e^{-2\kappa a}$$

This captures the dominant exponential behavior correctly but misses the algebraic prefactor. The WKB approximation is excellent for determining the order of magnitude of the tunneling probability (which is what matters physically, since the tunneling exponent varies over tens of orders of magnitude), but it does not capture the prefactor precisely.

✅ Checkpoint: Before proceeding, verify that you can: 1. Write down the WKB wavefunction in both allowed and forbidden regions 2. State the validity condition for the WKB approximation in words and equations 3. Identify where the WKB approximation breaks down (answer: at turning points) 4. Explain why $|\psi_{\text{WKB}}|^2 \propto 1/p(x)$ is the classical probability distribution

20.4 Connection Formulas at Turning Points

20.4.1 The Problem at Turning Points

The WKB approximation breaks down at classical turning points where $p(x) = 0$. The amplitude factor $1/\sqrt{p(x)}$ diverges, and the validity condition $|d\lambda/dx| \ll 2\pi$ is violated since $\lambda \to \infty$.

Physically, there is nothing singular about the turning point — the exact wavefunction $\psi(x)$ is perfectly smooth there. The singularity is an artifact of the approximation. The question is: how do we connect the WKB solutions on either side of the turning point?

20.4.2 The Airy Function Strategy

The standard approach is to approximate the potential near the turning point $x_0$ as linear:

$$V(x) \approx V(x_0) + V'(x_0)(x - x_0) = E + V'(x_0)(x - x_0)$$

since $V(x_0) = E$ at the turning point. With this linear approximation, the Schrödinger equation near $x_0$ becomes:

$$\frac{d^2\psi}{dx^2} = \frac{2m V'(x_0)}{\hbar^2}(x - x_0)\,\psi$$

Defining the dimensionless variable:

$$z = \left(\frac{2m V'(x_0)}{\hbar^2}\right)^{1/3}(x - x_0) = \alpha(x - x_0)$$

where $\alpha = (2m|V'(x_0)|/\hbar^2)^{1/3}$, this becomes the Airy equation:

$$\frac{d^2\psi}{dz^2} = z\,\psi$$

20.4.3 Airy Functions

The Airy equation has two linearly independent solutions, denoted $\text{Ai}(z)$ and $\text{Bi}(z)$:

For large positive $z$ (deep in the forbidden region): $$\text{Ai}(z) \approx \frac{1}{2\sqrt{\pi}\, z^{1/4}}\exp\!\left(-\frac{2}{3}z^{3/2}\right) \quad \text{(decays)}$$ $$\text{Bi}(z) \approx \frac{1}{\sqrt{\pi}\, z^{1/4}}\exp\!\left(\frac{2}{3}z^{3/2}\right) \quad \text{(grows)}$$

For large negative $z$ (deep in the allowed region): $$\text{Ai}(z) \approx \frac{1}{\sqrt{\pi}\, |z|^{1/4}}\sin\!\left(\frac{2}{3}|z|^{3/2} + \frac{\pi}{4}\right)$$ $$\text{Bi}(z) \approx \frac{1}{\sqrt{\pi}\, |z|^{1/4}}\cos\!\left(\frac{2}{3}|z|^{3/2} + \frac{\pi}{4}\right)$$

The key observation is that the Airy functions naturally interpolate between oscillatory behavior (allowed region) and exponential behavior (forbidden region). By matching the Airy solution near the turning point to the WKB solutions far from it, we obtain the connection formulas.

🔵 Historical Note: George Biddell Airy introduced these functions in 1838 in the context of optics — specifically, the intensity pattern near a caustic (a turning point for light rays). The mathematical connection between optical caustics and quantum turning points is not a coincidence: both involve wave phenomena at the boundary between "allowed" and "forbidden" propagation.

20.4.4 Deriving the Connection Formulas

Consider a turning point $x_0$ at which the potential increases to the right: $V'(x_0) > 0$. The classically allowed region is to the left ($x < x_0$) and the forbidden region is to the right ($x > x_0$).

In the forbidden region ($x > x_0$), we demand the physical (decaying) solution:

$$\psi(x) \to \frac{D}{\sqrt{\kappa(x)}}\exp\!\left(-\int_{x_0}^{x}\kappa(x')\,dx'\right)$$

Matching through the Airy function and continuing into the allowed region ($x < x_0$):

$$\psi(x) \to \frac{2D}{\sqrt{p(x)}}\sin\!\left(\frac{1}{\hbar}\int_{x}^{x_0}p(x')\,dx' + \frac{\pi}{4}\right)$$

This is the first connection formula (forbidden $\to$ allowed, right-side turning point):

$$\boxed{\frac{D}{\sqrt{\kappa}}\,e^{-\int_{x_0}^{x}\kappa\,dx'} \quad \longleftrightarrow \quad \frac{2D}{\sqrt{p}}\sin\!\left(\frac{1}{\hbar}\int_{x}^{x_0}p\,dx' + \frac{\pi}{4}\right)}$$

Similarly, for a turning point at $x_1$ where the potential increases to the left ($V'(x_1) < 0$), the classically allowed region is to the right and the forbidden region to the left. The second connection formula is:

$$\boxed{\frac{D}{\sqrt{\kappa}}\,e^{-\int_{x}^{x_1}\kappa\,dx'} \quad \longleftrightarrow \quad \frac{2D}{\sqrt{p}}\sin\!\left(\frac{1}{\hbar}\int_{x_1}^{x}p\,dx' + \frac{\pi}{4}\right)}$$

20.4.5 Step-by-Step: How the Matching Works

To make the derivation of the connection formulas concrete, let us trace through the matching procedure explicitly.

