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For the two $m=0$ branches of the Breit-Rabi formula, their splitting is $\sqrt{{\Delta }_{\mathrm{hf}}^2+[(g_J-g_I){\mu }_{B}B{]}^2}$. Expanding at small field gives the quadratic frequency shift displayed above. Vanishing linear frequency sensitivity at zero field therefore does not eliminate field-gradient dephasing at a finite bias field.

Recovering two orthogonal components of a signal by averaging its products with sine and cosine reference signals at the chosen frequency.

The 2002 experiment described in Manipulating the Quantum State of an Electrical Circuit demonstrates Ramsey interferometry in a circuit using superconductivity. An aluminum island between two small Josephson junctions and a larger readout Josephson junction formed the loop; a dilution refrigerator cooled it. Manipulation used the charge/flux bias point least sensitive to first-order fluctuations. Microwave pulse amplitude and duration were calibrated using Rabi oscillations. Two phase-coherent $\pi /2$ pulses then enclosed a variable free interval. A current pulse switched the large Josephson junction to finite voltage with state-dependent probability; a preamplifier and threshold discriminator counted repeated trials.^[ref]

As a readout control, switching was tested against pulse height at thermal equilibrium. The Ramsey interferometry oscillation period agreed with inverse frequency detuning, supporting relative quantum phase accumulation against the external oscillator. The decaying envelope indicated finite quantum coherence, while reduced contrast was attributed provisionally to population relaxation during readout. This supports controlled coherent evolution over a finite interval, without establishing perfect isolation or identifying one exclusive noise mechanism.^[ref]

A quantum harmonic chain has quadratic couplings between canonical coordinates and momenta. A normal mode transformation turns it into independent quantum harmonic oscillators. A continuum elastic action with time coefficient $\chi$ and spatial stiffness $\kappa$ has long-wave frequency ${\omega }^2=(\kappa /\chi )q^2$, the structure of a smooth condensate phase action.

For one quantum harmonic oscillator of frequency $\Omega$, $Z=e^{-x/2}/(1-e^{-x})$ with $x=\hslash \Omega /(k_{B}T)$. Differentiating $\ln Z$ gives $U=\hslash \Omega [1/2+1/(e^x-1)]$ and $C_V=k_B{x}^2{e}^x/(e^x-1{)}^2$. Zero-point energy leaves $U\to \hslash \Omega /2$ as $T\to 0$ but contributes no heat capacity. Excitation freeze-out makes $C_V$ exponentially small, while high temperature gives $U\sim k_{B}T$ and $C_V\to k_B$.

A lossless intermediate section of positive real characteristic impedance $Z_M$ matches unequal real exterior characteristic impedances $Z_L,Z_R$ when $Z_M^2=Z_L{Z}_R$ and its propagation wave phase is an odd multiple of $\pi /2$. The length is an odd quarter of the internal wavelength. This principle also applies to the characteristic impedance of a stretched string.

For a uniform intermediate string of length $l$, put $\delta =k_{M}l$. Continuity of displacement and transverse force gives the displacement reflection coefficientAt real positive mechanical impedances, the denominator never vanishes. Its numerator vanishes only if $(Z_L-Z_R)\cos \delta =0$ and $(Z_M^2-Z_L{Z}_R)\sin \delta =0$. Thus unequal exterior strings require the quarter-wave condition. Equal exterior strings with a different middle string instead transmit perfectly at integer half-wavelengths; if all three mechanical impedances agree, every frequency is matched.

An oscillatory feature with finite width in a source's variability power spectrum, rather than a perfectly periodic clock. Orbital, precession and disc-mode interpretations connect different frequencies to the same compact object. Identifying an observed peak with an innermost stable circular orbit orbital frequency is therefore a model assumption, not a direct black-hole mass or spin measurement.

In 1938, Isidor Isaac Rabi, Jerrold Zacharias, Sidney Millman and Polykarp Kusch introduced molecular-beam magnetic resonance. Their 1939 account describes a heated oven, vacuum slits, inhomogeneous A/B magnetic fields with opposite gradients, and a uniform static C magnetic field. Conductors supplied transverse oscillatory radiofrequency (RF) fields. Transitions spoiled refocusing, reducing the tungsten-filament detector signal. Alignment compared A-only and B-only attenuation; C-field sweeps at fixed RF frequency limited oscillator pickup. Matching frequency/field ratios across compounds checked nuclear assignments. Magnetic field calibration limited absolute magnetic moments; weak intervening fields caused unwanted transitions.^[ref]

The readout has a useful logical structure: A and the apertures admit trajectories correlated with a magnetic moment projection; B tests whether that projection has survived C. The two deflectors compensate for unchanged states, whereas a changed projection experiences a different force in B. A dip followed by recovery as the static magnetic field passes through resonance is therefore a population-sensitive signal. A comparison with RF absent or far from resonance is the natural baseline in this functional model; RF pickup, drift or background loss must not be mistaken for a state transition. A dip alone does not establish a pure initial quantum state or resolve an entire time-domain Rabi cycle.

