A particle with total energy E = 3 eV encounters a potential energy barrier of height V₀ = 5 eV and finite width. What does quantum mechanics predict?
AThe particle is reflected with 100% probability — it lacks sufficient energy to cross
BThe particle temporarily borrows 2 eV from the uncertainty principle to climb over the barrier
CThere is a nonzero probability that the particle is transmitted through the barrier, with its energy unchanged at 3 eV throughout
DThe particle's energy increases to 5 eV inside the barrier, then drops back to 3 eV on the far side
Tunneling does not involve energy change. The particle's energy is conserved at 3 eV throughout — inside the barrier, outside it, and in the transmitted wave. What changes inside the barrier is the character of the wavefunction: instead of oscillating, it decays exponentially. If the barrier is thin enough, this decaying tail has nonzero amplitude at the far wall, where it reconnects to a propagating wave. The 'borrowed energy' picture (options B and D) is a common but incorrect classical analogy. Quantum mechanically, the particle never 'climbs over' — it tunnels through.
Question 2 Multiple Choice
A scanning tunneling microscope can image individual atoms because tunneling current is exquisitely sensitive to gap distance. If the tip-to-surface gap doubles from 0.1 nm to 0.2 nm, what happens to the tunneling current?
AIt decreases by a factor of 2 — current is proportional to 1/distance
BIt decreases by a factor of 4 — current decreases as the square of distance
CIt decreases by roughly an order of magnitude or more — transmission probability decays exponentially with barrier width
DIt stays approximately the same — atomic-scale gap changes are too small to matter
Tunneling probability decays exponentially with barrier width: T ≈ e^{−2κL}. Doubling L from 0.1 to 0.2 nm roughly squares the transmission probability (e^{−2κ(0.2)} = (e^{−2κ(0.1)})²), resulting in a massive drop in tunneling current — roughly an order of magnitude per 0.1 nm of gap change in typical STM conditions. This extreme sensitivity (not a gentle linear or quadratic falloff) is precisely what gives the STM its sub-angstrom height resolution. Linear or inverse-square laws would give far too gradual a response to detect individual atomic steps.
Question 3 True / False
When a particle quantum-tunnels through a barrier, its total energy is momentarily higher than usual inside the barrier, allowing it to pass through the classically forbidden region.
TTrue
FFalse
Answer: False
This is the most common misconception about tunneling. The particle's total energy is constant throughout the process — it equals E before, during, and after tunneling. There is no energy borrowing or violation of energy conservation. Inside the barrier (where E < V₀), the wavefunction decays exponentially rather than oscillating, but this is a property of the mathematical solution to the Schrödinger equation in that region, not a sign that energy has changed. The Heisenberg uncertainty principle provides intuition for why tunneling is possible but does not license an energy-violation picture.
Question 4 True / False
A heavier particle tunnels through a given barrier more easily than a lighter particle with the same energy and the same barrier.
TTrue
FFalse
Answer: False
Mass appears in the exponent of the tunneling probability: κ = √(2m(V₀ − E))/ℏ. A larger mass means a larger κ, which means a more steeply decaying wavefunction inside the barrier, which means much less transmission. Electrons tunnel readily; protons (about 1800× heavier) tunnel far less; alpha particles (7000× heavier than electrons) tunnel least readily of all. The exponential sensitivity to mass is why alpha decay rates vary so dramatically across isotopes, and why nuclear fusion requires enormous temperatures or pressures to bring heavy nuclei close enough for their wavefunctions to overlap.
Question 5 Short Answer
Why does tunneling probability decrease exponentially (rather than linearly or gradually) as the barrier width increases?
Think about your answer, then reveal below.
Model answer: Inside the barrier, the wavefunction decays exponentially as ψ(x) ∝ e^{−κx}, because the Schrödinger equation in a classically forbidden region (E < V₀) has real exponential solutions rather than oscillating ones. The tunneling probability is proportional to |ψ|² at the far wall, so T ≈ e^{−2κL}. Each additional increment of barrier width multiplies the already-reduced amplitude by another exponential factor, compounding the suppression. This is inherent to the mathematics of exponential decay: unlike linear decay, each doubling of width squares the transmission probability rather than halving it.
This exponential sensitivity is not a nuisance — it is the feature that makes tunneling devices useful. The STM exploits it for atomic-resolution imaging; tunnel diodes exploit it for ultra-fast switching. It also explains why alpha decay rates span twenty orders of magnitude across different isotopes despite similar energies: small differences in the Coulomb barrier height and width translate to enormous differences in the exponential factor, producing half-lives that range from microseconds to billions of years.