Step 1: Identify the overlap regions. Far to the left of $x_0$ (deep in the allowed region), the exact Airy solution and the WKB solution both apply. Far to the right of $x_0$ (deep in the forbidden region), the same is true. The matching is done in these overlap regions, where both approximations are valid simultaneously.

Step 2: Express the Airy function in the WKB limit. Using $z = \alpha(x - x_0)$ and the asymptotic forms of $\text{Ai}(z)$:

For $z \gg 1$ (forbidden region, $x \gg x_0$):

$$\text{Ai}(z) \sim \frac{1}{2\sqrt{\pi}\, z^{1/4}}\exp\!\left(-\frac{2}{3}z^{3/2}\right)$$

The WKB forbidden-region solution is:

$$\psi = \frac{D}{\sqrt{\kappa}}\exp\!\left(-\int_{x_0}^x \kappa\,dx'\right)$$

Since $\kappa = \sqrt{2mV'(x_0)(x - x_0)}/\hbar = \alpha^{3/2}\sqrt{z}\hbar/\hbar = \alpha^{3/2}\sqrt{z}\cdot(\hbar/\hbar)$ (after careful unit tracking), and $\int_{x_0}^x\kappa\,dx' = \frac{2}{3}z^{3/2}$, these match with $D \propto 1/(2\sqrt{\pi}\alpha^{1/2})$.

Step 3: Continue the Airy function into the allowed region. For $z \ll -1$ ($x \ll x_0$), the asymptotic form gives:

$$\text{Ai}(z) \sim \frac{1}{\sqrt{\pi}\,|z|^{1/4}}\sin\!\left(\frac{2}{3}|z|^{3/2} + \frac{\pi}{4}\right)$$

The $\pi/4$ appears here from the intrinsic phase of the Airy function. Using $|z|^{3/2} = [\alpha(x_0 - x)]^{3/2}$ and $\frac{2}{3}[\alpha(x_0 - x)]^{3/2} = \frac{1}{\hbar}\int_x^{x_0}p\,dx'$, we match to the WKB allowed-region form and obtain the connection formula with its $\pi/4$ phase.

Step 4: Read off the connection formula. Equating the Airy-function coefficients in the two regions gives the connection formula stated in Section 20.4.4. The $\pi/4$ is not imposed by hand — it emerges inevitably from the asymptotic properties of the Airy function.

20.4.6 The Crucial Phase Shift of $\pi/4$

The $\pi/4$ phase shift in the connection formulas is the signature of the turning point. It arises from the asymptotic properties of the Airy function and has no analogue in a purely classical treatment. Physically, the turning point introduces a phase delay of $\pi/4$ relative to naive WKB — the wave "feels" the transition between allowed and forbidden regions and picks up a quarter-wavelength phase shift.

This $\pi/4$ shift is essential for the Bohr-Sommerfeld quantization condition, as we shall see in Section 20.5. Without it, the quantization rule would give the wrong answer.

🔴 Warning: The connection formulas are one-directional in a subtle sense. The formula connecting a decaying exponential in the forbidden region to a sinusoidal function in the allowed region is robust. But the reverse — deducing the behavior in the forbidden region from the allowed region — is exponentially sensitive to the coefficients. A tiny admixture of $\cos$ in the allowed region (which is the growing exponential in the forbidden region) can dominate the decaying solution. This asymmetry is the source of much grief in WKB calculations and must be handled with care.

20.4.6 Summary of Connection Formulas

For quick reference, here are the connection formulas in a compact notation. Let $x_0$ be a turning point. Define:

$$\Phi_L = \frac{1}{\hbar}\int_x^{x_0} p(x')\,dx', \quad \Gamma_R = \int_{x_0}^x \kappa(x')\,dx'$$

Right-side turning point (allowed on left, forbidden on right):

Left-side turning point (forbidden on left, allowed on right):

where $\Phi_R = \frac{1}{\hbar}\int_{x_1}^x p\,dx'$ and $\Gamma_L = \int_x^{x_1}\kappa\,dx'$.

20.5 Bohr-Sommerfeld Quantization Recovered

20.5.1 Bound States in the WKB Approximation

Consider a particle bound in a potential well with two classical turning points, $x_1$ (left) and $x_2$ (right), where $E = V(x_1) = V(x_2)$ and $E > V(x)$ for $x_1 < x < x_2$.

In the classically forbidden regions ($x < x_1$ and $x > x_2$), the wavefunction must decay. Applying the connection formula at the right turning point $x_2$:

$$\psi(x) = \frac{2D_R}{\sqrt{p(x)}}\sin\!\left(\frac{1}{\hbar}\int_x^{x_2} p(x')\,dx' + \frac{\pi}{4}\right) \quad (x_1 < x < x_2)$$

Applying the connection formula at the left turning point $x_1$:

$$\psi(x) = \frac{2D_L}{\sqrt{p(x)}}\sin\!\left(\frac{1}{\hbar}\int_{x_1}^x p(x')\,dx' + \frac{\pi}{4}\right) \quad (x_1 < x < x_2)$$

