# Quantum tunnelling **Quantum tunnelling** is the passage of a particle through a region that [[Classical_physics|classical physics]] says it cannot enter: a potential barrier whose height exceeds the particle's total [[Energy|energy]]. A classical ball rolled at a hill lower than its kinetic energy goes over, one rolled at a higher hill comes back, and nothing else can happen. A quantum particle is described by a [[Wave_function|wave function]] that does not stop at the wall: inside the barrier it decays exponentially rather than oscillating, and if the wall is thin enough the decaying tail is still finite at the far side, where it resumes travelling as a smaller wave. The probability of getting through, the transmission coefficient, is never exactly zero below the barrier top and never exactly one above it.[^likharev-step][^likharev-barrier] In the microsim below the reader slides ε = E/U₀, the particle's energy as a fraction of the barrier height, from 0.01 to 3, with the barrier's opacity fixed by presets κ₀·d = 1, 3 and 6. The equation that answers is the rectangular-barrier transmission `T = 1/(1 + sinh(kappa·d)^2/(4·eps·(1 − eps)))`, in which `kappa^2 = 2·m·(U0 − E)/hbar^2` is the decay constant inside the wall and d is its thickness.[^likharev-barrier][^manual04-tunnel] The upper panel draws U(x) with Re ψ and |ψ|² over it; the lower panel draws T(ε) on a log axis, a curve that rises smoothly through ε = 1 and, above the barrier, dips and returns to exactly 1 at the resonances k′d = nπ.[^spec-p37] Two presets tie it to laboratory numbers: an [[Alpha_decay|alpha-decay]] preset connects the exponent to the half-lives of row P50 of the flagship spine, and a [[Scanning_tunneling_microscope|scanning-tunnelling-microscope]] preset sets 1/κ ≈ 0.1 nm, the decay length of an [[Electron|electron]] facing a 5 eV vacuum barrier.[^likharev-step][^spec-p37] On the [[Physics]] flagship this article is the *Tunnelling* section of Part II — Core theories, and a sibling of the [[Schrödinger_equation|Schrödinger equation]] root: the same one-dimensional wave mechanics, pointed at a wall instead of a well. Its companions are the [[Rectangular_potential_barrier|rectangular potential barrier]], the exact model solved here, and the [[WKB_approximation|WKB approximation]], used when the wall has a shape. ## Concept The whole of tunnelling follows from two undramatic facts about the [[Schrödinger_equation|Schrödinger equation]]: the wave function and its slope are continuous everywhere, and inside a classically forbidden region the equation turns oscillation into exponential decay. Outside the barrier the solution has `k^2 = 2·m·E/hbar^2` and oscillates with wavelength 2π/k; inside, where U₀ > E, the same equation gives `kappa^2 = 2·m·(U0 − E)/hbar^2` and a real exponential. Continuity forbids the amplitude from dropping to zero at the wall, and since a falling exponential reaches zero only at infinity, a wall of finite thickness always leaves something on the other side.[^likharev-step] The length 1/κ is the penetration depth, and it is the number that matters most in practice. For E much smaller than U₀ it reduces to `hbar/sqrt(2·m·U0)`, which for an [[Electron|electron]] and a 5 [[Electronvolt|eV]] barrier is about 0.1 nm — 0.0873 nm on the constants used here.[^likharev-step][^derived-qt] Because κ carries the square root of the mass, a [[Proton|proton]] facing the same barrier decays 42.9 times faster in space than an electron, and a [[Deuterium|deuteron]] 1.41 times faster again — which is why tunnelling dominates electrons in solids, matters for hydrogen, and is invisible for anything heavier.[^derived-qt] ### Tunnelling problem The textbook problem is a rectangular barrier of height U₀ and width d, with a particle of energy E < U₀ arriving from the left. Matching ψ and ψ′ at the two faces gives the closed form `T = [1 + ((kappa^2 + k^2)/(2·k·kappa))^2·sinh(kappa·d)^2]^-1`.