Section of Intermediate Algebra 2e by Lynn Marecek and Andrea Honeycutt Mathis. No separate chapter author or section publication date is credited. The parent edition was published 6 May 2020.

Online text: openstax.org/books/intermediate-algebra-2e/pages/9-2-solve-quadratic-equations-by-completing-the-square

Cited passages: §9.2, Binomial Squares Pattern; Solve Quadratic Equations of the Form ax²+bx+c=0; Citation information.

Chapter of Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach. Whole-work author credits apply; no separate chapter author byline is established. No chapter publication or revision date is established; the URL path does not establish a date or edition.

Online text: interactivetextbooks.tudelft.nl/interactive-optics/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html

Cited passages: Scalar Diffraction Optics: Fresnel Approximation, equations (381)–(385), following paraxial equation remark; Fresnel and Fraunhofer Approximations, equations (386)–(392): omitted aperture quadratic phase, full-diameter Fresnel number; About the authors: biographies and final whole-book author credit.

Section of Interactive Linear Algebra by Dan Margalit and others by its whole-book authors Dan Margalit and Joseph Rabinoff; no separate chapter byline or date is asserted.

Online text: textbooks.math.gatech.edu/ila/one-to-one-onto.html

Cited passages: §3.2.1, One-to-one matrix transformations theorem and Wide matrices do not have one-to-one transformations; §3.2.2, Onto matrix transformations theorem and Tall matrices do not have onto transformations.

Section of Interactive Linear Algebra by Dan Margalit and others by its whole-book authors Dan Margalit and Joseph Rabinoff; no separate chapter byline or date is asserted.

Online text: textbooks.math.gatech.edu/ila/row-reduction.html

Cited passages: §1.2.1, Augmented Matrices and Row Operations: preservation of solutions; §1.2.2, Echelon Forms: leading entries and reduced form; §1.2.3, definitions of pivot position and pivot column.

Online text: textbooks.math.gatech.edu/ila/projections.html

Section of Interactive Linear Algebra by Dan Margalit and others, by Dan Margalit and Joseph Rabinoff; no separate chapter byline or date is asserted. The parent record's 2019 year dates the front matter; the colophon copyright is 2017.

Cited passages: §6.3.1, Orthogonal decomposition theorem and Closest vector and distance; §6.3.2, Properties of Orthogonal Projections.

The complement system comprises immune proteins; its components provide an interaction partner for Fc fragments.

Section 5.6 of Hyper-TVT by ETH Zurich. No personal chapter author is credited. ETH Zurich’s Institute of Process Engineering issued the teaching site; the ETH Web Archive hosts the preserved text. No publication or revision date is established.

Online text: www.hyper-tvt.ethz.ch/distillation-efficiency.html

Cited section: 5.6 Efficiency, opening paragraphs: equilibrium plates versus actual plates; Overall efficiency paragraph: hydrodynamics and mass transfer affect efficiency; Lower efficiency increases actual stage count; Site imprint, Publisher and Webmaster fields.

Hydrogen atom spectral lines terminating at principal quantum number $n=1$; see the Bohr model for the elementary energy difference calculation.

Hydrogen atom spectral lines terminating at principal quantum number $n=2$; see the Bohr model for the elementary wavelength calculation.

For the elementary hydrogen atom derivation, see the Bohr model.

For spherical charge density ${\rho }_e=-e{e}^{-2r/a_0}/(\pi a_0^3)$, integrate $4\pi s^2{\rho }_e(s)$ from zero to $r$ to obtain the displayed enclosed charge. Gauss's law then gives $E_e=Q_e(r)/(4\pi {ϵ}_0{r}^2)$, directed inward. Near the origin $E_e\simeq -er/(3\pi {ϵ}_0{a}_0^3)$; at large radius it approaches the field of charge $-e$.

Adding a point nucleus of charge $+e$ leaves the unperturbed total fieldIt is exponentially small rather than identically zero, because the electron cloud has no compact boundary. This frozen-density field alone does not describe the dominant long-range response to another charge: charge-induced dipole attraction requires allowing the atom to polarize.

A family of hydrogen emission lines terminating at a fixed lower principal quantum number $n_f$. In the nonrelativistic Bohr model, their inverse wavelengths are $1/\lambda =R_H(1/n_f^2-1/n_i^2)$ for integers $n_i>n_f$. Holding $n_f$ fixed and increasing $n_i$ gives lines approaching the series limit $1/\lambda =R_H/n_f^2$. Reduced mass shifts the Rydberg constant, while additional interactions produce corrections to this gross spectral pattern.

Model a proton as a uniformly charged thin spherical shell of radius $b$. Write $C=e^2/(4\pi {ϵ}_0)$. The shell theorem gives an electron potential energy $-C/b$ inside the shell and $-C/r$ outside. Relative to a point Coulomb potential, the perturbation is consequently $C(1/r-1/b)$ for $r<b$ and zero otherwise.

For the hydrogen atom ground state, time-independent perturbation theory gives the first-order energy correctionSince $E_{1s}=-C/(2a_0)$, the signed fractional change is $\Delta E/E_{1s}=-4b^2/(3a_0^2)+O((b/a_0{)}^3)$. The correction $\Delta E$ is positive: distributing the positive charge weakens the attraction near the origin and reduces the magnitude of the binding energy. Here $a_0$ is the Bohr radius and $b\ll a_0$ is the small-radius approximation.