The WKB Approximation

Perturbation theory and the variational method both start from a solvable problem. The WKB approximation, named for Wentzel, Kramers, and Brillouin, starts instead from the classical limit. When the potential changes slowly enough that the de Broglie wavelength is nearly constant over one wavelength, the wavefunction is locally sinusoidal with a wavelength set by the local classical momentum. The method builds the full wavefunction by stitching these local plane waves together, fails only at the turning points where the classical momentum vanishes, and is patched there by matching to an exact local solution. It yields the quantization condition of the old quantum theory with the correct half-integer offset, and the tunneling rate through a barrier of arbitrary shape.

The semiclassical wavefunction

In a region where $E > V (x)$ the classical momentum $p (x) = 2 m [E - V (x)]$ is real. Write the stationary wavefunction in amplitude–phase form $ψ (x) = A (x) e^{i ϕ (x)}$ and substitute into the time-independent Schrödinger equation $ψ^{''} = - (p^{2} / ℏ^{2}) ψ$ . Separating real and imaginary parts and dropping the term with $A^{''}$ (small when the amplitude varies slowly) gives two equations: the phase obeys $ϕ^{'} (x) = \pm p (x) /ℏ$ , and the amplitude obeys $(A^{2} ϕ^{'})^{'} = 0$ , so $A \propto 1/ p$ .

The amplitude law has a transparent classical reading. The probability density is $∣ ψ ∣^{2} \propto 1/ p \propto 1/ v$ : the particle is most likely to be found where it moves slowest, exactly as a classical particle spends the most time in the slow parts of its orbit. The quantum wavefunction inherits the classical dwelling time.

In the classically allowed region the WKB wavefunction oscillates with a local wavelength set by the momentum and an amplitude that swells toward the turning points where the particle slows and the classical dwelling time grows.

The validity condition

The neglected term $A^{''}$ is small precisely when the wavelength changes little over one wavelength. Quantifying that gives the domain of the method.

The condition fails wherever $p \to 0$ , that is at the classical turning points where $E = V (x)$ . There the wavelength diverges, the amplitude $1/ p$ blows up, and the local-plane-wave picture is meaningless. Every WKB calculation is organized around these points: the wavefunction is built from WKB pieces in the regions between them and repaired across them.

The classically forbidden region

Where $E < V (x)$ the momentum is imaginary. Writing $∣ p (x)∣ = 2 m [V (x) - E]$ , the same derivation gives real exponentials instead of oscillations,

ψ (x) \approx \frac{C}{∣ p ( x )∣} exp \pm \frac{1}{ℏ} \int x ∣ p (x^{'})∣ d x^{'} .

The decaying solution is the tunneling tail: a wavefunction penetrating a barrier is attenuated by the exponential of the accumulated $∣ p ∣$ . The growing solution is discarded for a barrier of finite width only after matching, since it would dominate; inside a wide barrier the decaying piece carries the transmitted amplitude.

Connection formulas

At a turning point the WKB forms on the two sides — oscillatory where allowed, exponential where forbidden — must be joined. Near an isolated turning point $x = a$ the potential is nearly linear, $V (x) \approx E + V^{'} (a) (x - a)$ , and the Schrödinger equation reduces to the Airy equation, whose solution is known exactly. Matching the Airy function's asymptotic forms to the WKB expressions on each side fixes the relative amplitudes and, decisively, a phase.

The $π /4$ phase shift at each turning point is the content of the connection formula and the origin of the half-integer in the quantization rule below. It records the smooth handoff, through the Airy region, between an oscillation and an exponential.

Across a turning point the linearized potential gives an exact Airy solution that interpolates the oscillatory allowed-side WKB wave and the decaying forbidden-side exponential, contributing a fixed phase lag of a quarter cycle.

The quantization condition

For a particle bound in a well with two turning points $a < b$ , the allowed region lies between them and the forbidden regions on either side demand decaying solutions. Applying the connection formula at each turning point yields a cosine referenced to $a$ and another referenced to $b$ ; consistency of the single wavefunction in the middle forces the total accumulated phase to be a multiple of $π$ .

The rule is the old quantum theory's $\oint p d x = nh$ corrected by the half-integer. Each of the two soft turning points contributes a $π /4$ phase, together shifting $n$ to $n + \frac{1}{2}$ . The integer $n$ counts the nodes of the wavefunction, so the quantization is a statement that a whole number of half-wavelengths, plus the two quarter-wave turning-point corrections, fits between the walls.

The quantization condition is the statement that the classical orbit encloses a half-integer number of units of Planck's constant in phase space; each closed loop in the x-p plane is an allowed state.

Tunneling and the Gamow factor

For a barrier where $V (x) > E$ between turning points $a$ and $b$ , the wavefunction decays across the classically forbidden region. The transmitted amplitude is suppressed by the exponential of the accumulated $∣ p ∣$ , and the transmission probability is its square.

The exponent $γ$ is the area under the forbidden-region momentum, so a wider or taller barrier suppresses tunneling exponentially. This is the semiclassical generalization of the rectangular-barrier result of the tunneling lesson to a barrier of any shape.

WKB tunneling accumulates the imaginary momentum across the forbidden region; the transmission is the exponential of twice the shaded area under the square-root of V minus E between the two turning points.

The Gamow factor governs processes from alpha decay, where the alpha particle tunnels through the Coulomb barrier of the daughter nucleus, to field emission and the ammonia inversion. In alpha decay the exponential dependence of $γ$ on energy converts a modest spread in decay energies into the enormous range of observed half-lives, the Geiger–Nuttall law, which the rectangular-barrier model cannot reproduce.

The three methods of this module cover the ground where exact solutions run out. Perturbation theory handles a small correction to a solved problem, the variational method bounds a ground state with no small parameter, and WKB works wherever the potential is smooth on the quantum scale. Together they turn the exactly solvable core of quantum mechanics into a toolkit for real atoms, molecules, and nuclei.