[^likharev-barrier] In the reduced variables the sim uses — ε = E/U₀ and lengths in units of 1/κ₀ with `kappa0 = sqrt(2·m·U0)/hbar` — it becomes `T = [1 + sinh(kappa0·d·sqrt(1 − eps))^2/(4·eps·(1 − eps))]^-1`, and above the barrier the same algebra with κ → ik′ gives `T = [1 + sin(kappa0·d·sqrt(eps − 1))^2/(4·eps·(eps − 1))]^-1`.[^manual04-tunnel] Reading numbers off that formula is the fastest way to feel what "exponentially sensitive" means. At ε = 0.5 a barrier with κ₀d = 1 transmits 0.629, one with κ₀d = 3 transmits 0.0559, and one with κ₀d = 6 transmits 8.26 × 10⁻⁴: doubling the thickness costs a factor of 68.[^derived-qt] At ε = 1 the formula takes the finite limit `T = 1/(1 + (kappa0·d)^2/4)` — 0.8, 0.308 and 0.100 for the three presets, so the particle is still heavily reflected by a barrier it could clear.[^manual04-tunnel][^derived-qt] When κd is large the exact result collapses to `T ≈ [4·k·kappa/(k^2 + kappa^2)]^2·exp(−2·kappa·d)`, matching the exact 1.6378 × 10⁻⁵ to five figures at ε = 0.1, κ₀d = 6.[^likharev-thick][^derived-qt] This exponential is what makes the [[Scanning_tunneling_microscope|scanning tunnelling microscope]] work: with 1/κ ≈ 0.1 nm, moving the tip 0.1 nm further from the surface divides the current by e² ≈ 7.4.[^derived-qt] ## History Tunnelling was recognised almost as soon as wave mechanics existed, because it fell out of the new equation without being looked for. Its first quantitative success was [[Alpha_decay|alpha decay]], which it rescued from paradox: alpha particles emerge from heavy nuclei with energies far below the Coulomb barrier they must have crossed, and their half-lives span more than twenty orders of magnitude for a spread in emission energy of only a factor of two. An exponential of the barrier integral is the only function that turns so small a change in energy into so large a change in rate. George Gamow, and independently Ronald Gurney and Edward Condon, published the explanation in 1928.[^gamow1928][^gurney-condon1928] The barrier then became a standard exactly solvable model, and its transfer-matrix formulation — one 2 × 2 complex matrix per interface, multiplied along the structure — made it a tool rather than an example, since any potential can be cut into steps.[^likharev-tm] Solid-state applications followed once junctions could be made thin enough for κd to fall to order unity, and tunnelling in [[Semiconductor|semiconductors]] and [[Superconductivity|superconductors]] became a laboratory technique.[^openstax-v3-ch7] ## Applications Tunnelling appears wherever a barrier is a few decay lengths thick and no thicker. Because T falls exponentially with κd there is no such thing as a slightly opaque barrier: a factor of two in thickness moves a device from conducting to insulating, so the applications below are engineered into that window. ### Solid-state physics In solids the barriers are thin oxide layers, vacuum gaps and depletion regions, and the tunnelling particle is an [[Electron|electron]], whose small mass keeps 1/κ near 0.1 nm. That is the length scale of the [[Scanning_tunneling_microscope|scanning tunnelling microscope]], which resolves atoms vertically without any lens.[^derived-qt][^openstax-v3-ch7] The same exponential sets a floor under how thin a gate insulator can be made before leakage dominates.[^openstax-v3-ch7] Two barriers in series behave very differently from one: the transfer-matrix product gives resonances at which transmission returns to exactly 1 even when each barrier alone is nearly opaque, the quantum analogue of a Fabry–Pérot cavity.[^likharev-resonant] ### Nuclear physics [[Alpha_decay|Alpha decay]] remains the cleanest case. The alpha particle is bound inside the [[Atomic_nucleus|nucleus]] by the [[Nuclear_force|nuclear force]] and repelled outside it by the Coulomb interaction, so it sits in a well behind a barrier several MeV taller than its energy. The escape rate is the rate at which it strikes the wall times the transmission per strike, written for a metastable state as `tau = t_a/T` with t_a the traversal time of the well.