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An energy level is an allowed energy eigenvalue of a Quantum Hamiltonian. Its corresponding energy eigenstates satisfy $H|n⟩=E_n|n⟩$. Several independent energy eigenstates may share one energy level; their number is the energy-level degeneracy. A quantum superposition of different energy eigenstates need not have a definite energy, even though an ideal energy measurement yields one of the allowed outcomes.

Quantum confinement often produces discrete energy levels: the quantum harmonic oscillator has $E_n=\hslash \omega (n+1/2)$. An unconfined free particle instead has a continuous spectrum. In spectroscopy, a transition between two energy levels exchanges approximately $|E_f-E_i|=h\nu$ with a photon; recoil and external fields can affect the measured frequency. Selection rules determine which energy differences produce observable spectral lines.

Indium arsenide, InAs, is a III-V semiconductor. An InAs island in a larger-gap host can provide quantum confinement for electrons and electron holes. The optical transition energy of a strained, confined island is not simply the bulk material's band gap.

Lead sulfide is $\mathrm{PbS}$, a compound of lead and sulfur that occurs naturally as galena. Its crystalline structure is related to that of sodium chloride.

Lead sulfide is also a semiconductor. Crystal size can strongly affect the optical response of very small particles, making the material an example of the connection between bulk band structure and quantum confinement.

Matjaž Humar explained how quantum confinement changes the optical properties of semiconductor quantum dots. Confinement raises electronic energy scales and modifies exciton states, so changing a dot's size can shift its absorption and fluorescence spectra. His treatment distinguishes regimes by comparing the dot size with the exciton's internal size and the wavelength of its motion. This prevents a bulk bound-pair model from being applied unchanged to a much smaller nanocrystal.^[ref]

A nanomaterial has constituent particles or structural features on the nanoscale, conventionally around $1$–$100\mathrm{nm}$. A collection of nanoparticles is one example; a bulk piece containing nanoscale internal structure can also be relevant. The scientific interest lies in how dimensions and interfaces affect behavior, rather than in treating $100\mathrm{nm}$ as a universal physical threshold.^[ref]

A simple geometric calculation explains one important effect. Suppose a fixed volume $V$ of solid is divided into identical spherical nanoparticles of radius $r$. Ignoring contact between particles, their number $N$ and total surface area $A$ areReducing $r$ from $50\mathrm{nm}$ to $5\mathrm{nm}$ makes $1000$ times as many particles and ten times as much surface area, at the same total volume. More material is consequently close to an interface. Surface reactions and interactions with the surrounding medium can become much more influential; this geometric result alone does not guarantee a faster useful reaction or a more stable material.^[ref]

Size can also affect electronic behavior through quantum confinement. Which effect matters depends on the material, shape and environment. A useful comparison therefore holds composition fixed, measures particle dimensions with electron microscopy, and compares a chosen property across a size series. Such a comparison separates a size-dependent change from a change caused merely by using a different substance.^[ref]

A nanostructure has structural features on the nanoscale. The relevant dimension may be the thickness of a thin film, the diameter of a nanowire, or the overall size of a nanoparticle. A material can therefore be macroscopically large while containing nanostructures internally.

The geometry matters because it changes the relative importance of surfaces, the distances over which charge carriers move, and sometimes the allowed quantum states. A thin film and a nanoparticle made from the same chemical material need not have the same properties: their quantum confinement dimensions and exposed surface areas differ. Characterizing a nanostructure requires specifying which dimensions are small, its shape and its chemical chemical composition, rather than reporting a single size alone.^[ref]

Nanotechnology studies and controls matter whose important dimensions are on the nanoscale, conventionally about $1$–$100$ nanometres. A nanometre is $10^{-9}$ metres. The scientific interest comes from changes in behaviour at these dimensions, including the increasing role of surfaces and, in suitable systems, quantum confinement.^[ref]

For example, subdividing a fixed volume into smaller nanoparticles increases the exposed surface area, changing how much material can participate in a surface reaction. Nanostructures can also alter electrical conductivity, interaction with light and mechanical properties by controlling geometry rather than changing only chemical chemical composition. Useful nanotechnology therefore combines fabrication with measurements of structure and properties, such as electron microscopy or atomic force microscopy.^[ref]

Two applications are ultraviolet photodetection and ultraviolet/near-band-edge light emission. The bulk edge corresponds approximately toAbove-gap absorption creates carriers that can be collected in a photodetector. Measure dark current and wavelength-dependent response to separate photocarriers from heating or leakage. An optically or electrically excited emitter instead uses electron-hole recombination. quantum confinement in sufficiently small dots can shift absorption and emission above the bulk band gap. Surface states can produce broad defect luminescence or nonradiative loss, so a direct bulk gap does not guarantee an efficient, narrow-band emitter.

