Questions: Quantum Tunneling and Barrier Penetration

Question 1

A hydrogen atom and a deuterium atom (mass ≈ 2× hydrogen) face identical energy barriers in an enzyme active site. Classical transition-state theory predicts nearly identical reaction rates. Which experimental observation is most consistent with quantum tunneling?

• A. Both atoms react at the same rate — particle mass does not affect tunneling probability
• B. Deuterium reacts faster because heavier particles have more momentum to push through the barrier
• C. Hydrogen reacts significantly faster, producing a kinetic isotope effect larger than classical theory predicts
• D. Hydrogen reacts slightly faster only at high temperatures where tunneling is negligible anyway

Answer: (C)

Tunneling probability depends exponentially on particle mass: T ∝ exp(−2a√(2m(V₀−E))/ℏ). Because mass appears under a square root in the exponent, doubling the mass substantially reduces tunneling probability. The result is an anomalously large kinetic isotope effect (kH/kD >> 7) that cannot be explained by classical transition-state theory — a hallmark signature of tunneling.

Question 2

What actually happens to a quantum particle's wave function when it encounters a barrier where E < V (classically forbidden region)?

• A. The wave function abruptly drops to zero at the barrier boundary, reflecting the classical impossibility
• B. The wave function oscillates within the barrier at a higher frequency to conserve energy
• C. The wave function decays exponentially inside the barrier but remains non-zero, allowing non-zero amplitude on the far side if the barrier is thin
• D. The particle momentarily gains kinetic energy from quantum fluctuations to surmount the barrier

Answer: (C)

In the classically forbidden region, the Schrödinger equation yields real exponential solutions rather than oscillating ones. The wave function decays as exp(−κx) where κ = √(2m(V₀−E))/ℏ, but it does not vanish — it threads through and emerges on the far side. Tunneling is a direct consequence of this non-zero amplitude, not of energy fluctuations or going around the barrier.

Question 3

A quantum particle with total energy less than the barrier height has a non-zero probability of appearing on the far side of a sufficiently thin barrier.

• T. True
• F. False

Answer: True

This is the defining statement of quantum tunneling. Because the wave function decays exponentially rather than vanishing inside the barrier, it emerges on the far side with reduced amplitude — which means non-zero probability of transmission. This probability approaches zero for very wide or tall barriers but is physically significant for light particles (especially hydrogen) tunneling through narrow barriers.

Question 4

Quantum tunneling allows a particle to bypass an energy barrier by briefly borrowing energy from its surroundings to surmount the barrier classically.

• T. True
• F. False

Answer: False

Tunneling does not involve energy borrowing. The particle penetrates the barrier without ever exceeding the barrier height — its total energy remains below V₀ throughout. The correct picture is wave-mechanical: the particle's wave function threads through the classically forbidden region with exponentially decaying amplitude. Energy is conserved; only the probabilistic interpretation of the wave function changes.

Question 5

Why does doubling the width of an energy barrier reduce tunneling probability far more dramatically than doubling its height, even though both intuitively make the barrier 'harder' to penetrate?

Model answer: Tunneling probability follows T ∝ exp(−2a√(2m(V₀−E))/ℏ), where a is the barrier width. Width enters the exponent linearly — doubling a doubles the exponent and squares T (e.g., if T = e⁻¹⁰, doubling width gives e⁻²⁰ = T²). Height appears only under a square root, so doubling (V₀−E) increases the exponent by only a factor of √2. This exponential sensitivity to width explains why tunneling is chemically relevant only for very thin barriers and very light particles.

The practical consequence: small changes in the distance between donor and acceptor atoms in enzyme active sites — even a fraction of an ångström — can change hydrogen transfer rates by orders of magnitude. Enzyme evolution can exploit this by precisely positioning reactive groups to minimize tunneling distances.