Tunnelling Through a Barrier

Tunnelling through a barrier is not a historical curiosity confined to textbooks; it is load-bearing machinery in modern chemistry, biology, and technology. In enzymology, hydrogen tunnelling explains rate enhancements and anomalously large kinetic isotope effects in enzymes such as alcohol dehydrogenase and lipoxygenase, where hydride or proton transfer occurs through, rather than over, an activation barrier — a phenomenon studied quantitatively using Marcus-like rate theories with tunnelling correction factors built directly on the transmission coefficient derived here. In DNA, proton tunnelling between base pairs has been proposed as a spontaneous mutation mechanism, since a tunnelled proton can transiently shift a base into a tautomeric form that mispairs during replication. In materials science and electronics, tunnel diodes, Josephson junctions, and flash memory (which relies on electrons tunnelling through a thin oxide layer to charge a floating gate) are direct engineering applications of the exponential $T(E)\propto e^{-2\kappa L}$ law. Scanning tunnelling microscopy, introduced in Example 3, remains the workhorse technique for imaging surfaces and molecules at atomic resolution, precisely because tunnelling current falls off so steeply with tip–sample distance that sub-angstrom height changes are directly readable as current changes. In astrophysics and nuclear chemistry, Gamow's original application persists: stellar nucleosynthesis rates (including the proton–proton fusion chain powering the Sun) depend on tunnelling through the Coulomb barrier between charged nuclei, and the "Gamow peak" energy distribution used in astrophysical reaction-rate calculations is a direct generalization of the transmission coefficient developed above. Finally, in the emerging area of quantum biology, electron tunnelling between redox cofactors in photosynthetic reaction centers and respiratory electron transport chains is now understood quantitatively via tunnelling pathway models (Dutton's ruler and related empirical relations), which are essentially the exponential barrier law recast with protein-specific decay constants replacing the free-particle $\kappa$.

Ask the tutor: Can you show how the WKB approximation generalizes this rectangular-barrier result to a smoothly varying, realistic reaction-coordinate barrier?

Consider a particle of mass $m$ moving along $x$ and encountering a rectangular potential energy barrier of height $V_0$ occupying the region $0 \le x \le L$, with $V=0$ elsewhere. A beam of particles with energy $E$ approaches from $x=-\infty$. Classically, if $E<V_0$, every particle reflects at $x=0$: the region inside the barrier is forbidden because kinetic energy would have to be negative there. Quantum mechanically the story is different because the Schrödinger equation is a linear differential equation for an amplitude, not a statement about classical trajectories. In region I ($x<0$) and region III ($x>L$), the potential is zero, and the time-independent Schrödinger equation gives plane-wave (oscillatory) solutions — these are the free-particle wavefunctions familiar from the box problem's exterior, except here there are no infinite walls forcing $\psi=0$. In region II ($0\le x \le L$), the equation becomes

Since $E<V_0$, the right-hand side is a negative number times $\psi$, so the second derivative has the same sign as $\psi$ itself. That is the mathematical signature of exponential behavior rather than oscillation: the general solution in region II is a combination of a growing exponential $e^{+\kappa x}$ and a decaying exponential $e^{-\kappa x}$, not $\sin$ and $\cos$. Because the barrier has finite width $L$, neither exponential is forced to vanish (unlike the infinite step, where only the decaying solution is physical) — both survive, matched at $x=0$ and $x=L$ to the oscillatory pieces outside. The crucial physical point is continuity: $\psi$ and $d\psi/dx$ must be continuous at $x=0$ and $x=L$ (finite potential jump means finite discontinuity in curvature but not in $\psi$ or $\psi'$). This continuity forces a nonzero amplitude to leak through to region III, where it re-emerges as an outgoing traveling wave carrying real transmitted probability current — the particle has tunnelled.

The wavefunction on the far side has a smaller amplitude than the incident wave — that ratio squared (with a velocity correction, trivial here since the outer regions have equal $V=0$) is exactly the transmission probability $T(E)$. Two limiting regimes are worth holding in mind before the algebra: if $L\to 0$ the barrier disappears and $T\to 1$; if $L\to\infty$ or $V_0\to\infty$, the decaying exponential kills the amplitude before it reaches $x=L$, and $T\to 0$, recovering the classical result and the infinite-wall limit of the box problem.