These must be the same function in the overlap region $x_1 < x < x_2$. Noting that:

$$\int_x^{x_2}p\,dx' = \int_{x_1}^{x_2}p\,dx' - \int_{x_1}^{x}p\,dx'$$

we can rewrite the right-turning-point expression as:

$$\psi \propto \sin\!\left(\frac{1}{\hbar}\int_{x_1}^{x_2}p\,dx' - \frac{1}{\hbar}\int_{x_1}^{x}p\,dx' + \frac{\pi}{4}\right)$$

Using $\sin(\Theta - \theta) = \sin\Theta\cos\theta - \cos\Theta\sin\theta$ and comparing with $\sin(\theta + \pi/4)$ from the left turning point, the two expressions are consistent if and only if:

$$\frac{1}{\hbar}\int_{x_1}^{x_2}p(x')\,dx' + \frac{\pi}{4} = n\pi - \frac{\pi}{4} + \pi, \quad n = 0, 1, 2, \ldots$$

Wait — let us be more careful. For two sine functions to be the same, their arguments must differ by an integer multiple of $\pi$ (allowing for sign changes). The condition is:

$$\frac{1}{\hbar}\int_{x_1}^{x_2}p(x)\,dx = \left(n + \frac{1}{2}\right)\pi, \quad n = 0, 1, 2, \ldots$$

or equivalently:

$$\boxed{\oint p(x)\,dx = \left(n + \frac{1}{2}\right) 2\pi\hbar = \left(n + \frac{1}{2}\right) h}$$

where $\oint$ denotes the integral over one complete classical period (from $x_1$ to $x_2$ and back), which is $2\int_{x_1}^{x_2}p\,dx$.

20.5.2 The Bohr-Sommerfeld Quantization Condition

This is the WKB quantization condition, also known as the Bohr-Sommerfeld quantization condition (with the crucial $+\frac{1}{2}$ correction from the turning points). Compare this with the "old quantum theory" (Bohr model, Chapter 1):

The WKB result differs from Bohr's original rule by the replacement $n \to n + \frac{1}{2}$. This correction comes entirely from the $\pi/4$ phase shifts at the two turning points. It is precisely the correction needed to produce the correct zero-point energy.

💡 Key Insight: Each classical turning point contributes a phase shift of $\pi/4$, or equivalently $\lambda/8$. Two turning points give a total phase shift of $\pi/2$, which shifts the quantum number by $1/2$. This is why the harmonic oscillator energies are $E_n = (n + \frac{1}{2})\hbar\omega$ rather than $E_n = n\hbar\omega$: the $1/2$ is the WKB turning-point correction to the Bohr rule.

20.5.3 Testing on the Harmonic Oscillator

Let us verify the WKB quantization condition on a system where we know the exact answer: the quantum harmonic oscillator, $V(x) = \frac{1}{2}m\omega^2 x^2$.

The classical turning points satisfy $E = \frac{1}{2}m\omega^2 x_0^2$, giving $x_0 = \sqrt{2E/(m\omega^2)}$.

The quantization integral:

$$\int_{-x_0}^{x_0} p(x)\,dx = \int_{-x_0}^{x_0}\sqrt{2m\left(E - \frac{1}{2}m\omega^2 x^2\right)}\,dx$$

Substituting $x = x_0\sin\theta$:

$$= 2m\omega x_0^2 \int_{-\pi/2}^{\pi/2}\cos^2\theta\,d\theta = \frac{\pi E}{\omega}$$

The WKB condition $\int_{-x_0}^{x_0}p\,dx = (n + \frac{1}{2})\pi\hbar$ gives:

$$\frac{\pi E}{\omega} = \left(n + \frac{1}{2}\right)\pi\hbar \quad \Rightarrow \quad E_n = \left(n + \frac{1}{2}\right)\hbar\omega$$

This is exact. The WKB quantization condition reproduces the harmonic oscillator energy levels perfectly — not approximately, but exactly. This is a remarkable and somewhat accidental result: for the harmonic oscillator, the higher-order WKB corrections all vanish identically.

📊 By the Numbers: WKB accuracy for selected potentials:

Potential	WKB Error ($n = 0$)	WKB Error ($n = 5$)	WKB Error ($n = 20$)
Harmonic oscillator	0% (exact)	0% (exact)	0% (exact)
Infinite square well	21%	0.8%	0.05%
Morse potential	~5%	~0.2%	~0.01%
Hydrogen ($l = 0$)	~8%	~0.3%	~0.02%

The pattern is clear: WKB is poor for the ground state (except for the oscillator) but improves rapidly with increasing quantum number. This makes sense — high quantum numbers mean short de Broglie wavelengths, which is precisely the WKB regime.

20.5.4 Worked Example: WKB 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, $r_e$ is the equilibrium bond length, and $\beta$ controls the width of the well. Near $r_e$, the Morse potential looks like a harmonic oscillator with frequency $\omega_e = \beta\sqrt{2D_e/m}$, but it becomes asymmetric for large displacements and supports only a finite number of bound states (unlike the harmonic oscillator).

Applying Bohr-Sommerfeld quantization to the Morse potential (the integral can be evaluated analytically using the substitution $u = e^{-\beta(r - r_e)}$):

$$E_n = \hbar\omega_e\left(n + \frac{1}{2}\right) - \frac{\left[\hbar\omega_e\left(n + \frac{1}{2}\right)\right]^2}{4D_e}, \quad n = 0, 1, 2, \ldots, n_{\max}$$

where $n_{\max}$ is determined by the condition $E_n < D_e$.

Remarkably, this result is exact — the Morse potential, like the harmonic oscillator, is one of the rare cases for which WKB gives the exact energy spectrum. This is because the Morse potential can be transformed into the Coulomb problem in an appropriate coordinate, and both belong to the class of "exactly solvable" potentials for which all WKB corrections vanish.

