For a wave penetrating a nonmagnetic good conductor in the $+z$ direction with time convention $e^{-i\omega t}$, its wavenumber is $k=(1+i)/\delta$, with skin depth $\delta =\sqrt{2/({\mu }_0\sigma \omega )}$. If $\mathbf{E}\parallel \hat{\mathbf{x}}$, Faraday's law of induction gives $\mathbf{H}=[k/({\mu }_0\omega )]E_x\hat{\mathbf{y}}$. Thus the field ratio is $Z_s=E_x/H_y=(1-i)/(\sigma \delta )$, equal to the displayed expression. Magnetic and electric fields have equal exponential penetration lengths but a phasor phase difference $\pi /4$; $H_y$ has the positive phase relative to $E_x$ in this convention.

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.

Insert a complete resolution of identity in energy eigenstates on either side of the time-evolution operator. Each state evolves by its scalar energy phase, yielding the displayed quantum propagator. Its spectral sum includes an integral over any continuous spectrum, with the measure fixed by the continuum normalization. Omitting the scattering states when bound states are also present destroys completeness and the delta-function short-time limit.

A leading principal minor is a principal minor using the first $k$ rows and columns of a square matrix in a specified ordering. Sylvester's criterion uses these minors to characterize positive-definite matrices.

For spin-degenerate electrons at atomic spacing $a$, alternate the hopping amplitudes $-(t+\delta )$ and $-(t-\delta )$, with $t>0$ and $|\delta |<t$. The doubled primitive unit cell gives an off-diagonal Bloch Hamiltonian entry $-(t+\delta )-(t-\delta )e^{-2ika}$. Its two eigenvalues are the displayed energy bands, for $|k|\le \pi /(2a)$. The band gap at the edge of the Brillouin zone is $4|\delta |$. At half filling there are two electrons per doubled primitive unit cell, so the lower band is full and the upper empty. This explicit model realizes the period-doubling form of a Peierls distortion.