Quantum Tunneling

Quantum Tunneling: Penetrating the Impenetrable Barrier

By Md Irfan | 2nd Year Physics (Hons)
Presented on National Science Day – 28th February 2023

Introduction The Mathematics Try It Yourself Applications Timeline Conclusion

Introduction

Quantum mechanics has revolutionized our understanding of nature at its most fundamental level. Among its most striking consequences is quantum tunneling: a particle can pass through an energy barrier that it does not have enough energy to climb over. Classically, a ball rolling towards a hill that is too high simply turns back. A quantum particle, described by a wavefunction rather than a point-like trajectory, has a small but non-zero probability of appearing on the other side.

Tunneling is not a curiosity. It powers the Sun, explains radioactive alpha decay, limits how small transistors can be made, and lets us image individual atoms.

Key idea
Inside the barrier the wavefunction does not vanish. It decays exponentially. If the barrier is thin enough, some amplitude is still left when the wave reaches the far side, and that leftover amplitude is the tunneling probability.

Theoretical Foundation: Mathematical Framework

Consider a non-relativistic particle of mass \(m\) and energy \(E\) approaching a rectangular potential barrier of height \(V_0\) and width \(a\), with \(E < V_0\). Classically the particle must be reflected. Quantum mechanically, the transmission probability is non-zero.

Potential Profile

\[ V(x) = \begin{cases} 0 & x < 0 \\ V_0 & 0 \le x \le a \\ 0 & x > a \end{cases} \]

Schematic (not to scale): the real part of the wavefunction oscillates, decays inside the barrier, and emerges on the right with a smaller amplitude.

Schrödinger Equation in Each Region

We solve the time-independent Schrödinger equation:

\[ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\,\psi = E\,\psi \]

For \(E < V_0\), the solutions in the three regions are:

• Region I (\(x < 0\)): \(\psi_I(x) = A e^{ikx} + B e^{-ikx}\), with \(k = \dfrac{\sqrt{2mE}}{\hbar}\)
• Region II (\(0 \le x \le a\)): \(\psi_{II}(x) = C e^{\kappa x} + D e^{-\kappa x}\), with \(\kappa = \dfrac{\sqrt{2m(V_0 - E)}}{\hbar}\)
• Region III (\(x > a\)): \(\psi_{III}(x) = F e^{ikx}\) (only a right-moving wave, since nothing comes in from the right)

Here \(A\) is the incident amplitude, \(B\) the reflected amplitude and \(F\) the transmitted amplitude. Requiring \(\psi\) and \(d\psi/dx\) to be continuous at \(x = 0\) and \(x = a\) gives four equations for the four unknowns \(B, C, D, F\) (in units of \(A\)), which can be solved exactly.

Tunneling Probability (Transmission Coefficient)

The transmission coefficient is the ratio of transmitted to incident probability current. Since the wavenumber is the same in Regions I and III, it reduces to \(T = |F|^2/|A|^2\). The exact result is

\[ T = \left[\,1 + \frac{V_0^2\,\sinh^2(\kappa a)}{4E\,(V_0 - E)}\,\right]^{-1}, \qquad R = 1 - T . \]

For a thick or tall barrier, \(\kappa a \gg 1\), this simplifies to

\[ T \approx \frac{16\,E\,(V_0 - E)}{V_0^2}\; e^{-2\kappa a} \;\sim\; \exp\!\left(-\frac{2a}{\hbar}\sqrt{2m(V_0 - E)}\right). \]

The exponential factor dominates, so the transmission probability falls off exponentially with the barrier width \(a\) and with \(\sqrt{m(V_0 - E)}\). It rises quickly as the particle’s energy \(E\) approaches \(V_0\). Notice also the dependence on mass: lighter particles tunnel far more easily than heavy ones.

Beyond the Rectangle: the WKB Approximation

For a smoothly varying barrier \(V(x)\), the Wentzel–Kramers–Brillouin (WKB) method treats the barrier as a stack of thin rectangular slices and multiplies their exponential suppressions:

\[ T \approx \exp\!\left(-2\int_{x_1}^{x_2} \frac{\sqrt{2m\,[V(x) - E]}}{\hbar}\,dx\right), \]

where \(x_1\) and \(x_2\) are the classical turning points where \(V(x) = E\). This is the formula behind the Gamow factor and the STM current discussed below.