The full problem is solved by writing wavefunctions in each of the three regions and enforcing matching conditions. Define wavenumbers

where:

• $k$ is the real wavenumber of the oscillatory wave in the field-free regions (I and III),
• $\kappa$ is the real decay constant governing exponential behavior inside the barrier (region II), valid for $0<E<V_0$,
• $m$ is the particle mass, $\hbar$ is the reduced Planck constant.

Reading. $k$ has the same form as in the particle-in-a-box; $\kappa$ plays an analogous role but multiplies a real exponential instead of $i$ times a wavenumber in a complex exponential — the entire tunnelling phenomenon is encoded in this single sign flip inside the Schrödinger equation. The wavefunctions are

where:

• $A$ is the incident wave amplitude, $B$ the reflected wave amplitude (region I, $x<0$),
• $C,D$ are the growing/decaying exponential amplitudes inside the barrier (region II),
• $F$ is the transmitted wave amplitude (region III, $x>L$); no wave travels back from $+\infty$, so there is no $e^{-ikx}$ term there.

Reading. This is a scattering boundary condition: particles are launched from the left only, so region III contains a purely outgoing wave. The transmission probability is defined through the ratio of probability currents,

where: the prefactor velocities cancel because regions I and III have the same $k$ (equal $V=0$ asymptotics on both sides). The final closed-form result, obtained below by matching, is

where:

• $V_0$ is the barrier height, $L$ the barrier width,
• $E$ is the incident kinetic energy, restricted to $0<E<V_0$ for this branch (the $E>V_0$ case replaces $\sinh$ with $\sin$ and $\kappa$ with a real wavenumber inside the barrier),
• $\sinh(\kappa L) = \tfrac{1}{2}(e^{\kappa L}-e^{-\kappa L})$.

Reading. As $\kappa L$ grows — through increasing width $L$, increasing mass $m$, or increasing $(V_0-E)$ — $\sinh^2(\kappa L)$ grows essentially as $\tfrac14 e^{2\kappa L}$, and $T(E)$ decays exponentially. This single formula is the quantitative core of every tunnelling application in chemistry.

Example 1 — Foundational: electron tunnelling through an idealized rectangular barrier. Take an electron ($m_e = 9.109\times10^{-31}$ kg) with kinetic energy $E = 2.0$ eV incident on a barrier of height $V_0 = 5.0$ eV and width $L = 0.50$ nm — deliberately asymmetric numbers, not the common textbook "half of $V_0$" case. First compute $\kappa$:

Evaluate the numerator inside the square root:

Then $\kappa L$ with $L = 0.50\times10^{-9}$ m:

This is comfortably in the "thick" regime ($\kappa L \gg 1$), so use the exponential approximation. First the prefactor:

Now $e^{-2\kappa L} = e^{-8.876}$. Since $e^{-8.876} = e^{-8}\cdot e^{-0.876} = (3.3546\times10^{-4})(0.4164) = 1.397\times10^{-4}$,

So roughly 1 electron in 1860 tunnels through this barrier — small, but very far from the classical zero, and easily large enough to dominate device physics at the nanometer scale. As a sanity check on the approximation, one can verify with the exact formula that the additive "1" is indeed negligible: $\sinh^2(4.438) \approx \left(\tfrac12 e^{4.438}\right)^2 \approx (42.4)^2 \approx 1800$, so $V_0^2\sinh^2(\kappa L)/[4E(V_0-E)] = 25(1800)/(24) \approx 1880 \gg 1$, confirming the thick-barrier approximation is excellent here.

Example 2 — Applied: proton tunnelling in a hydrogen bond and the isotope effect. Enzymatic and hydrogen-bonded proton transfers are often modeled with a barrier of modest height and width. Take $V_0 = 0.30$ eV, $L = 0.080$ nm ($= 0.80$ Å, a realistic proton-transfer distance), and $E = 0.10$ eV for the proton's effective kinetic energy along the transfer coordinate. Compare the transmission probability for a proton ($m_p = 1.673\times10^{-27}$ kg) versus a deuteron ($m_d = 3.344\times10^{-27}$ kg, almost exactly twice $m_p$) at the same $E$ and barrier — the essence of a kinetic isotope effect. For the proton:

