Questions: Tunneling Probability and Transmission Coefficient Calculations

Question 1

A particle tunnels through a barrier of width L with transmission coefficient T. The barrier width is doubled to 2L. How does T change?

• A. T is halved — T is proportional to 1/L
• B. T drops to T² — doubling L doubles the exponent in exp(−2κL), squaring the original T
• C. T drops by a factor of exp(−2κL) — the same factor as the original T
• D. T is unchanged if the particle energy stays the same

Answer: (B)

T ≈ exp(−2κL). If L doubles, the exponent becomes −2κ(2L) = −4κL = 2 × (−2κL). So the new T = exp(−4κL) = [exp(−2κL)]² = T². The transmission coefficient is squared, not halved. This exponential sensitivity is the defining feature of tunneling: linear changes in barrier parameters produce exponential changes in T. A naive linear-scaling intuition (T ∝ 1/L) would dramatically overestimate the tunneling probability for wider barriers.

Question 2

The scanning tunneling microscope achieves atomic resolution because of which property of quantum tunneling?

• A. Tunneling current depends linearly on tip-to-surface distance, giving a sensitive but smooth signal
• B. Only specific atoms at the surface are quantum-mechanically allowed to contribute to tunneling current
• C. Tunneling current depends exponentially on tip-to-surface distance, so even a single-atom height change produces a measurable current change
• D. The tunneling wavefunction is concentrated at atomic positions, creating a map of electron density

Answer: (C)

The STM's extraordinary resolution comes directly from the exponential dependence of tunneling current on distance. Moving the tip 0.1 nm closer roughly doubles the current; 0.1 nm farther roughly halves it. A single-atom bump — a height change of ~0.2 nm — changes the current by roughly a factor of four. This extreme sensitivity means the STM effectively traces the atomic-scale topography of the surface. If the dependence were linear, height changes of 0.1 nm would produce much smaller fractional current changes and atomic resolution would be impossible.

Question 3

A proton and an electron, both with the same kinetic energy, approach an identical potential barrier. The proton has much lower transmission probability than the electron.

• T. True
• F. False

Answer: True

The decay constant κ = √(2m(V₀−E))/ℏ depends on particle mass m. A proton is about 1836 times heavier than an electron, so its κ is larger by √1836 ≈ 43. The transmission coefficient T ≈ exp(−2κL) falls exponentially with κ, so the proton's T is astronomically smaller than the electron's T for the same barrier. This mass dependence explains why quantum tunneling is observable at atomic and nuclear scales for light particles (electrons, alpha particles) but completely negligible for macroscopic objects.

Question 4

For a particle with energy E approaching a barrier of height V₀, if E is very close to but still below V₀, the transmission coefficient T approaches 1.

• T. True
• F. False

Answer: False

T ≈ exp(−2κL) with κ = √(2m(V₀−E))/ℏ. As E → V₀ from below, κ → 0, so exp(−2κL) → exp(0) = 1. For a thin barrier, T does indeed approach 1 as E → V₀. However, for a thick barrier (large L), even with small κ, the exponent −2κL can still be large and T can remain small. The claim that T 'approaches 1' requires either a thin barrier or E very close to V₀. The exponential dependence on L means T can be very small even when E is nearly equal to V₀ if the barrier is thick enough.

Question 5

Explain why the exponential dependence of the transmission coefficient on barrier parameters means that quantum tunneling is observable at the atomic scale but completely negligible for macroscopic objects.

Model answer: T ≈ exp(−2κL) where κ = √(2m(V₀−E))/ℏ. For macroscopic objects, the mass m is enormous — say 10⁻³ kg versus 10⁻³⁰ kg for an electron. Since κ ∝ √m, the macroscopic κ is roughly 10¹³ times larger than for an electron. With even a nanometer-wide barrier, 2κL becomes a number so large that exp(−2κL) is effectively zero — far smaller than 1/10^(10^26). The exponential amplifies the mass difference into an utterly negligible probability. For electrons, κ is small enough that T is measurable for barriers a few nanometers wide.

The core insight is that the exponential function amplifies small differences in the exponent into astronomical differences in T. A factor-of-13 difference in κ (say, between a proton and an electron) leads to a factor of exp(−26L) difference in T — for any moderate L, this is an incomprehensibly small number. For macroscopic masses, the exponent is so large that tunneling is not merely rare but physically impossible on any timescale relevant to observation.