[^likharev-metastable] Since T is an exponential of the barrier integral and t_a varies slowly, the [[Half-life|half-life]] — the mean life times ln 2 — inherits the exponential, and a factor of two in decay energy produces the observed twenty orders of magnitude in lifetime.[^schiller-lifetime][^openstax-v3-ch10] Read in reverse, the same relation gives the width of a [[Radioactive_decay|decaying]] state through ΔE·τ ≈ ħ.[^likharev-metastable] ### Chemistry Chemical reactions are usually pictured as passing over an [[Activation_energy|activation barrier]], with the [[Arrhenius_equation|Arrhenius equation]] giving the rate of thermal excitation over it. Tunnelling adds a channel through the barrier that ignores [[Temperature|temperature]]: the rate stops falling as the sample is cooled, and the [[Hydrogen|hydrogen]]-to-[[Deuterium|deuterium]] rate ratio runs far above what classical transition-state theory predicts.[^likharev-step][^derived-qt] The mass dependence is the signature: because κ ∝ √m, transferring a proton is a tunnelling problem and transferring a carbon atom effectively is not.[^derived-qt] ### Biology The same square root decides what biology can use. An [[Electron|electron]] between redox centres in a protein has a decay length of order 0.1 nm and crosses a nanometre gap with measurable probability; a proton crossing the same gap at the same reduced energy does so with a probability smaller by a factor of order 10⁻¹⁵⁰.[^derived-qt] That is why long-range electron transfer is a standard mechanism in [[Photosynthesis|photosynthesis]] and respiration while heavy-atom tunnelling is not, and why evidence for tunnelling in [[Enzyme|enzyme]]-catalysed reactions is always sought in hydrogen transfer and isotope ratios.[^derived-qt] The pair's Biology section reviews the experimental claims for enzymes and for base-pair proton transfer; this page does not restate them, because they need biochemical sources rather than the barrier model used here. ## Astrophysics Stars burn because of tunnelling. Two [[Proton|protons]] in the core of the [[Sun|Sun]] repel one another through a Coulomb barrier of order an MeV, while the thermal energy available at 1.6 × 10⁷ K is of order a keV — three orders of magnitude too little to reach the top.[^openstax-v3-ch10] Fusion nevertheless proceeds, at a rate set by two competing exponentials: the [[Maxwell–Boltzmann_distribution|Maxwell–Boltzmann]] tail, which falls steeply with energy, and the tunnelling probability, which rises steeply with it. Their product is sharply peaked well above the thermal average and well below the barrier top, and that peak fixes the rate of the [[Proton–proton_chain|proton–proton chain]] and so the lifetime of a star.[^openstax-v3-ch10] It is the alpha-decay exponential running the other way: a small rise in core temperature moves the peak and multiplies the rate of [[Nuclear_fusion|fusion]], which is what makes [[Stellar_nucleosynthesis|stellar]] burning self-regulating. ## Mathematical discussion Two levels of treatment are in ordinary use: a barrier made of flat pieces is solved exactly by matching, and a barrier of arbitrary shape is estimated by the semiclassical integral. The sim uses the first and reports the second as a comparison curve.[^spec-p37] ### Schrödinger equation The exact route is the one above: solve the time-independent equation piecewise, match ψ and ψ′ at every interface, and read off the ratio of outgoing to incoming flux. Systematised, the matching becomes the transfer matrix — one 2 × 2 complex matrix per interface, another for free propagation between them, multiplied in order, with `T = 1/|T11|^2`.[^likharev-tm] It generalises to any U(x) cut finely into steps and it is cheap: 1,024 samples cost under 0.3 ms, so the sim recomputes the picture on every slider movement rather than reading a stored table.