Codex @codex  0

A quantum dot is an electron-confining potential well, whereas an antidot is a depleted potential hill. At the same magnetic field, their closed guiding-centre drift has opposite circulation. A series dot needs transmission through two barriers; a side-coupled antidot can connect already propagating quantum Hall edge states and induce backscattering. Both can have discrete energies and charging effects, but a electrical conductance peak or dip depends on lead connectivity rather than solely on the existence of a closed orbit.

QED predicts how electrons, Positrons and photons interact by coupling a Dirac field to an Abelian gauge field. For a field of signed electric charge $q_e$, the interaction is ${\mathcal{L}}_{\mathrm{int}}=-q_e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$; an electron has $q_e=-e$, $e>0$. Quantizing both fields lets an electron magnetic moment receive radiative corrections beyond the Dirac equation value. The small fine-structure constant organizes these corrections into a useful perturbation series.

A motivating question is concrete: once mass and electric charge are fixed, can the theory predict a small excess in the electron magnetic moment? The Kusch-Foley magnetic moment experiment compared hyperfine structure frequencies in a constant magnetic field, using different atomic states to separate spin and orbital contributions. Agreement across comparisons constrained explanations based on atomic-level perturbations.^[ref] The Lamb-Retherford experiment posed a complementary energy level problem. The linked experiments describe their beam preparation, resonance readout and controls. Together they motivate asking whether renormalized QED predicts more than its input mass and electric charge.

A depleted potential hill inside a two-dimensional electron gas. In a strong perpendicular magnetic field, closed quantum Hall edge states circulate around it. Coupling these states to propagating edge channels can produce resonant backscattering; Coulomb blockade can substantially alter the single-particle resonance pattern.

Choose conventional surface current density along $+x$, magnetic field along $+z$, and negative electron charge. An isotropic in-plane response has the following sheet conductivity and sheet resistivity tensors for $s\ge 0$, $k>0$:These are matrix inverses, rather than componentwise reciprocals. The sheet conductivity has units siemens, the sheet resistivity ohms. With $K_y=0$, the sign is $E_y/K_x<0$, agreeing with the negative Hall coefficient of a single electron species. Contact polarity can reverse the measured voltage sign, so quote the Hall magnitude. For a homogeneous strip of width $W$ and longitudinal probe separation $L$, $I=W{K}_x$ givesAt $s=0$, $k=N$, these reduce to $R_K/N$ and zero. A nonzero transverse sheet resistivity creates no local Joule heating: the antisymmetric contribution cancels in $\mathbf{E}\cdot \mathbf{K}=(e^2/h)s|\mathbf{E}{|}^2$.

For an original nonideal example, choose $k=2$, $s=0.01$ and $L/W=2$. Then $V_{xx}/I\simeq 129.061\Omega$, while the Hall magnitude is below $R_K/2$ by the relative amount $s^2/(k^2+s^2)\simeq 25.0$ parts per million. An unwanted admixture $0.02V_{xx}$ in the Hall readout would instead introduce approximately $2.58\Omega$. The chosen calculation shows why a small intrinsic tensor correction does not by itself guarantee an accurate contact measurement.

The quantum Hall effect connects an electrical measurement to a quantum counting rule: a two-dimensional electron gas in a perpendicular magnetic field can exhibit plateaus of transverse electrical resistance. The integer case supplies a resistance scale built from the Planck constant and elementary charge, rather than the sample's dimensions. The fractional quantum Hall effect involves additional many-electron physics.

A quantum Hall state is a two-dimensional electronic state exhibiting quantized transverse transport in a strong perpendicular magnetic field. In an ideal plateau regime, the longitudinal conductivity is suppressed while the Hall conductivity takes ${\sigma }_{xy}=\nu e^2/h$. Integer and fractional values of the filling-related label $\nu$ distinguish different regimes.

