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In Ammonia, the nitrogen chemical atom can lie on either side of the plane of the three hydrogen chemical atoms. Quantum tunnelling couples these configurations; the resulting pair of energy levels permits an electric dipole transition. This geometrical inversion is distinct from population inversion, which concerns how many molecules occupy each level.^[ref]

For an idealized pair of energy levels with equal quantum degeneracy, the Boltzmann distribution gives $N_u/N_l=\exp [-h{\nu }_0/(k_{\mathrm{B}}T)]<1$ at positive temperature $T$. A warm source therefore supplies upper-state molecules together with an unwanted lower-state population. Selecting the former can produce population inversion in the transmitted beam without exciting each transmitted molecule during selection. The energy per net downward transition isHere $h$ is the Planck constant. Taking the reported ${\nu }_0=23.87\mathrm{GHz}$ as a rounded input gives $ϵ\approx 1.582\times 10^{-23}\mathrm{J}$. This calculation is a conversion of the reported frequency, not a new measurement of the ammonia inversion transition.

BCS theory describes a superconducting instability caused by attractive interactions between opposite-momentum fermions near a Fermi surface.

John Bardeen, Leon Neil Cooper and John Robert Schrieffer developed BCS theory in 1957, using attraction mediated by phonons.^[ref]

Weak-coupling Cooper pairs overlap. The Cooper instability reorganizes an unpaired Fermi surface; it is not Bose-Einstein condensation of compact, pre-existing molecules. The original paper explicitly distinguishes these pictures.^[ref]

Two complementary clues made the microscopic problem concrete. The superconducting isotope effect connected the transition to phonons.^[ref]

A 1957 transmission experiment used thin lead and tin films as samples in far-infrared and millimeter-microwave measurements. The measured transmission was converted to complex electrical conductivity, normalized to the normal-state value. Comparing film thickness, resistance, annealing and the two metals tested the response's reproducibility. Weak low-photon-energy absorption followed by an onset supported a superconducting energy gap; its extraction required a thin-film electrodynamic model.^[ref]

The later Giaever superconducting tunnelling experiment provided another way to resolve the superconducting energy gap through quantum tunnelling.^[ref] The derivations below explain what the model predicts, including the distinction between a single quasiparticle's minimum energy and the threshold for making two of them.

Why can a star shine for so long, and why do hotter stellar cores favor different nuclear reactions? A useful theory must supply both an adequate energy reservoir and the right stellar energy-generation rate. Nuclear binding energy supplies the reservoir; the thermonuclear reaction rate, stellar composition and stellar structure determine how quickly it is spent. The proton–proton chain and CNO cycle are two routes for hydrogen burning with very different temperature sensitivities.

In his 1967 lecture, Hans Albrecht Bethe contrasted the short Kelvin-Helmholtz cooling time with long ages inferred from terrestrial and stellar evidence.^[ref]

For an original solar-scale estimate, choose mass $M=1.99\times 10^{30}\mathrm{kg}$, radius $R=6.96\times 10^8\mathrm{m}$ and luminosity $L=3.83\times 10^{26}\mathrm{W}$. The characteristic Kelvin-Helmholtz cooling time $G{M}^2/(RL)$ is about $31\mathrm{Myr}$; the precise gravitational reservoir has a structure-dependent coefficient. If a fraction $f=0.10$ of the mass can burn its initial hydrogen mass fraction $X=0.70$, with a chosen rest-energy conversion fraction $\eta =0.007$, the fuel estimate isThis assumes fixed luminosity and neglects neutrino losses. It demonstrates the reservoir advantage, rather than predicting the Sun's age or its full stellar evolution. Having enough fuel still does not identify a reaction: the fuel must release energy at a rate compatible with hydrostatic equilibrium and radiative transfer.

Hans Albrecht Bethe and Charles Louis Critchfield calculated deuterium formation from two protons in 1938, combining quantum tunnelling with a weak conversion of a proton into a neutron. Their calculated nuclear reaction cross-section was a theoretical input to hydrogen burning.^[ref] In 1939, Hans Albrecht Bethe analysed the catalytic carbon–nitrogen route and compared required stellar core temperatures with stellar structure estimates in Table VIII. His reported solar agreement used historical abundance and structure assumptions; it should not be read as a modern demonstration that the CNO cycle dominates the Sun.^[ref]

Hans Albrecht Bethe's 1967 lecture credits Carl Friedrich von Weizsäcker's independent 1938 carbon–nitrogen cycle discovery, distinguishing it from calculation of rates and temperature dependence.^[ref] The 1967 Nobel Prize in Physics recognized Hans Albrecht Bethe's contributions to nuclear reaction theory, especially stellar energy production.^[ref]

Laboratory nuclear reaction cross-sections were inputs, not measurements of an artificial star. The 1939 paper converted thick-target yields to nuclear reaction cross-sections using a range–energy relation and tabulated capture data.^[ref] A concrete apparatus example is the Cockcroft-Walton lithium disintegration experiment, with an accelerated proton beam, lithium target, zinc sulfide screen and beam-blocking control. Its nuclear reaction differs from both stellar burning routes. The 1967 lecture stresses extrapolation below laboratory access and tests for nearby nuclear levels.^[ref]

