For an isolated quantum system, pure states are normalized rays in a complex Hilbert space, observables are self-adjoint operators, and their probabilities obey the Born rule. A spectral projector $P_a$ has outcome probability $⟨\psi |P_a|\psi ⟩$; an ideal repeatable projective measurement has conditional state $P_a\psi /\sqrt{⟨\psi |P_a|\psi ⟩}$. The Schrödinger equation specifies continuous unitary evolution between such ideal measurements, given a Hamiltonian operator. General measurements need not have the projective state-update rule.

Composite systems use the tensor product of their state spaces. Identical particles additionally obey the bosonic or fermionic exchange symmetry appropriate to their species. The photoelectric effect, Davisson–Germer experiment and double-slit experiment motivate energy quanta, matter-wave phases and probability interference, respectively; these experiments constrain the framework but do not separately prove every abstract postulate or specify every system's Hamiltonian.

Second quantization describes identical-particle quantum states in Fock space using creation operators and annihilation operators. Relative to an orthonormal basis of one-particle states, a one-body operator becomes ${\sum }_{\alpha \beta }{h}_{\alpha \beta }{a}_{\alpha }^{†}{a}_{\beta }$. The canonical commutation relation implements bosonic statistics, while the canonical anticommutation relations implement fermionic statistics. Expanding the quantized field in a complete basis gives a delta-function equal-time bracket; restricting the expansion to finitely many modes replaces that delta function by the corresponding projection operator kernel.

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.

A virtual particle is an internal contribution to a perturbative scattering amplitude, commonly represented by an internal line of a Feynman diagram. The line contributes a propagator, and its internal four-momentum is summed or integrated as required by the diagram. It need not satisfy the on-shell relativistic energy-momentum relation of an external detected particle.

For example, a virtual photon connects charged currents in electron scattering. This is a calculational contribution to the full interaction amplitude, rather than a directly observed photon flying between the electrons. Energy conservation and momentum conservation still hold at the vertices; off-shell propagation does not mean energy has been borrowed in violation of conservation. Individual virtual contributions can depend on the gauge or perturbative organization, whereas an observable prediction requires the appropriate complete amplitude.

A quantum dot of InAs embedded in gallium arsenide confines injected electrons and electron holes. Strain-driven island formation supplies dots with lateral dimensions of order tens of nanometres and heights of a few nanometres. In an electrically driven implementation, a dot lies in a junction's intrinsic region. Selecting one dot spatially and one radiative line spectrally turns discrete exciton recombination into a single-photon source. Pulsed injection must avoid re-excitation during the same collection window.

The pioneering electrically driven device used an areal dot density around $5\times 10^8\mathrm{c}{\mathrm{m}}^{-2}$, a GaAs p-i-n structure and an aperture selecting individual-dot light.^[ref]

Rounding a voltage to the nearest level in a uniform grid of spacing $q$ gives a quantization error of magnitude at most $q/2$, provided the input is within the converter's range. This follows because each grid point represents the interval bounded by the midpoints to its two neighbors. Clipping invalidates the bound.

The bound concerns recorded samples, whereas the Nyquist–Shannon sampling theorem concerns reconstruction from exact samples of a bandlimited function. If a reconstruction is $\tilde{v}(t)={\sum }_n{Q}_q(v_n)w_n(t)$, its error is ${\sum }_n{ϵ}_n{w}_n(t)$, so a general bound involves $(q/2){\sum }_n|w_n(t)|$, not simply $q/2$. Thus sampling bandwidth and per-sample rounding precision are distinct specifications. Under an additional uniform independent-error model, the quantization-noise standard deviation is $q/\sqrt{12}$, an rms statement rather than a worst-case reconstruction guarantee.

A stop-transfer sequence interrupts ongoing passage of a polypeptide chain across a membrane and retains a membrane-spanning segment. Protein targeting, insertion and subsequent sorting involve distinguishable sequence instructions.^[ref]

In the figure's cleaved-leader example, assume the N-terminus of the mature chain has entered the ER lumen, a stop-transfer sequence remains in the membrane, and the remaining C-terminus is made on the cytosol side. The outcome differs from releasing the entire mature chain into the ER lumen. The latter is soluble cargo, potentially retained there or transported onward; entry alone does not establish secretion.

A signal anchor combines targeting or initiation of membrane passage with a retained membrane-spanning segment. Its orientation determines which flanking part enters the ER lumen; a single internal signal can yield either orientation.^[ref]

In the figure's stipulated internal-anchor example, the N-terminus remains in the cytosol and the C-terminus enters the ER lumen. Reversing that orientation would reverse which end is protected from an externally added protease. Merely counting one membrane-spanning segment cannot choose between the two.