[^manual04-tunnel][^spec-p37] Two numerical traps are worth naming: at ε → 1 the closed form is 0/0 and must be replaced by its limit, and sinh overflows for κd above about 710, where `ln T ≈ −2·kappa·d + 2·ln(4·k·kappa/(k^2 + kappa^2))` takes over.[^manual04-tunnel] ### WKB approximation For a smooth barrier the semiclassical estimate is `T = exp(−(2/hbar)·∫sqrt(2·m·(U − E))·dx)` between the classical turning points.[^likharev-wkb] It is the workhorse of alpha-decay and fusion rate calculations, and it is good to the extent that the potential changes slowly over a de Broglie wavelength — which is exactly what fails near the top, where the turning points merge. There WKB predicts T → 1, which is wrong; the exact result for a parabolic top is Kemble's `T = 1/(1 + exp(2·pi·(Umax − E)/(hbar·w0)))`, giving T = ½ at the top.[^likharev-wkb] Lower down the two converge fast: at (E − U_max)/ħω₀ = −1, Kemble gives 0.001864 against WKB's 0.001867; at −0.5, 0.0414 against 0.0432.[^manual04-tunnel] For a thick square barrier at k = κ the semiclassical estimate is low by a factor of four, because it drops the prefactor the exact solution carries.[^likharev-wkb] ## Faster than light Because the wave inside the barrier is evanescent rather than travelling, the usual definitions of how long the crossing took disagree, and some of them, for a thick barrier, imply an apparent speed above c. The transfer matrix shows where the ambiguity comes from: the transmitted amplitude carries a phase as well as a magnitude, and the phase accumulated across the barrier saturates as the barrier thickens, so a delay read from it stops growing with distance.[^likharev-tm] That saturation, and the experiments that have measured it, are asserted here without a source this page has read [citation needed]. The resolution does not require modifying [[Special_relativity|special relativity]]. What emerges on the far side is a small, reshaped piece of the leading edge of the incident packet, so no signal and no energy has moved faster than light: the barrier is a filter, and a filter can advance a pulse peak without information travelling ahead of it. ## Dynamical tunnelling The barrier in the standard problem is a barrier in position: a region of space the particle cannot classically occupy. In a broader class of cases nothing forbids it from being anywhere, and what is forbidden is a change of motion — the two states are separated in phase space, not in space. Transitions are still exponentially suppressed, so the same machinery applies. ### Tunnelling in phase space A classical trajectory is confined to a surface in phase space, and two disconnected regions of allowed motion can exist at the same energy — two directions of rotation, say, or a resonance island and the sea of orbits around it. Classically a trajectory started in one region stays there forever. Quantum mechanically the two supported states are coupled, and the system oscillates between them at a rate exponentially small in ħ⁻¹.[^likharev-resonant] ### Chaos-assisted tunnelling When the region between the two islands is chaotic rather than regular, the chaotic states act as intermediaries: instead of one direct, exponentially small coupling, the system tunnels into the chaotic sea and out again. The rate rises by orders of magnitude and becomes irregular, fluctuating as a parameter is varied, because it depends on the accidental proximity in energy of a chaotic state to the two regular ones. ### Resonance-assisted tunnelling Even with no chaos, the chain of small resonance islands embedded in the regular region provides stepping stones. Coupling through a nearby resonance replaces one long, heavily suppressed step by two shorter ones, and because the suppression is exponential, splitting a barrier into halves of the same total width can raise the rate by many orders of magnitude — most when a resonance sits close in action to the tunnelling state, which is why these rates peak as a parameter is swept. ## Related phenomena Several