This beam method measures transmitted particles after state-dependent deflection. The later bulk nuclear magnetic resonance methods of Felix Bloch and Edward Mills Purcell instead use a macroscopic sample and detect nuclear induction or RF absorption; NMR spectroscopy builds on that bulk setting. The common resonant coupling does not make their preparations or readouts interchangeable.^[ref][ref]

For an ideal single transit, let the oscillatory magnetic field occupy a region of length $L$, take the longitudinal speed $v$ constant, and neglect edge fields. Each molecule then experiences one square pulse of duration $\tau =L/v$. This is one contiguous interaction region, distinct from the two separated pulses of Ramsey interferometry. The static magnetic field sets the selected transition gap; the transverse drive sets Rabi angular frequency. A change of RF frequency or static magnetic field changes Rabi detuning.

For an original ideal detector model, suppose every successful transition removes an otherwise accepted molecule and the incident beam occupies the lower state of the selected pair. With accepted no-transition signal $S_0$, constant background $S_{\mathrm{bg}}$, and effective loss contrast $0\le \eta \le 1$, writeFor identical trajectories $\overline{P}=P_{g\to e}(L/v)$, giving an intensity dip where transfer is largest. For real trajectory variation, a normalized joint probability density $w(v,ϵ,{\Omega }^{\prime })$ gives the static-offset modelHere $ϵ$ is the trajectory's transition-frequency offset and ${\Omega }^{\prime }$ its drive strength, each in angular-frequency units. The weighting is for molecules accepted by the apparatus, rather than an assumed unfiltered oven distribution. In particular, designing $\Omega L/v_0=\pi$ does not produce a Rabi pi pulse for every speed: at $0.8v_0$ and $1.2v_0$ the resonant transfer probabilities are respectively ${\sin }^2(5\pi /8)\simeq 0.8536$ and ${\sin }^2(5\pi /12)\simeq 0.9330$.

Spread in transit times, static magnetic fields and drive amplitudes can reduce or smear oscillations without quantum decoherence on any individual trajectory. Even the single-transit square pulse has finite spectral width of order $1/\tau$ in angular frequency. If the magnetic field varies substantially along one path, its Hamiltonian operator changes during the passage and the constant-offset average above must be replaced by a time-ordered exponential. Actual molecules may also have several coupled energy levels; the isolated pair and perfect loss discriminator are explicit idealizations of this model.

For a rotating-frame two-state Hamiltonian operator $H=\hslash \mathbf{W}\cdot \mathbf{\sigma }/2$, the Bloch vector obeys $\dot{\mathbf{R}}=\mathbf{W}\times \mathbf{R}$. Thus $\mathbf{W}$ specifies a rotation axis and angular speed. With detuning defined as drive frequency minus transition frequency and the lower state at positive $R_z$, a phase-zero transverse drive can give $\mathbf{W}=(\Omega ,0,\delta )$. Changing basis or detuning convention requires changing the corresponding signs consistently.

For central potential $V(r)=\beta r^k$ and nonzero angular momentum $J$, a circular orbit satisfies $\beta k{r}_0^{k+2}=J^2/m$, so attraction requires $\beta k>0$. Twice differentiating the radial effective potential $V+J^2/(2m{r}^2)$ gives $V_{\mathrm{eff}}^{\prime \prime }(r_0)=(k+2)J^2/(m{r}_0^4)$. Its harmonic approximation therefore has radial angular frequency ${\omega }_r=\sqrt{k+2}|J|/(m{r}_0^2)$, whereas $|{\omega }_{\theta }|=|J|/(m{r}_0^2)$. This proves the ratio and shows stability precisely when $k>-2$. At $k=-2$ the tuned effective potential is flat; at $k<-2$ the radial disturbance grows exponentially. Rationality of this local frequency ratio is only a necessary condition for all nearby orbits to close; Bertrand's theorem makes a stronger finite-amplitude assertion.

Radiant exitance is total outward emitted radiant flux per actual emitting area, measured in $\mathrm{W}{\mathrm{m}}^{-2}$. Integrate radiance over the outward hemisphere, keeping the projected area factor:If the radiance is direction-independent, the spherical coordinates element is $d\Omega =\sin \theta d\theta d\phi$, soThe weighted solid angle is $\pi \mathrm{sr}$, although the whole hemisphere covers $2\pi \mathrm{sr}$. Directions near grazing contribute less because of the projected area. This geometric factor is the bridge between Lambert's cosine law and the total output of a diffuse patch. ^[ref].

For the uniform patch with $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, radiant exitance is $M=10\pi \mathrm{W}{\mathrm{m}}^{-2}$. Its area $0.020{\mathrm{m}}^2$ gives total radiant flux $\Phi =MA=0.20\pi \mathrm{W}\approx 0.628319\mathrm{W}$ over the outward hemisphere. This total differs from its radiant intensity $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$, which describes only one direction.