The first term, $\hbar\omega_e(n + 1/2)$, is the harmonic approximation. The second term, proportional to $(n + 1/2)^2$, is the anharmonic correction that causes the energy levels to converge as $n$ increases and eventually reach the dissociation limit. This convergence is readily observed in molecular spectroscopy — the vibrational overtones of diatomic molecules like HCl are not equally spaced, and the departure from equal spacing is precisely the anharmonic WKB correction.

20.5.5 Why the Infinite Square Well Is the Worst Case

The infinite square well provides an instructive failure. The potential jumps discontinuously from $0$ to $\infty$ at the walls — about as far from "slowly varying" as one can get. There are no smooth turning points; instead, the wavefunction is forced to zero at the walls. The correct boundary condition is Dirichlet ($\psi = 0$), not the WKB connection formula. If we naively apply WKB quantization:

$$\int_0^{a} p\,dx = \sqrt{2mE}\cdot a = \left(n + \frac{1}{2}\right)\pi\hbar$$

$$E_n^{\text{WKB}} = \frac{(n + \frac{1}{2})^2 \pi^2\hbar^2}{2ma^2}$$

The exact answer is $E_n = (n+1)^2\pi^2\hbar^2/(2ma^2)$ (with $n = 0, 1, 2, \ldots$). For $n = 0$: WKB gives $E_0 = \pi^2\hbar^2/(8ma^2)$, exact gives $E_0 = \pi^2\hbar^2/(2ma^2)$ — off by a factor of 4. But for $n = 20$: WKB gives $(20.5)^2 = 420.25$ vs. exact $(21)^2 = 441$ — only 4.7% error. The lesson: WKB struggles with hard walls but improves with quantum number, as always.

🔗 Connection: In Chapter 3, we solved the infinite square well exactly and noted the energy levels $E_n = n^2\pi^2\hbar^2/(2ma^2)$. The WKB result converges to this for large $n$ because the hard-wall boundary conditions become less important relative to the accumulated phase across the well.

20.6 Tunneling Rates and Barrier Penetration

20.6.1 The WKB Transmission Coefficient

Consider a particle of energy $E$ incident from the left on a potential barrier, with turning points $x_1$ and $x_2$ where $E = V(x_1) = V(x_2)$. In the region $x_1 < x < x_2$, the particle is classically forbidden and the wavefunction decays exponentially.

The WKB transmission coefficient is:

$$\boxed{T \approx \exp\!\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m[V(x) - E]}\,dx\right) = e^{-2\gamma}}$$

where:

$$\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2}\sqrt{2m[V(x) - E]}\,dx$$

is the Gamow exponent. This is the central result of WKB tunneling theory. The transmission probability depends exponentially on the integral of $\kappa(x)$ across the forbidden region. Any factor that reduces this integral — a narrower barrier, a thinner barrier, a lower barrier, a lighter particle — exponentially increases the tunneling rate.

20.6.2 Worked Example: Tunneling Through a Parabolic Barrier

Consider a parabolic barrier:

$$V(x) = V_0\left(1 - \frac{x^2}{a^2}\right)$$

with $V_0 > E$. The turning points are at $x = \pm a\sqrt{1 - E/V_0}$. Let $b = a\sqrt{1 - E/V_0}$. Then:

$$\gamma = \frac{1}{\hbar}\int_{-b}^{b}\sqrt{2m\left[V_0\left(1 - \frac{x^2}{a^2}\right) - E\right]}\,dx$$

$$= \frac{\sqrt{2m(V_0 - E)}}{\hbar}\int_{-b}^{b}\sqrt{1 - \frac{x^2}{b^2}}\,dx$$

The integral $\int_{-b}^{b}\sqrt{1 - x^2/b^2}\,dx = \frac{\pi b}{2}$, so:

$$\gamma = \frac{\pi a}{2\hbar}\sqrt{2m(V_0 - E)}\sqrt{1 - E/V_0} = \frac{\pi a}{2\hbar}\frac{2m(V_0 - E)}{\sqrt{2mV_0}}$$

For $E \ll V_0$:

$$T \approx \exp\!\left(-\frac{\pi a\sqrt{2mV_0}}{\hbar}\right)$$

This result illustrates how the tunneling rate depends exponentially on the barrier width $a$, the barrier height $V_0$, and the particle mass $m$.

20.6.3 Physical Applications of WKB Tunneling

WKB tunneling is ubiquitous in physics:

Nuclear fusion in stars: Protons must tunnel through the Coulomb barrier to fuse. The barrier height is $\sim 1$ MeV but stellar thermal energies are only $\sim 1$ keV. Without tunneling, the Sun would not shine.

Scanning tunneling microscope (STM): Electrons tunnel through the vacuum gap between a sharp metallic tip and a surface. The tunneling current depends exponentially on the gap width, giving sub-angstrom vertical resolution.

Josephson junctions: Cooper pairs tunnel through a thin insulating barrier between superconductors, producing a supercurrent with no voltage drop — the basis of SQUID magnetometers and superconducting qubits.

Field emission: In a strong electric field, the potential barrier at a metal surface becomes thin enough for electrons to tunnel through, producing a measurable current even at zero temperature.