Hubbard-Stratonovich decoupling of an exponentially coupled Ising chain gives a scalar-field action with a nearest-neighbor gradient term and $U(m)=\tanh (\kappa /2)m^2/J-\log [2\cosh (2m+h)]$. Its continuum partition function is the Euclidean path integral of the displayed quantum particle. At zero field the potential can have two wells, but a finite barrier still permits quantum tunnelling and gives a unique symmetric ground state. The excitation gap produces a finite classical correlation length, so there is no finite-temperature spontaneous magnetization for fixed positive $\kappa$. One classical spatial coordinate becomes imaginary time; a $d$-dimensional classical model corresponds to a quantum model with $d-1$ spatial dimensions.

A concrete quantum quench experiment loaded a Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$ chemical atoms from a magnetic field trap into an optical lattice: three orthogonal $838\mathrm{nm}$ laser standing waves, with approximately $125\mu \mathrm{m}$ beam waists. An $80\mathrm{ms}$ loading ramp preceded a $50\mu \mathrm{s}$ increase in lattice depth. This suppressed quantum tunnelling while retaining each well's vibrational ground state. The switch preserved atom-number populations rather than freezing every motional degree of freedom.^[ref]

After variable hold times, releasing the traps and taking photon absorption images after $16\mathrm{ms}$ of flight revealed disappearance and return of interference peaks. Background-subtracted peak counts supplied a coherent fraction. Varying final depth tested the predicted revival period $h/U$; varying initial depth tested the collapse's sensitivity to atom-number statistics. Peak broadening outside counting windows and spatial variation of $U$ limited later signals.^[ref]

Here $U$ is the on-site interaction energy. The approximate isolated-well Hamiltonian operator $H=U\hat{n}(\hat{n}-1)/2$, with number operator $\hat{n}$, gives different quantum phases to its number states, which later rephase. This is a many-level realization of the distinction between fixed populations and changing quantum phases.^[ref]

Choose a fixed tip-sample electric potential difference and a positive target electric current magnitude $I_0$. At each lateral position $(x,y)$, feedback control adjusts $z$ until the measured electric current approaches $I_0$. Since increasing the gap reduces quantum tunnelling, excessive electric current calls for retraction; insufficient electric current calls for approach. The image is the calibrated vertical displacement $z(x,y)$, while the residual electric current error tests how closely the loop followed its target.

This is a contour of equal electric current, not necessarily a contour of equal separation. The electronic contrast in scanning tunneling microscopy model below shows why. It also explains the value of vibration isolation: an unintended displacement changes the same sensitive gap as the feature being measured. A scan must be slow enough for the finite bandwidth of feedback control; otherwise $z(x,y)$ lags the desired contour.

In the plane wave basis of a ring, $V(x)=-\lambda \cos (2\pi x/L)$ has matrix elements $V_{mn}=-(\lambda /2)({\delta }_{m,n+1}+{\delta }_{m,n-1})$. Near $\alpha =1/2$, degenerate perturbation theory on $n=0,1$ givesup to corrections from remote states. Thus the avoided crossing gap is $|\lambda |$ to first order when $|\lambda |\ll E_L$.

For $|\lambda |\gg E_L$, a parabolic approximation about a potential minimum instead gives a local quantum harmonic oscillator spacing $\sqrt{2|\lambda |E_L}$. A local gauge transformation removes the flux from this approximation. Actual ring levels retain exponentially small flux dependence through quantum tunnelling, while nonquadratic corrections to their mean energies are algebraic in $\sqrt{E_L/|\lambda |}$.

Repulsion between positively charged nuclei creates this electrostatic energy barrier outside the short-range nuclear interaction. Quantum tunnelling allows reaction at thermal energies below its classical height and contributes the penetration factor in the Gamow peak.

A Josephson junction can make a collective quantum phase behave as a measurable dynamical coordinate. In current-biased Josephson phase dynamics, the Gauge-invariant Josephson phase $\varphi$ moves in a tilted Josephson washboard potential; the junction's capacitance supplies its effective inertia. A trapped phase gives zero mean electric potential difference, while sustained phase advance produces a nonzero mean electric potential difference. Quantized phase levels in a Josephson junction and quantum escape of a current-biased Josephson junction test quantum mechanics for this collective coordinate.

Begin with a stipulated parallel electrical circuit: an ideal sinusoidal Josephson junction, capacitance $C>0$, and effective electrical resistance $R>0$. The Kirchhoff node law and Josephson effect giveHere $I_b$ is the constant bias electric current, $I_{\mu }$ is a deliberately applied microwave electric current, and $I_N$ represents unwanted fluctuating electric current. Multiplication by $\hslash /(2e)$ puts the undriven equation into the mechanical formThe effective inertia is a coefficient for a dimensionless quantum phase, not a literal material mass moving through the insulating layer. The capacitor's energy, $C{V}^2/2=M_{\varphi }{\dot{\varphi }}^2/2$, provides a direct check of that identification. The effective $R$ includes loading by the external electrical circuit; a constant parallel $R$ is an approximation over the relevant frequency range.