With $L = 0.080\times10^{-9}$ m,

Prefactor:

Since $e^{-15.71} = e^{-15}\cdot e^{-0.71} = (3.059\times10^{-7})(0.4916) = 1.504\times10^{-7}$,

For the deuteron, mass doubles, so $\kappa_d = \sqrt{2}\,\kappa_p$ (since $\kappa\propto\sqrt m$):

The prefactor is unchanged (it depends only on $E$ and $V_0$, not mass):

Since $e^{-22.22} = e^{-22}\cdot e^{-0.22} = (2.74\times10^{-10})(0.8025) = 2.199\times10^{-10}$,

The ratio, the essence of a primary kinetic isotope effect from tunnelling alone,

This shows how doubling the mass (H → D) can suppress the transmission probability by nearly three orders of magnitude, which is why tunnelling-dominated proton transfers show anomalously large H/D rate ratios (often far above the "semiclassical" ceiling of about 7 expected from zero-point energy differences alone) — direct experimental evidence that transfer is occurring by tunnelling rather than classical barrier-crossing. This connects directly to the zero-point energy ideas developed for Harmonic Oscillator Energy Levels and Zero-Point Energy, since the effective barrier height $V_0-E$ used here already reflects a zero-point-corrected energy along the transfer coordinate.

Example 3 — Exam/competition level: extracting barrier width from a measured STM tunnelling current ratio. A scanning tunnelling microscope tip is retracted from $d_1 = 0.30$ nm to $d_2 = 0.45$ nm above a metal surface, and the tunnelling current drops by a measured factor of 42.0 (i.e., $I_1/I_2 = 42.0$), at a fixed effective barrier height (work function) corresponding to $V_0 - E = 4.0$ eV for the tunnelling electron. Because tunnelling current is proportional to transmission probability, and the thick-barrier form gives $T\propto e^{-2\kappa d}$ with the same prefactor at both distances (only $d$ changes, not $E$ or $V_0$),

Solve for $\kappa$ from the measured ratio:

Now check this against the value expected from the stated barrier height $V_0-E = 4.0$ eV $= 6.408\times10^{-19}$ J, using the electron mass:

The measured $\kappa = 1.246\times10^{10}$ m$^{-1}$ is somewhat larger than the "clean metal" expectation of $1.025\times10^{10}$ m$^{-1}$, a roughly 22% discrepancy — physically this is the kind of deviation used to flag that the effective barrier height at the surface is a little higher than the nominal work function value (surface states, adsorbates, or tip chemistry can raise the effective barrier), illustrating how the tunnelling formula is used inversely, to extract barrier information from measured current-distance curves — the operating principle of STM as an atomic-resolution probe. This exponential dependence on distance (roughly a factor of $e^{2}\approx 7.4$ current change per 0.1 nm here) is exactly why STM achieves sub-angstrom vertical sensitivity: no other accessible physical effect is this exponentially sharp over such short length scales.

The rectangular-barrier tunnelling calculation is a direct descendant of the wave mechanics that Erwin Schrödinger introduced in 1926, but its chemical and physical exploitation is most closely associated with George Gamow, who in 1928 used essentially this mathematics — a particle penetrating a potential barrier — to explain alpha-particle radioactivity, showing that the enormous range of observed half-lives (from microseconds to billions of years) follows from the exponential sensitivity of $T(E)$ to barrier width and height. Ronald Gurney and Edward Condon independently reached the same explanation the same year, and the coincidence of three groups solving the identical piecewise Schrödinger equation within months of each other is a testament to how naturally the mathematics falls out of matching boundary conditions once the wave picture of matter is accepted. Gamow's calculation is historically the first quantitative triumph of tunnelling theory: it converted a bewildering empirical regularity (the Geiger–Nuttall law relating decay energy to half-life) into a direct consequence of an exponential $e^{-2\kappa L}$ term, with $\kappa$ set by the alpha particle's mass and the nuclear Coulomb barrier. The same functional form, transplanted into chemistry, underlies Rudolph Marcus's later treatment of electron-transfer reactions (for which Marcus received the 1992 Nobel Prize in Chemistry), where the "barrier" is a reorganization-energy landscape and tunnelling through or near the barrier top governs electron-transfer rates in biological and inorganic redox chemistry.