🧪 Experiment: The scanning tunneling microscope, invented by Gerd Binnig and Heinrich Rohrer in 1981 (Nobel Prize, 1986), exploits WKB tunneling directly. The tunneling current through a vacuum barrier of width $d$ is approximately $I \propto e^{-2\kappa d}$ with $\kappa \approx \sqrt{2m\phi}/\hbar$, where $\phi \approx 4$–$5$ eV is the work function. For a typical work function, $\kappa \approx 10^{10}$ m$^{-1}$, so the current decreases by roughly a factor of $e^2 \approx 7.4$ for every angstrom of additional gap width. This extreme sensitivity to distance is what gives the STM its extraordinary resolution.

20.6.4 Worked Example: The Scanning Tunneling Microscope

Let us estimate the tunneling current in a scanning tunneling microscope with concrete numbers.

Given: A tungsten tip hovers $d = 5$ \AA above a gold surface. The effective barrier height is the average work function, $\phi \approx 4.5$ eV. The electron energy is at the Fermi level ($E = 0$ relative to the barrier).

Step 1: Compute $\kappa$.

$$\kappa = \frac{\sqrt{2m_e\phi}}{\hbar} = \frac{\sqrt{2(9.11 \times 10^{-31})(4.5 \times 1.60 \times 10^{-19})}}{1.055 \times 10^{-34}}$$

$$= \frac{\sqrt{1.31 \times 10^{-48}}}{1.055 \times 10^{-34}} = \frac{1.145 \times 10^{-24}}{1.055 \times 10^{-34}} = 1.085 \times 10^{10}\,\text{m}^{-1}$$

Step 2: Compute the tunneling probability.

$$T = e^{-2\kappa d} = e^{-2(1.085 \times 10^{10})(5 \times 10^{-10})} = e^{-10.85} \approx 1.95 \times 10^{-5}$$

Step 3: Sensitivity to distance.

If the tip moves 1 \AA closer ($d = 4$ \AA):

$$T' = e^{-2\kappa(4\,\text{\AA})} = e^{-8.68} \approx 1.69 \times 10^{-4}$$

The ratio $T'/T = e^{2\kappa \cdot 1\,\text{\AA}} = e^{2.17} \approx 8.7$. Moving the tip one angstrom closer increases the current by a factor of $\sim 9$. This extraordinary sensitivity — roughly one order of magnitude per angstrom — is what gives the STM its ability to image individual atoms on a surface.

Step 4: Absolute current estimate.

The tunneling current can be estimated as $I \approx n_e e v_F T A$, where $n_e$ is the electron density, $v_F$ is the Fermi velocity, and $A$ is the effective tunneling area (roughly the tip area, $\sim 1$ nm$^2$). For typical parameters, this gives $I \sim 1$ nA, consistent with experimental STM operating conditions.

20.6.5 Multiple Barrier Tunneling and Resonances

When a particle encounters two barriers in succession (a double-barrier structure), something remarkable happens: at certain energies, the transmission probability jumps to nearly unity, even though each individual barrier is nearly opaque. This is resonant tunneling, and it occurs when the energy of the incident particle matches a quasi-bound state of the well between the barriers.

The phenomenon is analogous to a Fabry-Pérot interferometer in optics: the waves bouncing back and forth between the two barriers interfere constructively at resonant energies. The WKB analysis of resonant tunneling combines the single-barrier transmission formula with a quantization condition for the inter-barrier well, yielding:

$$T_{\text{double}} \approx \frac{T_1 T_2}{|1 - \sqrt{R_1 R_2}\, e^{2i\phi}|^2}$$

where $\phi$ is the WKB phase accumulated in one round trip between the barriers. At resonance ($\phi = n\pi$), $T_{\text{double}} \approx 1$ if the two barriers are identical ($T_1 = T_2$). This is the principle behind the resonant tunneling diode, a real semiconductor device.

20.7 Alpha Decay: Gamow's Theory

20.7.1 The Puzzle of Alpha Decay

By the late 1920s, nuclear physics faced a baffling puzzle. Certain heavy nuclei — uranium, thorium, radium, and others — spontaneously emit alpha particles ($^4_2$He nuclei) with well-defined kinetic energies. But the energies of the emitted alphas (typically 4–9 MeV) were far below the height of the Coulomb barrier surrounding the nucleus (typically 25–30 MeV). Classically, an alpha particle with 5 MeV of kinetic energy simply cannot escape from a 30-MeV barrier. It would be like a ball rolling partway up a hill, stopping, and then appearing on the other side.

Even more puzzling was the Geiger-Nuttall law, an empirical relation discovered in 1911:

$$\log_{10} t_{1/2} = a + b/\sqrt{E_\alpha}$$

where $t_{1/2}$ is the half-life and $E_\alpha$ is the kinetic energy of the emitted alpha. Small changes in $E_\alpha$ produced enormous changes in half-life: a factor of 2 change in energy could change the half-life by 20 orders of magnitude. No classical mechanism could explain such extreme sensitivity.

20.7.2 Gamow's Model

In 1928, George Gamow (and, independently, Ronald Gurney and Edward Condon) solved this puzzle using quantum tunneling. The model is elegant in its simplicity.

The potential: Inside the nucleus ($r < R$), the alpha particle is bound by the strong nuclear force, modeled as a constant potential $-V_0$ (a deep well). Outside the nucleus ($r > R$), the alpha particle feels only the Coulomb repulsion:

$$V(r) = \frac{2Ze^2}{4\pi\epsilon_0 r} = \frac{kZe^2}{r} \quad (r > R)$$

where $Z$ is the atomic number of the daughter nucleus (the nucleus left behind after the alpha escapes) and the factor of 2 is the alpha particle charge. Here we use the notation $k = 1/(4\pi\epsilon_0)$ for Coulomb's constant.

