An electron with energy E encounters a potential barrier where V > E over a region of width L. According to the WKB approximation, which factor most strongly governs the tunneling probability?
AThe kinetic energy of the electron at the center of the barrier
BThe frequency of the electron's wavefunction oscillation outside the barrier
CThe group velocity of the electron's wavepacket approaching the barrier
DThe exponential factor exp(−2∫|p|dx/ℏ), which depends on both the barrier height and width
The WKB tunneling probability is T ∝ exp(−2∫|p|dx/ℏ), where |p| = √(2m(V−E)) is the imaginary local momentum inside the barrier. This exponential factor depends on both the height (V−E, which determines |p|) and the width (the integration range L) of the barrier. Both parameters enter exponentially, so even modest increases in barrier height or width drastically reduce tunneling probability. This exponential suppression is why tunneling is a quantum phenomenon with no classical analog.
Question 2 Multiple Choice
A physics student says the WKB approximation is best applied when a particle's potential energy changes very rapidly — varying significantly over distances much shorter than the de Broglie wavelength. Is this correct?
AYes — rapidly varying potentials require an approximation method, which is exactly what WKB provides
BYes — the WKB method was developed specifically for step-function and rapidly varying potentials
CNo — WKB requires the potential (and hence the de Broglie wavelength) to change SLOWLY over one de Broglie wavelength; rapidly varying potentials violate the approximation's validity condition
DNo — WKB is only valid for constant potentials where exact solutions exist
The validity condition for WKB is precisely the opposite of what the student claims. WKB works when the de Broglie wavelength λ = h/p(x) varies slowly over one wavelength — formally, |dλ/dx| ≪ 1. This is the 'semiclassical limit': the potential changes so slowly that the wavefunction locally resembles a plane wave. When the potential changes rapidly (on scales comparable to λ), the approximation breaks down because the local momentum p(x) cannot be treated as nearly constant. Exact solutions are used at abrupt steps; WKB is used for smooth, slowly varying potentials.
Question 3 True / False
In the classically allowed region (E > V), the WKB wavefunction has amplitude proportional to 1/√p — larger where the particle moves slowly and smaller where it moves fast — which reflects conservation of probability current.
TTrue
FFalse
Answer: True
Probability current J = |ψ|² × v must be constant for a steady-state solution. Since v ∝ p, we need |ψ|² ∝ 1/p, giving |ψ| ∝ 1/√p. Where the particle moves slowly (small p, near a turning point), the wavefunction amplitude is large — the particle 'spends more time' there. Where it moves fast (large p), the amplitude is small. This is the quantum analog of how a classical particle decelerates near turning points and spends more time in low-kinetic-energy regions.
Question 4 True / False
The WKB approximation remains accurate at classical turning points where E = V(x), since the potential is varying smoothly and continuously at those locations.
TTrue
FFalse
Answer: False
Classical turning points are precisely where WKB breaks down, even though the potential is smooth there. At a turning point, E = V(x), so p(x) = √(2m(E−V)) = 0. The WKB amplitude formula 1/√p diverges at p = 0, signaling failure of the approximation. Physically, the de Broglie wavelength λ = h/p also diverges — the 'slowly varying wavelength' condition collapses. Airy functions are required to connect the oscillating (classically allowed) and decaying (classically forbidden) WKB solutions across turning points.
Question 5 Short Answer
Why does the WKB approximation break down at classical turning points, and what happens to the wavefunction amplitude formula at those points?
Think about your answer, then reveal below.
Model answer: At a classical turning point, E = V(x), so the local kinetic energy is zero and the local momentum p(x) = √(2m(E−V)) = 0. The WKB amplitude formula 1/√p diverges as p → 0, predicting infinite amplitude — which is unphysical. The approximation's validity condition (wavelength changes slowly over one wavelength) also fails here because λ = h/p → ∞. This breakdown requires using Airy functions — exact solutions to the Schrödinger equation near a linear turning point — to bridge between the oscillating WKB solution in the allowed region and the exponentially decaying solution in the forbidden region.
Turning points are the price WKB pays for being a local approximation. It works wherever the potential is nearly constant over one wavelength, but at turning points the wavelength itself becomes infinite and the local-momentum description fails. The connection formulas derived from Airy function analysis are what make WKB quantization possible: they enforce consistent boundary conditions around a bound state, yielding the Bohr-Sommerfeld quantization rule.