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By Frank W. J. Olver, Leonard C. Maximon.

Online text: dlmf.nist.gov/10.12

This section belongs to Chapter 10, Bessel Functions, credited to Frank W. J. Olver and Leonard C. Maximon. They are chapter authors rather than authors of the entire edited reference. The inspected site version 1.2.8 was released 15 September 2026; that is not a chapter publication date.

By Tom M. Apostol.

Online text: dlmf.nist.gov/25.6

Part of Chapter 25, Zeta and Related Functions, credited to T. M. Apostol. The site release date is not treated as the chapter publication year.

Cited passages: §25.6(i), equations 25.6.1–25.6.2: positive even arguments.

Online text: neuronaldynamics.epfl.ch/online/Ch2.S2.html

Cited passages: §2.2.1, Activation and De-inactivation; §2.2.3, Figure 2.6 and Ion channel dynamics during spike generation; §2.2.3, Refractoriness, Figure 2.9: a stronger second pulse can evoke an earlier second spike; increased conductance shortens the voltage response.

A mutation affecting a house mouse circadian clock.

Changes in behavioural timing can be measured as altered free-running rhythms under constant conditions. Such measurements distinguish an internally generated circadian rhythm from a daily pattern imposed directly by alternating light and darkness. ^[ref]

A vibrational overtone excites more than one vibrational quantum: $v=0\to 2$ is the first overtone, and $0\to 3$ the second. Such transitions vanish in the combined harmonic-potential and linear-dipole approximation. Anharmonicity changes wavefunctions and level spacing; nonlinear terms in the electric dipole moment can also supply otherwise missing transition matrix elements.^[ref]

For an original illustrative level model, setThen $E_1-E_0=\hslash \omega (1-2ϵ)$ and $E_2-E_0=2\hslash \omega (1-3ϵ)$. With chosen $ϵ=0.02$, the fundamental gap is $0.96\hslash \omega$ and the first-overtone gap $1.88\hslash \omega$, below twice the fundamental, $1.92\hslash \omega$. This low-level illustration does not determine transition intensity or justify extrapolation to arbitrarily large $v$. It shows why even a well-resolved fundamental band origin need not equal the harmonic curvature scale ${\tilde{\nu }}_{\mathrm{harm}}$.

The University of Florida's Rotation Vibration Spectrum of the HCl Molecule, University of Florida laboratory protocol, printed pp. 2–3 and 10–11, uses a scanning Michelson interferometer for Fourier transform spectroscopy. An air-filled silicon-window cell supplies the background. Supervised preparation adds concentrated hydrogen chloride liquid; the cleaned cell subsequently receives DCl liquid. Acquisition settings are identical.

The readout is transmission or absorbance against spectroscopic wavenumber, resolving molecular rotation alongside stretching. Background addresses instrument and cell contributions; cleaning addresses carryover. Excessive absorption impairs analysis. Assigning isotopes and rotational branches precedes fitting band parameters to test isotope scaling of diatomic vibration. These are prescribed measurements, not numerical observations supplied here.^[ref]

For a transparent ideal model, add the linear rigid rotor to the harmonic vibrational energy levels and assume the same frequency-valued rotational constant $B$ in both vibrational states:For a spin-free diatomic molecule with no electronic angular momentum, electric-dipole selection rules give $\Delta J=\pm 1$ for this vibrational absorption. Subtracting initial from final energy eigenvalues yieldsThese are two rotational branches about the vibrational band origin, not a single stretch line. For stipulated dimensionless values ${\tilde{\nu }}_{\mathrm{harm}}=100$ and $\tilde{B}=1$, the first three R-branch positions are $102,104,106$, and the first three P-branch positions are $98,96,94$. There is no $J=0\to -1$ transition. The numbers illustrate level subtraction; they are not HCl measurements. Real upper and lower rotational constants differ, and centrifugal distortion shifts lines further.

In the ideal electric-dipole interaction, a vibrational transition needs a nonzero matrix element of the electric dipole moment. For a diatomic bond stretch, expand the body-axis dipole component as $d(q)=d_e+d_e^{\prime }q+\cdots$. A changing electric dipole moment, rather than a permanent value $d_e$ alone, permits the harmonic fundamental. Keeping only the linear term and harmonic wavefunctions gives $\Delta v=\pm 1$.^[ref]

The derivation is sharper than a picture of a shaking molecule. Orthogonality removes the constant term for $v^{\prime }\ne v$, and creation and annihilation operators giveHere $\delta$ is the Kronecker delta. Thus absorption raises $v$ by one when the dipole derivative and angular factor are nonzero, and its photon satisfiesA nonzero vibrational matrix element is only one factor: rotational electric-dipole selection rules, initial populations, and spectral resolution also constrain visibility.

