A semiconductor state is approximated by a slowly varying envelope multiplying a cell-periodic Bloch function from a selected band. effective masses govern the envelope equation. Interband optical matrix elements then factor approximately into a cell-scale momentum matrix element and an envelope overlap; abrupt interfaces, band mixing and nonparabolicity limit the separation.

In an infinite square well, electron and hole envelopes are the same normalized sine functions even when their effective masses differ. In the electric dipole and envelope approximations, their overlap is $\int F_{n^{\prime }}^{\ast }(z)F_n(z)dz={\delta }_{n^{\prime }n}$. The same-index rule is an ideal-envelope rule, separate from Bloch-state polarization selection rules.

For parabolic bands confined by an infinite well of width $L$, a same-index transition has edge $E_{g,j}+{\pi }^2{\hslash }^2{n}^2(1/m_e^{\ast }+1/m_{h,j}^{\ast })/(2L^2)$. Above it the in-plane pair kinetic energy is ${\hslash }^2{k}_{\parallel }^2/(2{\mu }_j)$ with reduced mass ${\mu }_j^{-1}=(m_e^{\ast }{)}^{-1}+(m_{h,j}^{\ast }{)}^{-1}$.

An electric field normal to a semiconductor quantum well shifts the electron and hole confinement energies and displaces their envelopes in opposite directions. The exciton optical transition usually redshifts and its oscillator strength falls as their overlap decreases. In a symmetric unbiased well the permanent dipole $p_0$ vanishes, so the leading weak-field shift is quadratic. This optical shift includes confinement and binding changes; it is not solely a change in exciton binding energy.

A transport expression relating current to occupied incident modes and their transmission probabilities. For spin-degenerate independent electrons, $I=(2e/h){\sum }_i\int T_i(E)[f_L(E)-f_R(E)]dE$, with the direction convention chosen consistently. In linear response a perfectly transmitted mode contributes $2e^2/h$.

A narrow conductor has discrete transverse modes and nearly continuous longitudinal motion. Its conductance depends on the transmissions of occupied quantum wire subbands, spin counting and contacts. The Landauer formula connects these transmissions to two-terminal conductance. Disorder, interference, dephasing and interactions distinguish ballistic, diffusive, localized and interacting regimes.

The characteristic distance over which phase-breaking processes suppress electron-path interference. For diffusion, $L_{\varphi }\sim \sqrt{D{\tau }_{\varphi }}$; ballistic phase propagation instead gives a scale $v{\tau }_{\varphi }$. This differs from the elastic mean free path, since elastic scattering can change momentum while retaining coherence.

A coherent disordered mesoscopic conductor has reproducible, sample-dependent conductance variations when magnetic field, carrier energy or disorder configuration changes. Their typical scale is of order $e^2/h$ in a fully coherent diffusive sample, with numerical factors determined by symmetry, dimensionality and geometry. Thermal averaging and finite phase coherence length reduce the measured amplitude.

An eigenstate of an interacting system that combines degrees of freedom which would be separate in the uncoupled model.

For a real point coupling $g$ between a three-dimensional continuum and one discrete state, $A=m{g}^2/(2\pi {\hslash }^2)$. After the short-distance level shift is absorbed, the isotropic amplitude has the displayed form. The imaginary part is fixed by the outgoing boundary condition, and $\mathrm{Im}(1/f_k)=-k$ supplies elastic unitarity.

For a constant-mass one-dimensional particle propagating through three regions of potentials $0,V_1,V_0$, let $k_j$ be the real positive wavevectors. A layer with $k_{1}a=\pi /2$ makes the two interface reflection amplitudes cancel exactly when $k_1^2=k_0{k}_2$. Equivalently $E-V_1=\sqrt{E(E-V_0)}$. For an increasing intermediate step this requires $E>V_0$ and $V_0/2<V_1<V_0$. The result follows from $r=(r_{01}+r_{12}{e}^{2i{k}_{1}a})/(1+r_{01}{r}_{12}{e}^{2i{k}_{1}a})$ and $r_{ij}=(k_i-k_j)/(k_i+k_j)$.

A particle with a periodic angular coordinate undergoing free rotational motion interrupted by impulsive angular potentials. A single-period unitary combines free propagation and a position-dependent kick. Its momentum-space interference can give resonant ballistic growth or other transport regimes.

For the Gaussian-unit flux term $e{\Phi }_B\dot{\varphi }/(2\pi c)$, canonical momentum is shifted by $e{\Phi }_B/(2\pi c)$. Periodic wavefunctions then give the displayed spectrum for integer $n$. The flux period is $hc/|e|$ in these units; the corresponding SI expressions omit $c$. A gauge change transfers the flux to a twisted boundary condition rather than removing it from the physical problem. Half-integer $f$ gives a pair of degenerate nearest-momentum levels.

The quantum asymmetric top has different rotational-energy coefficients on its principal axes. Its Hamiltonian commutes with $L^2$ but generally not with an individual component, so diagonalization proceeds within fixed total-angular-momentum multiplets. For a mechanical rigid rotor the positive coefficients are reciprocals of twice its principal moments of inertia. The component sign convention for body-fixed generators does not affect this squared-component energy expression.

A phase-selected Hermitian field observable. With annihilation operator $a$ and $[a,a^{†}]=1$, define $X_1=(a+a^{†})/2$ and $X_2=(a-a^{†})/(2i)$. Then $[X_1,X_2]=i/2$ and the Fock vacuum has variance $1/4$ in either quadrature. This dimensionless convention gives $X_{\vartheta }=X_1\cos \vartheta +X_2\sin \vartheta$. Balanced homodyne detection selects $\vartheta$ through the reference wave phase.

Interference of a signal with a coherent reference at the same optical carrier frequency to measure phase-sensitive field information. The reference fixes an optical quadrature. The inferred noise depends on reference strength, detector efficiency and the detection arrangement.