The ground-state energies are $1.50\mathrm{eV}$ and $0.376\mathrm{eV}$, respectively. For a particle in a box of width $L$, the wave function must vanish at both infinite walls. The lowest nonzero standing wave therefore has wavelength $2L$, giving the kinetic energy eigenvalueUsing the electron mass and converting joules to electronvolts givesDoubling the confinement length divides this kinetic energy by four. A zero potential energy inside the box does not imply zero total energy: a constant, zero-momentum wave function cannot satisfy both wall conditions unless it vanishes everywhere. This is a concrete manifestation of quantum confinement and the Heisenberg uncertainty principle.

A self-assembled InAs quantum-dot single-photon source uses small indium arsenide islands inside a larger-gap gallium arsenide host. Strain during epitaxial growth favors islands after a thin wetting layer has formed. Their characteristic diameter is of order $10$–$30\mathrm{nm}$ and their height a few nanometres, for example $2$–$5\mathrm{nm}$. Growth conditions determine the areal density: low-density samples suitable for isolating dots can be around $10^8$–$10^9\mathrm{c}{\mathrm{m}}^{-2}$, while ordinary dense layers can be much denser. The particular electrically driven device underlying the supplied image had about $5\times 10^8\mathrm{c}{\mathrm{m}}^{-2}$. These are characteristic scales, not dimensions deducible from the emission image.^[ref]

Quantum confinement in the dot traps both electrons and electron holes in three directions, giving discrete energy levels. In a junction device, injected carriers relax into these levels. Electron-hole recombination then emits a photon, and the dot must be repopulated before emitting again on that same transition. Spatial selection of one dot, spectral selection of one line, and well-separated injection pulses make this a single-photon source. Photon antibunching, rather than low average brightness alone, establishes suppression of simultaneous photons. In particular, collecting both photons of a cascade without selecting a line would not produce exactly one collected photon per excitation.

The lower-current feature $X$ is the transition from one confined exciton to the empty dot. The higher-current feature $X_2$ is the first step of the biexciton cascade $XX\to X\to 0$: it becomes strong when the dot can contain two electron-hole pairs. In the weak-pumping limit the probabilities of one and two pairs scale approximately as $I$ and $I^2$, respectively. Coulomb and confinement energies shift the two transitions relative to one another; $X_2$ is not a photon of twice the $X$ energy. In this particular image the $X_2$ photon is about $4$–$5\mathrm{meV}$ higher in energy. With $B_{XX}=2E_X-E_{XX}$, this is a negative biexciton binding energy, since $E_{XX}-E_X=E_X-B_{XX}$.

For the mass estimate, the original plot places the $X$ line at approximately $E_X=1.3945\mathrm{eV}$, with graph-reading precision of order $0.5\mathrm{meV}$. Under the prescribed identification of photon energy with the full lowest exciton excitation energy,The three-dimensional screened Wannier-Mott exciton has the hydrogenic exciton binding energyThereforeReading the line as $1.395\mathrm{eV}$ instead gives $0.331m_0$; about $0.33$–$0.34m_0$ is an appropriate precision. This inference uses the supplied pair-creation energy and a three-dimensional screened hydrogenic model. In a real dot, strain and confinement also alter transition energies, so it is an effective model estimate and does not independently determine either individual carrier mass.

Quantum confinement restricts a particle's wavefunction over a length comparable with its characteristic quantum length scales, altering the allowed energy levels relative to a bulk material. For a parabolic band confined over width $L$, a characteristic spacing is of order ${\hslash }^2/(m^{\ast }{L}^2)$, with boundary-dependent numerical factors. A quantum dot confines all three directions; confinement in only one direction can leave continuous motion in the other two. Finite barrier height and interactions modify the simplest infinite-wall estimates.

Quantum transport studies the flow of particles, electric charge or energy when quantum states, their occupation and their transmission probabilities determine the flow. In a short quantum wire, transverse quantum confinement permits only discrete transverse electronic modes, even though motion along the wire remains possible.

The Landauer formula treats two contacts as reservoirs supplying incident electrons. Their different occupations drive an electric current; a potential barrier changes the transmission probability of each incident mode. At low temperature, opening one perfectly transmitted spin-resolved mode increases the electrical conductance by the spin-resolved conductance quantum $e^2/h$. Quantum tunnelling, interference and Coulomb blockade explain departures from a simple classical resistor picture.