A signal peptide is a short targeting segment of a newly made polypeptide chain. A familiar ER signal sequence lies near the N-terminus and is removed during entry; Günter Blobel and Bernhard Dobberstein's 1975 experiments linked a larger immunoglobulin light chain precursor to a processed product.^[ref]

Cleavage is a separate question from recognition. Not every targeting signal is a cleaved signal peptide: an internal signal anchor can remain part of a transmembrane protein. Conversely, loss of a segment does not by itself prove membrane crossing. Compare the independent size and accessibility readouts in the microsomal protein import experiment.

A serine protease uses a serine side chain as a catalytic nucleophile in peptide-bond hydrolysis. In the chymotrypsin family, a catalytic arrangement involving serine, histidine and aspartate promotes formation and hydrolysis of a covalent acyl-enzyme intermediate. The catalytic mechanism and substrate-recognition pocket are distinct: related serine proteases can prefer different cleavage sites. ^[ref]

Hoare logic proves properties of programs through Hoare triples $\{P\}C\{Q\}$. A precondition $P$ constrains initial states and a postcondition $Q$ constrains returned states. The basic rules establish partial correctness; termination requires a further argument.^[ref]

For a program expression $E$ whose evaluation is defined and has no side effects, the assignment rule isThe substitution replaces the free occurrences of $x$ in the desired postcondition. For example, obtaining $x\ge 1$ after x := x+1 requires $x+1\ge 1$ beforehand, so $x\ge 0$ suffices. Integer overflow must be excluded or handled by the stated arithmetic model.

Sequential composition joins a first postcondition to the next precondition. For a while loop, proving $\{I\wedge b\}C\{I\}$ establishesHere $I$ is the loop invariant. For while x > 0 do x := x-1, starting with an exact nonnegative integer, $x\ge 0$ is preserved and the false exit guard yields $x=0$. The decreasing nonnegative integer $x$ is also a loop variant, establishing total correctness for this example.^[ref]

An integer arithmetic result outside the range of its chosen representation. A programming language may specify wrapping, trapping or another behavior; a proof over mathematical integers must account for that difference.

A description of the information needed to determine a program's next permitted step. In the summation loop verification, this includes the current variable values and the position in the instructions.

Section of Precalculus 2e by Jay Abramson. The citation panel credits Jay Abramson as whole-book author; the preface gives no individual chapter allocation. OpenStax publishes and hosts the book. The parent record establishes whole-book publication on 21 December 2021; no separate section revision date is asserted.

Online text: openstax.org/books/precalculus-2e/pages/11-2-arithmetic-sequences

Cited passages: Arithmetic Sequence definition; Recursive Formula for an Arithmetic Sequence; Explicit Formula for an Arithmetic Sequence.

Online text: openstax.org/books/precalculus-2e/pages/10-4-rotation-of-axes

Section of Precalculus 2e by Jay Abramson. The citation panel names Jay Abramson as whole-book author and gives publication on 21 December 2021; the preface supplies no individual chapter allocation. No separately credited section author or revision date is established.

Cited passages: Identifying Conics without Rotating Axes; Using the Discriminant to Identify a Conic.

Online text: openstax.org/books/precalculus-2e/pages/10-1-the-ellipse

Section of Precalculus 2e by Jay Abramson. The citation panel names Jay Abramson as whole-book author and gives publication on 21 December 2021; the preface supplies no individual chapter allocation. No separately credited section author or revision date is established.

Cited passages: Writing Equations of Ellipses in Standard Form, Figure 2; Ellipse definition and Deriving the Equation of an Ellipse Centered at the Origin, Figure 5; Citation panel authors/title/publication date.

The radius of gyration of a dilute linear polystyrene coil in a good solvent depends on its molar mass as $R_g=K{M}^{\nu }$, with a material- and solvent-dependent amplitude. A manufacturer-supplied light-scattering calibration for polystyrene in benzene or toluene near room temperature is the displayed relation, quoted over $M\simeq 2\times 10^5$ to $5\times 10^7\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The exponent is close to the Flory approximation $3/5$. Values outside the quoted mass range are extrapolations, not extra measured data. Using $c^{\ast }=M/[N_A(4\pi /3)R_g^3]$ then gives the polymer overlap concentration scaling $c^{\ast }\propto M^{1-3\nu }$. ^[ref]

A nascent chain is a polypeptide chain still being synthesized by a ribosome. Its exposed targeting information can direct entry into the endoplasmic reticulum before biological translation finishes.