For an ideal blackbody, integrate Planck's law over frequency as well: $M=M_e=j^{\ast }=\sigma T^4$ by the Stefan–Boltzmann law. An ideal $6000\mathrm{K}$ emitting patch of area $1\mathrm{m}{\mathrm{m}}^2$ emits about $73.4881\mathrm{W}$ outward. Irradiance counts incoming radiation separately; for purely emitted output the net outward exchange per area is $M-E$. Reflected output must also be included if the surface reflects incident radiation.

Radiation pressure is the normal momentum flux transferred by electromagnetic radiation to a surface. A photon travelling in vacuum carries momentum $p_{\gamma }=E_{\gamma }/c=h\nu /c$, where $E_{\gamma }$ is its energy, $\nu$ its frequency, $h$ the Planck constant and $c$ the speed of light. Photon absorption transfers that momentum; specular reflection reverses its normal component. This lets light exert a force even when almost no energy becomes heat in the receiver.

Two useful formulas describe different preparations. A directed beam normally incident on a stationary perfect absorber gives pressure $I/c$, and on a stationary perfect mirror gives $2I/c$. Here $I$ is light intensity, the incident power per beam cross-sectional area perpendicular to propagation, in $\mathrm{W}{\mathrm{m}}^{-2}$. An isotropic photon gas has pressure $u/3$, where $u=U/V$ is its radiation internal energy per volume, in $\mathrm{J}{\mathrm{m}}^{-3}$. Stored energy density and travelling energy flux must be distinguished: an isotropic field can have zero net energy flux and positive radiation pressure.

The physical appeal is that the same momentum conservation connects a single photon collision, a mirror's measurable force, and a radiation-filled cavity's equation of state. The directional calculation below complements radiation pressure on a perfectly absorbing surface; the equilibrium spectrum and its temperature dependence belong to blackbody radiation and the photon gas.

The distribution of emitted or received electromagnetic power, radiative flux or specific intensity across frequency or wavelength. The spectral quantity and interval convention must be specified: $F_{\nu }$ and $F_{\lambda }$ differ by the change-of-variable factor, while $\nu F_{\nu }$ gives energy flux per logarithmic frequency interval. This physical spectrum is distinct from the spectrum of a linear operator or a spectrum in topology.

Radiative flux is the first angular moment $F_{\nu }=\int I_{\nu }\cos \theta d\Omega$ of specific intensity and measures net radiant power crossing unit area. Integrating over frequency gives bolometric radiative flux. The receiving surface's orientation matters through $\cos \theta$; observed radiative flux from a compact distant source conventionally refers to an area perpendicular to its line of sight.

Radio communication conveys information using radio waves coupled into and out of space by antennas. Amplitude modulation and frequency modulation vary a carrier wave; wireless telegraphy can instead encode symbols in transmission intervals. The useful separation is between the rapid electrical oscillation and the slower message timing. A receiver needs both frequency selection and a way to turn its selected signal into a readable output.

A radio frequency is a frequency in the range used for radio waves. It describes the rapid oscillation of a radio wave, which can differ from the rate at which information is transmitted in radio communication.

A radiofrequency field induces transitions from trapped weak-field-seeking magnetic sublevels to untrapped states at a resonance surface. Atoms with sufficient energy reach that surface and leave. Sweeping the frequency changes the energy cutoff without equating every lost atom with the sample-average energy.

Radio interferometry measures the spatial coherence of received radio waves. At two antennas, a receiver preserves the relative wave phase, compensates the geometric delay for a chosen phase centre, and averages a product of one signal with the complex conjugate of the other. In a complex-signal description this correlation is $⟨E_1{E}_2^{\ast }⟩$. Its real and imaginary parts carry two complementary components of the complex visibility; measuring only the received powers discards the relative wave phase needed for general imaging.

A finite frequency band makes delay compensation essential. If the residual delay is $\Delta \tau$, the residual wave phase varies by $2\pi \Delta \nu \Delta \tau$ across a band of width $\Delta \nu$. Averaging these differently oriented complex contributions reduces the correlation. For an ideal rectangular band the multiplicative factor is $\sin (\pi \Delta \nu \Delta \tau )/(\pi \Delta \nu \Delta \tau )$. A phase correction at just the central frequency cannot remove this variation across the band. Long time averages can likewise smear a changing signal.

Radiometry measures the energy and power carried by electromagnetic radiation, including light. It asks how much radiation a source supplies, where it travels, and how much reaches a receiving surface. The measurements are physical rather than weighted by the sensitivity of the human eye: equal radiant fluxes at different wavelengths need not look equally bright. ^[ref].

The five quantities below answer different questions about the same radiation. The International System of Units makes the distinctions visible: a watt is a joule per second, while a steradian measures solid angle. ^[ref].

Resolving wavelength or frequency adds a further “per unit interval”, as in spectral radiance. The notation is consistent with NIST's table of spectral quantities. In particular, radiant flux is power, whereas the existing astronomical radiative flux is power per area. ^[ref].

This bookkeeping makes an optical camera's collected signal calculable from a source's radiance and the geometry of the receiving aperture. It also lets radiation measurements from different instruments be compared. For example, NIST describes separate calibrations of spectral radiance and spectral irradiance for instruments used to observe Earth. ^[ref].