The Coulomb barrier height at the nuclear surface is:

$$V(R) = \frac{2kZe^2}{R}$$

For $^{238}$U: $Z_{\text{daughter}} = 90$ (thorium), $R \approx 7.4$ fm, giving $V(R) \approx 35$ MeV. The emitted alpha has energy $E_\alpha = 4.27$ MeV — far below the barrier.

The turning points: The inner turning point is at the nuclear radius $r_1 = R$. The outer turning point is where $E = V(r)$:

$$r_2 = \frac{2kZe^2}{E_\alpha}$$

For $^{238}$U: $r_2 \approx 61$ fm, so the forbidden region extends from $\sim 7$ fm to $\sim 61$ fm.

20.7.3 Computing the Gamow Factor

The WKB tunneling exponent is:

$$\gamma = \frac{1}{\hbar}\int_R^{r_2}\sqrt{2m_\alpha\left(\frac{2kZe^2}{r} - E\right)}\,dr$$

where $m_\alpha = 4 \times 1.66 \times 10^{-27}$ kg $= 6.64 \times 10^{-27}$ kg is the alpha particle mass. This integral can be evaluated analytically by substituting $r = r_2\cos^2\theta$:

$$\gamma = \frac{\sqrt{2m_\alpha E}}{\hbar}\, r_2\left[\arccos\sqrt{\frac{R}{r_2}} - \sqrt{\frac{R}{r_2}\left(1 - \frac{R}{r_2}\right)}\right]$$

In the limit $R \ll r_2$ (which is typically a good approximation), this simplifies to:

$$\gamma \approx \frac{\pi kZe^2}{\hbar}\sqrt{\frac{2m_\alpha}{E}} - \frac{2}{\hbar}\sqrt{2m_\alpha \cdot 2kZe^2 R}$$

The first term dominates and depends on $1/\sqrt{E}$. This is the origin of the Geiger-Nuttall law.

20.7.4 The Decay Rate

The alpha particle bounces back and forth inside the nucleus with a frequency:

$$f \approx \frac{v}{2R} = \frac{\sqrt{2E/m_\alpha}}{2R}$$

Each time it hits the barrier, it has a probability $T = e^{-2\gamma}$ of escaping. The decay rate is therefore:

$$\lambda = f \cdot T = \frac{v}{2R}\,e^{-2\gamma}$$

and the half-life is:

$$\boxed{t_{1/2} = \frac{\ln 2}{\lambda} = \frac{2R\ln 2}{v}\, e^{2\gamma}}$$

20.7.5 The Geiger-Nuttall Law Derived

Since $\gamma \propto Z/\sqrt{E}$, we have $\log t_{1/2} \propto 2\gamma \propto Z/\sqrt{E}$, which is precisely the Geiger-Nuttall law. For a series of isotopes with similar $Z$ and $R$:

$$\log_{10} t_{1/2} = A + \frac{B}{\sqrt{E_\alpha}}$$

where $A$ and $B$ are approximately constant within a decay series. This explains the extraordinary sensitivity of the half-life to the alpha energy: a small change in $E_\alpha$ changes the argument of the exponential, and the exponential amplifies that change enormously.

20.7.6 Numerical Results

Let us compute the half-life of $^{238}$U.

Given: $Z = 90$, $E_\alpha = 4.27$ MeV, $R = 7.4$ fm, $m_\alpha = 3727.4$ MeV/$c^2$.

Outer turning point:

$$r_2 = \frac{2kZe^2}{E_\alpha} = \frac{2(1.44\,\text{MeV}\cdot\text{fm})(90)}{4.27\,\text{MeV}} = 60.7\,\text{fm}$$

(using $ke^2 = 1.44$ MeV$\cdot$fm).

Gamow exponent (evaluating the integral numerically or using the analytical formula):

$$2\gamma \approx 88.2$$

Bounce frequency:

$$v = \sqrt{\frac{2E_\alpha}{m_\alpha}}c = \sqrt{\frac{2 \times 4.27}{3727.4}}\, c \approx 0.048c \approx 1.43 \times 10^7\,\text{m/s}$$

$$f = \frac{v}{2R} = \frac{1.43 \times 10^7}{2 \times 7.4 \times 10^{-15}} \approx 9.7 \times 10^{20}\,\text{Hz}$$

Decay rate:

$$\lambda = f \cdot e^{-2\gamma} = 9.7 \times 10^{20} \times e^{-88.2} \approx 9.7 \times 10^{20} \times 3.3 \times 10^{-39} \approx 3.2 \times 10^{-18}\,\text{s}^{-1}$$

Half-life:

$$t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{3.2 \times 10^{-18}} \approx 2.2 \times 10^{17}\,\text{s} \approx 6.9 \times 10^9\,\text{yr}$$

The experimental value is $4.47 \times 10^9$ years. Our WKB estimate is off by a factor of $\sim 1.5$ — remarkably good considering the $10^{39}$ factor in the exponential. The WKB result captures the correct order of magnitude of a number spanning 40 decades.