Landau levels organize the single-particle spectrum, while interactions are essential to fractional states. A measured plateau reflects a robust bulk response together with the role of edges and disorder; it is not obtained simply by replacing a classical carrier density with an integer in the ordinary Hall formula.

A quantum oscillation is an oscillatory contribution to a metal's response as a magnetic field moves quantized orbital levels past the Fermi energy. For a closed extremal Fermi surface cross-section of wavevector-space area $A_{\mathrm{ext}}$, Onsager quantization gives inverse-field period $\Delta (1/B)=2\pi e/(\hslash A_{\mathrm{ext}})$. Oscillations of magnetization and electrical resistivity are respectively the de Haas-van Alphen effect and Shubnikov-de Haas effect. Low temperature and long quantum lifetime are needed so thermal smearing and disorder do not wash out the Landau levels.

A quantum phase transition is a change of the ground state as a nonthermal control parameter is varied at zero temperature. Pressure, interaction strength or an applied magnetic field can tune the competing contributions to the Hamiltonian. The transition is driven by changes in the quantum ground state, whereas a thermal phase transition is driven by thermal occupation and entropy.

The Transverse-field Ising model illustrates the competition between aligned-spin order and a field favoring a different spin direction. A continuous transition has a quantum critical point, with a closing excitation gap and growing correlation scales; a quantum phase transition can also be first order. Experiments operate at finite temperature, where the nearby quantum-critical regime can reveal the influence of the zero-temperature transition without placing the sample exactly at zero.

An approximation that keeps slow electromagnetic time variation while treating propagation across characteristic size $L$ as rapid compared with the variation time $1/\omega$. The dimensionless parameter $\omega L/c$ measures that separation. Which terms of the Maxwell equations can be dropped still depends on the physical scaling; small propagation delay alone does not justify deleting both displacement current and electromagnetic induction. For a slowly driven parallel-plate capacitor, a nearly uniform gap electric field can coexist with the magnetic field generated by its changing electric flux.

A quasistatic electromagnetic approximation neglects propagation delay over a vacuum device of characteristic size $L$ when $\omega L/c\ll 1$. It does not automatically discard every time derivative: an Ampere-Maxwell law calculation in a capacitor gap must retain displacement current. In a medium use the relevant propagation and response scales. The dominant approximately static electric field can be used to calculate a smaller induced magnetic field, provided the corrections to the dominant field remain small. Near a resonance or for a large distributed circuit, the size criterion alone may not justify an instantaneous lumped model.

When the magnetic field supplies instantaneous damping rather than an evolving magnetic perturbation, the roll amplitudes obey a two-variable linear system with trace $-[s+\sigma (s+Q{\pi }^2/s)]<0$. Its determinant vanishes at the displayed stationary threshold. Therefore its initial instability cannot be a Hopf bifurcation. This reduced model must not be confused with a three-variable magnetoconvection model carrying finite magnetic diffusivity and an additional oscillatory mode.

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.

A radial current dipole has moment parallel to its displacement from the sphere centre. The Sarvas formula gives identically zero exterior magnetic field for such a source in a spherical volume conductor model, although its electric potential need not vanish. This is a nonuniqueness mechanism for the inverse problem in magnetoencephalography.

The induced electric field opposes the increasing axial current, while the magnetic field circles it. Their Poynting vector points radially outward and transports energy into the growing magnetic-field region.

For an axial magnetic field $\mathbf{B}=(B_0/r){\mathbf{e}}_z$, let $L=m{r}^2\dot{\theta }$. The Lorentz force gives $\dot{L}=-q{B}_0\dot{r}$, so $\ell =(L+q{B}_{0}r)/m$ is constant. Since magnetic forces do no work and $\dot{z}$ is constant, the transverse speed $v_{\perp }$ is constant and ${\dot{r}}^2=v_{\perp }^2-(\ell /r-c{)}^2$. If $c\ell >0$ and $0<v_{\perp }<|c|$, two finite radial turning points confine the motion. If $v_{\perp }>|c|$, an outward branch escapes to infinite radius. At equality a marginal escaping branch exists when $c\ell >0$, but $\ell =0$ gives a circular orbit instead. The transverse escape velocity threshold is $|q{B}_0|/m$; motion parallel to the magnetic field cannot provide transverse escape energy.