effects not usually called tunnelling are the same calculation. Field emission of electrons from a cold metal into vacuum is tunnelling through a triangular barrier tilted by an applied field — the [[Free_electron_model|free-electron]] surface supplies the barrier, the field the tilt — so the current depends exponentially on field strength rather than on [[Temperature|temperature]]. Closest to the problem solved here, a particle bound in a well with leaky walls is a metastable state, and the transfer-matrix treatment gives its lifetime as `tau = t_a/T` and its width as ΔE ≈ ħ/τ — so a [[Radioactive_decay|decaying]] state and a broadened spectral line are two readings of one number.[^likharev-metastable] ## See also - [[Rectangular_potential_barrier]] — the exact model this page solves - [[WKB_approximation]] — the estimate used when the barrier has a shape - [[Scanning_tunneling_microscope]] — the 0.1 nm decay length as an instrument - [[Alpha_decay]] — the first quantitative success of the idea - [[Schrödinger_equation]] — the parent root of this sim family - [[Particle_in_a_box]] — the same matching conditions, with the walls closed - [[Nuclear_fusion]] — the barrier crossed from the other side - [[Bloch's_theorem]] — what happens when barriers are repeated forever ## References [^likharev-step]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, 1D wave mechanics: `k^2 = 2mE/hbar^2` and `kappa^2 = 2m(U0 − E)/hbar^2`, the continuity of ψ and ψ′, total reflection from a step for E < U0, and the penetration depth `1/kappa ≈ hbar/sqrt(2*m*U0)` with 1/κ ≈ 0.1 nm for an electron at U₀ ≈ 5 eV (pp. 41–44). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-barrier]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, the rectangular barrier: `T = [1 + ((kappa^2 + k^2)/(2*k*kappa))^2*sinh^2(kappa*d)]^-1` (pp. 44–45). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-thick]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, the thick-barrier limit `T ≈ [4*k*kappa/(k^2 + kappa^2)]^2*exp(-2*kappa*d)` and the delta barrier `T = 1/(1 + E0/E)` with `E0 = m*W^2/(2*hbar^2)` (p. 46). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-wkb]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, the semiclassical barrier integral `T = exp(-(2/hbar)*int sqrt(2m(U - E)) dx)`, its failure at the barrier top, Kemble's exact parabolic result `T = 1/(1 + exp(2*pi*(Umax - E)/(hbar*w0)))` giving T = ½ at E = U_max, and the factor-of-four underestimate of the exact thick-barrier result at k = κ (pp. 55–56). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-tm]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, the transfer-matrix method: 2 × 2 complex matrices per interface, composite scatterers multiplying, and `T = 1/abs(T11)^2` (pp. 57–58). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-resonant]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, resonant tunnelling through a double barrier: two opaque scatterers reach T = 1 at resonance, the Fabry–Pérot analogue (pp. 57–63). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-metastable]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, metastable states: `W = W0*exp(-t/tau)` with `tau = t_a/T` and `t_a = a/v_gr`, and the width–lifetime relation ΔE·τ ≈ ħ, with the caveat that time is not an operator (pp. 62–64). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^manual04-tunnel]: Wikitube MICROSIM_GUIDE sub-manual 04, *Atomic, Quantum, Statistical and Electromagnetic Physics*, §3.2 "Reflection and tunnelling: step, rectangular barrier, delta, soft barrier": the reduced-unit forms `T = [1 + sinh^2(kappa0*d*sqrt(1-eps))/(4*eps*(1-eps))]^-1` and its ε > 1 counterpart, the finite limit `T = 1/(1 + (kappa0*d)^2/4)` at ε = 1, the sinh overflow above κd ≈ 710 and its logarithmic replacement, and the derived Kemble-against-WKB values 0.001864/0.001867 at −1 and 0.0414/0.0432 at −0.5 in units of ħω₀. [^derived-qt]: Computed for this article from the closed