📊 By the Numbers: WKB predictions for alpha-decay half-lives:

Nucleus	$E_\alpha$ (MeV)	$t_{1/2}$ (WKB)	$t_{1/2}$ (expt)	Ratio
$^{238}$U	4.27	$6.9 \times 10^9$ yr	$4.5 \times 10^9$ yr	1.5
$^{232}$Th	4.08	$2.0 \times 10^{10}$ yr	$1.4 \times 10^{10}$ yr	1.4
$^{226}$Ra	4.87	$2.3 \times 10^3$ yr	$1.6 \times 10^3$ yr	1.4
$^{212}$Po	8.95	$0.6 \times 10^{-6}$ s	$0.3 \times 10^{-6}$ s	2

The WKB/Gamow model works across 24 orders of magnitude in half-life, from microseconds to tens of billions of years. The ratio of WKB to experiment is consistently of order unity. This is one of the great triumphs of quantum mechanics.

🔵 Historical Note: George Gamow was 24 years old when he published his alpha decay theory in 1928 — just two years after Schrödinger's equation. He had traveled from Leningrad to Göttingen on a fellowship and, during a conversation with Fritz Houthermans, realized that quantum tunneling could explain the puzzle of alpha decay that had baffled nuclear physicists for decades. The paper electrified the physics community. It was one of the first applications of quantum mechanics to nuclear physics and remains one of the most dramatic successes of the theory. Gamow later applied the same tunneling idea in reverse — to proton-proton fusion in stars — and became a pioneer of nuclear astrophysics.

20.7.7 Why Alpha Particles?

A natural question: why do nuclei preferentially emit alpha particles rather than individual protons, neutrons, or other light nuclei?

The answer lies in the exceptional stability of the alpha particle. The $^4$He nucleus has a binding energy of 28.3 MeV — the highest binding energy per nucleon of any light nucleus. This means the alpha particle can form as a quasi-bound cluster inside the heavy nucleus with enough kinetic energy to have a reasonable tunneling probability. A single proton, by contrast, faces a lower Coulomb barrier (charge $e$ vs. $2e$) but has less kinetic energy available from the nuclear rearrangement, and the lighter mass further suppresses the tunneling integral. The alpha particle hits the sweet spot: enough energy to have a thin-enough barrier, and enough structural stability to exist as a coherent entity inside the nucleus.

⚖️ Interpretation: The Gamow model treats the alpha particle as pre-formed inside the nucleus, bouncing back and forth like a billiard ball. This is a simplification — in reality, the alpha particle assembles and disassembles continuously through correlations among the nucleons. A more sophisticated treatment introduces a "spectroscopic factor" $S_\alpha \leq 1$ representing the probability that the alpha cluster exists at any given time. This factor modifies the decay rate by an order-unity amount, consistent with the factor-of-2 discrepancies in our table.

20.7.8 Running the Gamow Model Backward: Stellar Nucleosynthesis

Gamow himself recognized that his tunneling calculation could be applied in reverse. In alpha decay, a particle tunnels out of a potential well through the Coulomb barrier. In nuclear fusion, a particle must tunnel into the well by penetrating the same barrier from the outside.

In the core of the Sun, hydrogen nuclei (protons) have thermal energies of $\sim k_BT \approx 1.3$ keV at $T = 1.5 \times 10^7$ K. The Coulomb barrier for two protons at nuclear contact ($r \approx 1$ fm) is:

$$V(r) = \frac{ke^2}{r} = \frac{1.44\,\text{MeV}\cdot\text{fm}}{1\,\text{fm}} = 1.44\,\text{MeV}$$

The protons must tunnel through a barrier that is a thousand times their kinetic energy. The tunneling probability is exponentially small — of order $e^{-2\gamma}$ with $2\gamma \sim 20$ for the most probable energy — but the Sun contains $\sim 10^{57}$ protons, and they collide $\sim 10^{36}$ times per second per cubic meter. The product of an astronomically large number of attempts and an astronomically small probability per attempt is a modest, steady fusion rate that powers the Sun at exactly the observed luminosity of $L_\odot = 3.8 \times 10^{26}$ W.

Without quantum tunneling, the Sun's core temperature would need to be $\sim 10^{10}$ K (instead of $10^7$ K) for protons to classically surmount the Coulomb barrier. At its actual temperature, the classical fusion rate would be identically zero. The Sun shines because of quantum tunneling — and the WKB formula gives us the quantitative rate.

🧪 Experiment: The nuclear reactions that power the Sun produce neutrinos (the "solar neutrinos" that Davis, Koshiba, and others detected starting in the 1960s). The predicted neutrino flux depends sensitively on the tunneling rate, which enters through the Gamow factor. The solar neutrino problem — the discrepancy between predicted and observed neutrino fluxes — was ultimately resolved not by revising the tunneling calculation (which was correct) but by discovering neutrino oscillations (Chapter 11 of any modern particle physics text). The WKB tunneling rate was vindicated.

20.8 Summary and Progressive Project

20.8.1 Chapter Summary

The WKB approximation is the bridge between classical and quantum mechanics. It gives us:

1. WKB wavefunctions — approximate solutions valid when the potential varies slowly compared to the local de Broglie wavelength: - Classically allowed: $\psi \propto p^{-1/2}\exp(\pm i\int p\,dx/\hbar)$ - Classically forbidden: $\psi \propto \kappa^{-1/2}\exp(\pm\int\kappa\,dx)$

2. Connection formulas — relations linking the WKB solutions across classical turning points, derived via Airy function matching. Each turning point contributes a $\pi/4$ phase shift.

3. Bohr-Sommerfeld quantization — $\oint p\,dx = (n + \frac{1}{2})h$ — a semiclassical quantization rule that is exact for the harmonic oscillator and increasingly accurate for all potentials at high quantum numbers.