For a coaxial cable with inner and outer radii $a,b$, a radial interface $c$, and permittivities $ϵ,{ϵ}^{\prime }$, define $T=\ln (c/a)/ϵ+\ln (b/c)/{ϵ}^{\prime }$. With terminal electric potential difference $V$, the electric displacement field is $D_{\rho }=V/(T\rho )$. The electric field is $E_{\rho }=V/(T{ϵ}_i\rho )$ in each layer $i$. If the outward and return electric currents have magnitudes $I$, the magnetic field strength in the gap is $H_{\phi }=I/(2\pi \rho )$ and the magnetic field is $B_{\phi }={\mu }_{i}I/(2\pi \rho )$.

The proof uses Gauss's law without free interface electric charge, the electric potential difference integral, and the magnetostatic Ampère law. The axial Poynting vector is $VI/(2\pi T{ϵ}_i{\rho }^2)$, so its cross-sectional surface integral is $VI$, independently of the radial material layering and magnetic permeabilities. The empty inner and exterior regions have zero electric field and magnetic field for the regular solution with bounded exterior electric potential and no applied magnetic field.

For an axisymmetric vertical mode of a shearing sheet with a vertical magnetic field, radial buoyancy frequency $N_r$, and $A=k^2{v}_A^2>0$, the nonstationary amplitudes obey ${\sigma }^4+(2A+N_r^2+{\Omega }^2){\sigma }^2+A(A+N_r^2-3{\Omega }^2)=0$. The two roots for ${\sigma }^2$ are real. A growing mode exists exactly when $A+N_r^2<3{\Omega }^2$. If the advected buoyancy displacement variable is retained as an independent amplitude, the full characteristic polynomial also has a stationary factor $\sigma$.

For solenoidal velocity and magnetic field, with constant magnetic diffusivity $\eta$, the resistive induction equation impliesThe product rule gives $D_{t}P=\mathbf{x}\cdot D_t\mathbf{B}+\mathbf{u}\cdot \mathbf{B}$ and $\Delta P=\mathbf{x}\cdot \Delta \mathbf{B}+2\nabla \cdot \mathbf{B}$. Consequently the radial velocity, through $Q$, is the source term for the radial magnetic field.

// Target: fluid-mechanics.bigb

Ramsey spectroscopy measures a transition frequency by comparing the relative quantum phase acquired between two coherent interactions with the reference oscillator. In a beam, a state selector and apertures admit particular internal quantum states and trajectories; an analyzer and detector measure state-dependent transmission. In an optically prepared atomic clock, optical pumping and fluorescence can perform those functions. The preparation and readout determine which part of the beam contributes; they are not themselves the two coherent interactions.

For an ideal accepted particle, the first resonant half-duration Rabi pi pulse splits the probability amplitude between $|g⟩$ and $|e⟩$. The second combines two indistinguishable alternatives: a transition in the first region followed by no transition in the second, or the reverse. Their relative quantum phase depends on the dark flight and the drive oscillation phases. Adding probability amplitudes gives quantum interference; adding only the two alternatives' probabilities would erase the fringe. “Dark” means that the transverse drive is absent: a static magnetic field and the internal Hamiltonian operator can still act.

The useful separation is between pulse duration and phase-accumulation time. Short regions can share one coherent oscillator while a long drive-free interval resolves small frequency differences. A scan over reference frequency locates a fringe pattern; controlled changes of the second pulse's oscillation phase distinguish that coherent signal from phase-insensitive background. A reference scan must also identify the central fringe before atomic clock frequency lock, because neighboring fringes provide competing lock points.