Worked example
An electron with \(E = 1\,\mathrm{eV}\) meets a barrier of height \(V_0 = 2\,\mathrm{eV}\). Here \(\kappa = \sqrt{2m(V_0-E)}/\hbar \approx 5.12\,\mathrm{nm}^{-1}\).

• Width \(a = 0.5\,\mathrm{nm}\): \(T \approx 2.4 \times 10^{-2}\), about 1 electron in 40 gets through.
• Width \(a = 1.0\,\mathrm{nm}\): \(T \approx 1.4 \times 10^{-4}\). Doubling the width cut the probability by a factor of about 170.
• A proton with the same \(E\), \(V_0\) and \(a = 0.5\,\mathrm{nm}\) has \(\kappa\) larger by \(\sqrt{m_p/m_e} \approx 43\), giving \(T \sim 10^{-95}\). That is effectively zero, which is why electrons tunnel in everyday devices while protons need nuclear-scale barriers.

Common misconceptions
The particle does not “borrow” energy or travel faster than light. Its energy \(E\) is the same before and after the barrier, and tunneling cannot be used to send signals faster than light. What happens is that the wavefunction, which is a spread-out probability amplitude, leaks through the barrier.

Try It Yourself: Tunneling Calculator

Change the particle, its energy, the barrier height and the barrier width, and see the exact transmission probability for a rectangular barrier.

Particle Electron Proton Particle energy E: 1.0 eV Barrier height V₀: 2.0 eV Barrier width a: 0.50 nm

Transmission probability T = –

Decay constant κ = – nm⁻¹  |  2κa = –

Deep Dive into Applications

1. Nuclear Fusion in Stars

The Sun and other stars generate energy by nuclear fusion. Two protons must come within about a femtometre of each other for the strong force to act, but their Coulomb repulsion creates a barrier of roughly 1 MeV. The thermal energy of a proton in the solar core (\(T \approx 1.5 \times 10^7\,\mathrm{K}\)) is only \(k_BT \approx 1.3\,\mathrm{keV}\), about a thousand times too small. Quantum tunneling lets the protons fuse anyway, starting the proton–proton chain:

\[ p + p \;\xrightarrow{\text{tunneling}}\; D + e^+ + \nu_e \]

The tunneling probability is the Gamow factor, which governs the reaction rate:

\[ P \propto \exp\!\left(-\sqrt{\frac{E_G}{E}}\right),\qquad E_G = 2\mu\left(\frac{\pi Z_1 Z_2 e^2}{\hbar}\right)^2 = 2\mu c^2\,(\pi\alpha Z_1 Z_2)^2 , \]

where \(\mu\) is the reduced mass, \(Z_1, Z_2\) are the charges of the nuclei, \(\alpha\) is the fine-structure constant, and the first form of \(E_G\) uses Gaussian units. For two protons \(E_G \approx 0.49\,\mathrm{MeV}\). Because tunneling is rare, and the first step also needs a weak-interaction transformation, the Sun burns its fuel slowly enough to shine for billions of years. In practice, the fusion rate peaks at an intermediate “Gamow peak” energy, where the falling Boltzmann tail and the rising Gamow factor balance.

2. Alpha Decay in Nuclear Physics

An alpha particle inside a heavy nucleus is held in by the nuclear potential well and a Coulomb barrier of roughly 25 MeV, yet it typically has only 4–9 MeV. Classically it is trapped forever. In 1928, George Gamow, and independently Ronald Gurney and Edward Condon, showed that alpha decay is quantum tunneling through this barrier. Their theory explained the empirical Geiger–Nuttall law:

\[ \log_{10} T_{1/2} = \frac{a\,Z}{\sqrt{E_\alpha}} + b , \]

where \(T_{1/2}\) is the half-life, \(E_\alpha\) is the energy of the emitted alpha particle, \(Z\) is the atomic number of the daughter nucleus, and \(a, b\) are constants. The effect is dramatic: polonium-212 (\(E_\alpha \approx 8.9\,\mathrm{MeV}\)) has a half-life of under a microsecond, while uranium-238 (\(E_\alpha \approx 4.3\,\mathrm{MeV}\)) lives for about 4.5 billion years. A factor of two in energy changes the half-life by more than 23 orders of magnitude.

3. Quantum Biology

Tunneling is well established in biochemistry. Enzymes can move hydrogen atoms and protons by tunneling, and electrons tunnel along chains of proteins in photosynthesis and cellular respiration.