For an isolated diatomic molecule with identical atomic nuclei, such as molecular hydrogen in its symmetric electronic ground state, stretching preserves zero electric dipole moment. By symmetry, this homonuclear stretch is infrared inactive in the ideal dipole model, although it still vibrates. Other interactions or environments can permit detection.^[ref]

Let $r$ be the instantaneous bond length, $r_e>0$ its equilibrium value, and $q=r-r_e$ the stretching displacement. Assume one fixed electronic quantum state, a smooth potential energy $V(r)$ with a nondegenerate local minimum at $r_e$, and omit molecular rotation initially. The Taylor series has no linear term because $V^{\prime }(r_e)=0$:Keeping only the quadratic term is the harmonic approximation near a potential minimum. The bond force constant $k$ has units $\mathrm{N}{\mathrm{m}}^{-1}$. It measures local stiffness; it is not the bond-dissociation energy, which concerns the depth and distant shape of the entire well.

The two-particle relative coordinate explains the reduced mass. For stretching along a fixed axis in a stationary centre of mass frame, let $u_1,u_2$ be the nuclear displacements from equilibrium. Then $u_2-u_1=q$ and $m_1{u}_1+m_2{u}_2=0$, soHere $T$ is the internal kinetic energy, and $m_1,m_2>0$ are the two nuclear masses. The lighter partner moves farther; the reduced mass is smaller than either individual mass. The restoring force $-dV/dq\simeq -kq$ gives simple harmonic motion:The angular frequency is $\omega$, the ordinary frequency is $\nu$, and ${\tilde{\nu }}_{\mathrm{harm}}$ is the corresponding spectroscopic wavenumber; $c$ is the speed of light. Use $c$ in $\mathrm{cm}{\mathrm{s}}^{-1}$ to obtain ${\tilde{\nu }}_{\mathrm{harm}}$ in $\mathrm{c}{\mathrm{m}}^{-1}$. The symbol $k$ here is a bond force constant, not an angular wavenumber.

For an original dimensionless example, choose $m_1=m_2=2$ and $k=4$ in consistent model units. Then $\mu =1$ and $\omega =2$. Releasing at $q(0)=0.1$ with $\dot{q}(0)=0$ gives $q(t)=0.1\cos (2t)$ and $u_1=-q/2$, $u_2=q/2$. Each atomic nucleus moves only half the stretching amplitude, while the conserved internal mechanical energy above $V_0$ is $\frac{1}{2}k(0.1{)}^2=0.02$. Fixing one partner instead would give $\omega =\sqrt{2}$, demonstrating why a one-body calculation changes the model.

Reverse transcription makes DNA from an RNA template using a reverse transcriptase. It reverses the template/product relationship of ordinary transcription, which makes RNA from DNA. The resulting Complementary DNA can supply a template for PCR amplification.^[ref]

Chelex extraction releases DNA from heated biological cells in the presence of a resin that binds metal ions. The PV92 teaching experiment transfers the supernatant, leaving resin and debris behind.

An Alu element is a repeated DNA element; the PV92 teaching experiment examines one particular insertion site rather than amplifying every copy across the genome.

Alu elements are primate-associated nonautonomous retrotransposons, typically about 300 bases long. A copy can persist at a genomic location after insertion, making presence or absence at a corresponding site informative about inheritance. The family designation and the allelic state of one insertion locus are different levels of description. ^[ref]

The PV92 locus is a genetic locus on human chromosome 16 used in the PV92 teaching experiment to compare the presence and absence of an Alu element.

Online text: www.ncbi.nlm.nih.gov/books/NBK26824/

Cited passages: Components of the Cell-Cycle Control System: cyclically activated protein kinases, Figures 17-15–17-17, and “Cdk Activity Can Be Suppressed Both by Inhibitory Phosphorylation and by Inhibitory Proteins”; Components of the Cell-Cycle Control System: inhibitory phosphorylation and Figure 17-18, Cdc25-mediated removal of inhibitory phosphates; Molecular Biology of the Cell, 4th edition, Components of the Cell-Cycle Control System: “The Cell-Cycle Control System Is Based on Cyclically Activated Protein Kinases”, Figures 17-15–17-16.

A molecular beam skimmer is a small aperture, commonly at the tip of a cone, that transmits the central part of an expanding molecular beam into a downstream chamber. It separates source and measurement regions so that differential pumping can maintain a lower background pressure after the source. Its geometry selects directions; that selection alone does not give all transmitted molecules the same speed.

The surface also perturbs the beam. At high source number density, particles reflected from the skimmer walls can collide with the incoming flow and create shock waves, reducing and heating the transmitted beam. A 2017 experiment cooled the skimmer and measured a beam-flux increase of about eightfold under its tested conditions. A pulsed plasma glow made the disturbed flow visible; cooling reduced reflected momentum and, at lower temperatures, promoted adsorption. This tests why a narrower geometric opening is not a complete model of a real skimmer.^[ref]