4. WKB tunneling — $T \approx e^{-2\gamma}$ where $\gamma = \hbar^{-1}\int\sqrt{2m(V-E)}\,dx$ — an exponentially accurate estimate of the transmission coefficient through an arbitrary barrier.

5. Gamow's alpha decay theory — the WKB tunneling formula applied to the nuclear Coulomb barrier explains half-lives spanning 24 orders of magnitude through a single integral.

20.8.2 What We Have Learned and What It Means

This chapter represents a philosophical turning point in our study of approximation methods. Perturbation theory (Chapter 17) and the variational method (Chapter 19) are powerful but they require input — a solvable reference system or a clever trial wavefunction. The WKB approximation requires only that the potential vary slowly, and it gives you everything: wavefunctions, energy levels, tunneling rates, and a deep connection to classical mechanics.

The physical content of the WKB approximation can be distilled to three ideas:

1. Quantum mechanics remembers classical mechanics. The phase of the WKB wavefunction is the classical action. The amplitude is the classical dwell time. The quantization condition is the old Bohr-Sommerfeld rule with a quantum correction. Classical mechanics is not separate from quantum mechanics — it is the $\hbar \to 0$ skeleton that quantum mechanics dresses with wave phenomena.

2. Turning points are where the magic happens. All the specifically quantum corrections in the WKB framework — the $\pi/4$ phase shifts, the zero-point energy, the very existence of tunneling — come from the turning points, the boundaries between allowed and forbidden regions. Away from turning points, WKB is essentially classical. The turning points are where quantum mechanics asserts its independence from the classical limit.

3. Tunneling is exponentially sensitive. The most important practical result of this chapter — the tunneling formula $T \approx e^{-2\gamma}$ — tells us that barrier penetration depends exponentially on an integral over the forbidden region. This single fact explains phenomena spanning 40 orders of magnitude in alpha-decay lifetimes, the operation of the scanning tunneling microscope, the fusion reactions that power stars, and the proton-transfer reactions that drive enzyme catalysis.

20.8.3 The Big Picture

The WKB approximation occupies a special place in the hierarchy of approximation methods:

WKB is the method of choice when you need tunneling rates through complicated barriers, when you want to understand the classical limit of a quantum system, or when you need quick estimates of high-lying energy levels.

🔗 Connection: In Chapter 22, we will study scattering theory, where the WKB method provides an excellent framework for understanding the phase shifts of partial waves in the semiclassical regime. The WKB phase shift is $\delta_l \approx \int [p(r)/\hbar - k]\,dr$, connecting tunneling and scattering in a unified semiclassical picture.

20.8.4 Looking Forward

The WKB approximation reveals a deep truth about quantum mechanics: classical mechanics is not separate from quantum mechanics but contained within it as a limiting case. The path from WKB to Feynman's path integral formulation (Chapter 31) is direct — the WKB phase $\int p\,dx/\hbar$ is the classical action divided by $\hbar$, which is precisely the phase that dominates the path integral in the semiclassical limit. The principle of stationary phase applied to the path integral is the WKB approximation. We will return to this beautiful connection in Chapter 31.

The WKB method also connects naturally to scattering theory (Chapter 22). In the partial-wave expansion for scattering, each angular momentum channel involves a one-dimensional radial equation with an effective potential. The WKB approximation gives the phase shift for each partial wave in terms of a classical integral — the deviation of the classical scattering trajectory from a straight line. At high energies and large angular momenta, these WKB phase shifts become increasingly accurate, providing a bridge between quantum scattering and classical deflection functions.

Finally, the semiclassical ideas of this chapter will reappear in a more sophisticated form when we study the adiabatic approximation and Berry's phase (Chapter 32). The WKB phase $\int p\,dx/\hbar$ is the dynamical phase; Berry's phase is its geometric counterpart. Together, they constitute the full semiclassical description of quantum evolution.

20.8.5 Progressive Project: WKB Module for the Quantum Simulation Toolkit

Toolkit version: v2.0 (Ch 20 checkpoint)

In this chapter's project component, you will build the WKB approximation module for the Quantum Simulation Toolkit. The module should include:

1. turning_points(V, E, x_grid) — Given a potential function and energy, find the classical turning points.

2. wkb_wavefunction(V, E, x_grid, region='allowed') — Compute the WKB wavefunction in allowed or forbidden regions.

3. gamow_exponent(V, E, x_grid) — Compute the Gamow tunneling exponent $\gamma = \hbar^{-1}\int_{x_1}^{x_2}\sqrt{2m(V-E)}\,dx$.

4. tunneling_rate(V, E, x_grid) — Return $T = e^{-2\gamma}$.

5. bohr_sommerfeld(V, x_grid, n_max) — Find the first $n_{\max}$ energy levels using the WKB quantization condition.

6. alpha_decay(Z, A, E_alpha, R=None) — Compute the Gamow factor and estimated half-life for alpha decay.

Verification tests: - Harmonic oscillator: WKB energies should be exact (to numerical precision) - Compare WKB tunneling with exact rectangular barrier result from Chapter 3 - Reproduce the $^{238}$U half-life to within a factor of 2

See code/project-checkpoint.py for the implementation.

Key Equations Summary

In the next chapter, we turn to time-dependent perturbation theory (Chapter 21), where we will study how quantum systems respond to time-varying potentials — absorbing and emitting radiation, transitioning between states, and obeying Fermi's golden rule.