A more speculative idea concerns DNA. In 1963 Per-Olov Löwdin proposed that a proton could tunnel across a hydrogen bond between base pairs (adenine–thymine or guanine–cytosine), producing a rare tautomeric form of a base that may be mispaired during replication and cause a point mutation. Calculations show this is physically possible, but whether it contributes significantly to real mutation rates is still debated, so it is best described as an active research question rather than settled fact.

4. Scanning Tunneling Microscope (STM)

Invented in 1981 by Gerd Binnig and Heinrich Rohrer at IBM Zürich (Nobel Prize in Physics, 1986), the STM images surfaces at the atomic level. A sharp conducting tip is brought within about a nanometre of a conducting sample. With a bias voltage applied, electrons tunnel across the vacuum gap, and the current depends exponentially on the gap width:

\[ I \propto V\,e^{-2\kappa d}, \qquad \kappa = \frac{\sqrt{2m\phi}}{\hbar}, \]

where \(\phi\) is the work function (typically 4–5 eV, so \(\kappa \approx 1\,\mathrm{\AA}^{-1}\)) and \(d\) is the tip–sample separation. Moving the tip by just 1 Å changes the current by roughly a factor of ten, which gives vertical resolution far below the size of an atom. In 1990, Don Eigler and Erhard Schweizer at IBM used an STM tip to drag 35 xenon atoms into the letters “IBM”, the first deliberate arrangement of individual atoms.

More Places Where Tunneling Matters

💾 Flash memory

Electrons tunnel (Fowler–Nordheim tunneling) through a thin oxide layer onto a floating gate. Whether the gate holds charge stores your bit of data.

🔌 Tunnel diodes

Leo Esaki’s 1957 diode uses tunneling across a very thin, heavily doped p–n junction, giving fast switching and negative differential resistance.

⚛️ Josephson junctions & qubits

Cooper pairs tunnel across a thin insulating layer between superconductors. Such junctions are the heart of SQUID magnetometers and superconducting quantum computers. The 2025 Nobel Prize in Physics honoured the discovery of macroscopic quantum tunneling in an electric circuit.

📉 The limit of Moore’s law

As transistor gates and oxide layers shrink to a few atoms, electrons tunnel through them unintentionally, causing leakage current and wasted power.

🕳️ Black holes (theory)

One theoretical picture describes Hawking radiation as particles tunneling out through the event horizon. It is a model used in calculations, not something we have observed.

A Short Timeline

1927 – Friedrich Hund notices tunneling-type behaviour in molecular double-well potentials.

1928 – Gamow, and independently Gurney and Condon, explain alpha decay as tunneling.

1929 – Atkinson and Houtermans apply tunneling to nuclear reactions in stars.

1957–58 – Esaki demonstrates the tunnel diode.

1962 – Josephson predicts tunneling of Cooper pairs between superconductors.

1981 – Binnig and Rohrer build the scanning tunneling microscope.

1973, 1986, 2025 – Nobel Prizes connected to tunneling: Esaki, Giaever and Josephson (1973); Binnig and Rohrer (1986); Clarke, Devoret and Martinis (2025).

Conclusion

Quantum tunneling defies classical intuition and shows that particles are better described by waves of probability than by tiny billiard balls. From the nuclear core of stars to the decay of heavy elements, from enzymes in living cells to the microscope that lets us see atoms, tunneling is a quiet but constant quantum effect shaping the world. As we gain more control over the quantum world, we will keep finding ways to use it, in quantum computing, ever smaller electronics and new sensors, and we will keep having to design around it.

“Where classical mechanics sees a wall, quantum mechanics sees a door—though a narrow one.” — Scientificirfan

Further Reading

1. D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, Cambridge University Press (the rectangular barrier and WKB are treated in the chapters on the Schrödinger equation and on the WKB approximation).
2. G. Gamow, “Zur Quantentheorie des Atomkernes”, Z. Phys. 51, 204 (1928).
3. R. W. Gurney and E. U. Condon, “Wave Mechanics and Radioactive Disintegration”, Nature 122, 439 (1928).
4. L. Esaki, “New Phenomenon in Narrow Germanium p–n Junctions”, Phys. Rev. 109, 603 (1958).
5. G. Binnig, H. Rohrer, Ch. Gerber and E. Weibel, “Surface Studies by Scanning Tunneling Microscopy”, Phys. Rev. Lett. 49, 57 (1982).
6. P.-O. Löwdin, “Proton Tunneling in DNA and its Biological Implications”, Rev. Mod. Phys. 35, 724 (1963).