The useful design freedom is that $I_b$ changes the confinement while $C$ changes its inertia. The same manufactured device can therefore connect a classical oscillation, discrete energy levels, and escape to an electrical readout. The Martinis-Devoret-Clarke circuit experiments separate these questions experimentally. John Clarke, Michel H. Devoret and John M. Martinis shared the 2025 Nobel Prize in Physics for circuit experiments establishing macroscopic quantum tunnelling and discrete energy levels.^[ref]

For $q\alpha k>0$, let $x_0=\sqrt{2\hslash k/(q\alpha )}$. Expansion of the magnetic-gradient quartic confinement at either minimum gives $V(x_0+\delta x)=(q\alpha {)}^2{x}_0^2(\delta x{)}^2/(2m)+\cdots$. The local oscillator frequency is therefore $|q\alpha |x_0/m$. Both lowest parity states have leading energy $\hslash {\omega }_d/2$ if $x_0$ greatly exceeds the oscillator width. Local anharmonicity changes their common energy, while quantum tunnelling gives their much smaller splitting.

A potential energy landscape with two local minima separated by a barrier. In a symmetric one-dimensional case, its two lowest wavefunctions are an even function and an odd function, with their energies split by quantum tunnelling. A two-mode approximation for a Bose gas retains these two orbitals when higher energy levels are sufficiently remote.

The zero-field Hubbard–Stratonovich transformation of an exponentially coupled Ising model produces $U(m)=A{m}^2-\log [2\cosh (2m)]$, where $A=\tanh (\kappa /2)/J>0$. Since $U^{\prime \prime }(0)=2A-4$ and $U$ tends to positive infinity, a symmetric double well exists exactly when $A<2$. Its nonzero stationary points solve $Am=\tanh (2m)$; the ratio $\tanh (2m)/m$ decreases from 2 to 0 for $m>0$, proving that there is exactly one positive minimum. At $A=2$ the minimum is a single quartic minimum. A small conjugate field adds $-h\tanh (2m)+O(h^2)$ and tilts the wells, while a sufficiently large field can remove the metastable well. A finite barrier permits quantum tunnelling; two classical minima alone do not establish spontaneous symmetry breaking.

A symmetric two-well system has nearly localized left/right states coupled by a positive quantum tunnelling magnitude $\Delta =\hslash K{e}^{-S_I/\hslash }$. Its effective Hamiltonian is $E_{\mathrm{well}}I-\Delta {\sigma }_x$, so the superposition with an even spatial wavefunction is lower, and the level splitting is $2\Delta$. The dilute instanton gas reproduces these two exponents through even/odd crossing sums.

A local minimum of the scalar potential that has higher energy than a global minimum. It can be stable against infinitesimal classical fluctuations but metastable to a transition to a lower-energy vacuum, including quantum tunnelling. The stability of a cubic-quartic scalar potential gives an elementary example distinguishing local curvature from global stability.

Field electron emission releases electrons from matter through electric-field-assisted quantum tunnelling at the surface. The emission mechanism and its rate are distinct from the subsequent acceleration and Larmor formula radiation of an electron already travelling through an approximately uniform electric field.

The increase in a nucleus's collective deformation potential energy between a local minimum and a saddle on a path toward separated fragments. A positive final-state Q value does not remove this intermediate barrier. A nucleus can cross it after sufficient excitation energy, or undergo quantum tunnelling. The relevant barrier for neutron-induced fission belongs to the compound nucleus formed after capture, not automatically to the target before capture.

An instanton is a localized finite-action classical solution in Euclidean spacetime, used in a semiclassical description of quantum tunnelling and other nonperturbative effects. In one-dimensional quantum mechanics it can connect distinct degenerate potential minima over infinite Euclidean time. A quartic double-well instanton is an elementary example; a Yang-Mills instanton is a gauge-field example with different geometric structure.

For a one-dimensional quantum tunnelling instanton, the prefactor combines nonzero Gaussian fluctuation eigenvalues with the translation zero-mode Jacobian. Here $L_I=-{\partial }_t^2+V^{\prime \prime }(q_I)/m$, $L_0=-{\partial }_t^2+{\omega }^2$, and the prime excludes the zero mode. Common endpoint regularization and reference normalization define the determinant ratio. The resulting $K$ has frequency units, while $K{e}^{-S_I/\hslash }$ is the dilute crossing rate. This determinant/collective-coordinate structure is also discussed in Dunne, Sulejmanpasic and Ünsal's treatment; unequal neighbouring well frequencies require care and need not share the symmetric-double-well result.

Quantum tunnelling between different energy bands, including transfer between the valence band and conduction band in a tunnel diode.

A physicist whose research includes semiconductors and quantum tunnelling.^[ref]