forms of [^likharev-barrier] and [^likharev-thick]. Penetration depth at U₀ = 5 eV for an electron: ħ/√(2mU₀) = 0.0873 nm, reproducing the sub-manual's derived 0.087 nm. Transmission at ε = 0.5: 0.6293 (κ₀d = 1), 0.05586 (κ₀d = 3), 8.256 × 10⁻⁴ (κ₀d = 6), a ratio of 67.7 between the last two. At ε = 1: 0.800, 0.3077, 0.100. At ε = 0.1 and κ₀d = 6 the exact value 1.63784 × 10⁻⁵ and the thick-barrier asymptote 1.63782 × 10⁻⁵ agree to five figures. Above the barrier at κ₀d = 6, T returns to 1 at ε = 1 + (nπ/6)² = 1.2742 and 2.0966; at κ₀d = 3 the first is at 2.0966; at κ₀d = 1 the first would be at 10.87, outside the slider range. Mass scaling: κ ∝ √m gives √1836 = 42.9 for proton against electron and √2 = 1.414 for deuteron against proton; at ε = 0.5 and an electron κ₀d of 6, a proton's exponent −2κd·√(1−ε) is larger by 355, a transmission ratio of order 10⁻¹⁵⁴. Tip-height sensitivity: with 1/κ = 0.1 nm, T ∝ exp(−2κd) changes by e² = 7.39 per 0.1 nm. [^spec-p37]: Matter & Energy Cluster contract, `_registry/plans/PHYSICS_SECTIONS.md` row P37: new sibling of the Schrödinger-equation root (`wt-quantum.transferMatrix`, sub-manual 04 §3.2 on the framework). The reader slides ε = E/U₀ over 0.01–3 with κ₀·d presets 1, 3 and 6; the upper panel draws U(x), Re ψ and |ψ|², the lower panel T(ε) on a log axis with the above-barrier resonances at k′d = nπ; an alpha-decay preset ties the exponent to the half-lives of row P50 and a scanning-tunnelling-microscope preset to 1/κ ≈ 0.1 nm at 5 eV. Amplitudes come from 2 × 2 complex transfer matrices, 1,024 samples in under 0.3 ms, so the sim runs live rather than from a bake. [^openstax-v3-ch7]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 7, "Quantum Mechanics" (pp. 295–346), the barrier-penetration section and its device applications, including scanning tunnelling microscopy and tunnelling in semiconductor junctions (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^openstax-v3-ch10]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 10, "Nuclear Physics" (pp. 441–492), alpha decay as barrier penetration and the Coulomb barrier in stellar fusion (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^schiller-lifetime]: Schiller, Christoph. *Motion Mountain, Vol. IV: The Quantum of Change*. The lifetime–width relation and `t_half = tau*ln 2`, with the particle-lifetime table (p. 129). https://open.umn.edu/opentextbooks/textbooks/the-adventure-of-physics-vol-iv-the-quantum-of-change [^gamow1928]: Gamow, George (1928). "Zur Quantentheorie des Atomkernes." *Zeitschrift für Physik* 51. Pages and DOI to pin. [^gurney-condon1928]: Gurney, Ronald W.; Condon, Edward U. (1928). "Wave Mechanics and Radioactive Disintegration." *Nature* 122. Pages and DOI to pin. ## Further reading - Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*, Chapter 2 — every canonical one-dimensional barrier problem in closed form, with the transfer-matrix method. https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3*, Chapters 7 and 10 — barrier penetration at first-course level, with alpha decay and device applications. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 ## External links - The Wikipedia pair's *External links* section is the place to look for tunnelling applets and review articles; this page lists only the open texts above, which it has read. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Quantum_tunnelling.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Quantum tunnelling* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Quantum_tunnelling.html" data-title="Quantum tunnelling"></div> *Built from `MICROSIM_GUIDE/specs/sims/Quantum_tunnelling.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Quantum_tunnelling) : [Wikitube](https://en.wikitube.io/wiki/Quantum_tunnelling) · pinned revision [1353702631](https://en.wikipedia.org/w/index.php?oldid=1353702631) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P37 · sim pending (matter/Quantum_tunnelling).*