A Penning trap uses a static electric quadrupole field for axial confinement and a static axial magnetic field for transverse confinement. The Lorentz force bends a moving electric charge's trajectory without doing work. An RF probe may interrogate a trapped particle, but an RF confining voltage is unnecessary in this ideal model. This makes the Penning trap useful for relating cyclotron frequencies and spin precession to a common magnetic field.

The Paul trap alternates focusing and defocusing electric forces. Their time average along a moving trajectory need not vanish, even when the electric field averages to zero at every fixed point. A spatial gradient makes the fast displacement correlate with the driving electric field; the RF-averaged trapping potential calculates the resulting restoring force.

Wolfgang Paul's lecture describes hyperbolic ring/endcap electrodes, loading by electron-impact ionization of low-pressure gas, and resonant ejection through an endcap hole into an electron multiplier. Sweeping RF voltage produces an air mass spectrum.^[ref]

Trapped-ion quantum control prepares an ion's internal quantum state, couples it to quantized motion with laser pulses, and measures the result through state-dependent fluorescence. Its appeal is access to both a controllable two-level system and a quantum harmonic oscillator in one apparatus.

Confinement, preparation, control and readout solve different problems. The Paul trap's RF electric field confines the ion through its RF-averaged trapping potential; it still produces Paul trap micromotion. Doppler cooling and resolved sideband cooling reduce motional excitation. Optical pumping selects an internal quantum state. A coherent pulse then changes amplitudes, while electron shelving converts an internal population into a detectable stream of photons. A bright/dark count by itself does not reveal a quantum superposition's relative phase; an additional controlled rotation before detection can make that phase affect the measured population.

In thermal equilibrium, nondegenerate conduction electrons and valence electron holes share one chemical potential. Multiplying $n=N_c{e}^{-(E_c-\mu )/(k_{B}T)}$ by $p=N_v{e}^{-(\mu -E_v)/(k_{B}T)}$ cancels that chemical potential:This relation remains valid with semiconductor doping under the stated assumptions. Equality of the two individual densities, $n=p=n_i$, additionally requires the intrinsic neutral case. Separate electron and hole quasi-Fermi levels in nonequilibrium need not obey this equilibrium product.

With one parabolic valley and two spin states at each band edge, the prefactor multiplying $(k_{B}T{)}^3$ in the product is $4(m_c{m}_h{)}^{3/2}/(2\pi {\hslash }^2{)}^3$. Both populations must be nondegenerate; the Pauli reduction of the pinned-edge mass-action product quantifies a failure of the classical approximation.

For fixed band effective masses and band gap $E_g$, the intrinsic carrier concentration obeys $n_i=C{T}^{3/2}{e}^{-E_g/(2k_{B}T)}$. ThusA base-ten plot against $x=1000\mathrm{K}/T$ instead has slope $-E_g/(2000k_B\mathrm{K}\ln 10)$ after the prefactor correction. The intrinsic branch must be used, rather than the extrinsic plateau or dopant freeze-out branch. A changing gap or effective mass makes this an activation-energy estimate, and a small gap uncertainty can produce a much larger multiplicative uncertainty in a room-temperature minority density.

In a nondegenerate parabolic-band semiconductor with fixed density-of-states mass, $n(T)=C{T}^{3/2}{e}^{-\Delta (T)/(k_{B}T)}$, where $\Delta (T)=E_c(T)-\mu (T)$. ThusA single measured value of $\Delta$ does not determine its temperature dependence. A fixed-gap intrinsic model instead has $n_i=C_i{T}^{3/2}{e}^{-E_g/(2k_{B}T)}$, with the two effective density of states factors in $C_i$. Identifying a crossover by $n_i(T_c)\simeq n_{\mathrm{ext}}$ gives an order-of-magnitude dopant-density estimate once that activation law is specified. At exact charge neutrality with $n_i=n_{\mathrm{ext}}$, the total mobile-electron density is $(1+\sqrt{5})n_{\mathrm{ext}}/2$, so a crossover is not a sharp equality of every density definition.

Weighted interval scheduling assigns each job a reward $w_i$ and maximizes their sum over compatible jobs. Earliest finish can fail for this different objective function; dynamic programming can compare alternatives.^[ref]

For a separate two-job example, let $X=[0,2)$ have reward one and $Y=[1,3)$ reward ten. Earliest finish accepts $X$ and rejects $Y$, earning one; the optimum accepts $Y$, earning ten. Replacing $Y$ by $X$ preserves job count but loses reward. Thus the interval scheduling exchange argument proves the unit-value theorem only: its preserved quantity must match the intended objective function.