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A finite optical pupil spreads a point into a point spread function. In incoherent imaging an extended intensity distribution is convolved with that function; its Fourier transform is the optical transfer function, which attenuates and ultimately removes high spatial frequencies. For an ideal circular pupil the point response is an Airy disk. The Rayleigh criterion for two equally bright incoherent points isA microscope's object-space scale is of order $\lambda /(2\mathrm{NA})$, with the numerical factor depending on illumination and the criterion; $\mathrm{NA}=n\sin \alpha$ measures the accepted ray cone. A larger pupil or shorter wavelength improves the diffraction scale. Aberrations, defocus, detector pixel sampling, motion, background and photon noise can worsen practical performance. Merely magnifying the same blurred image adds no information. Coherent sources require amplitude superposition instead of the incoherent intensity rule. Rayleigh resolution is a convention for separating a pair, distinct from the precision of localizing one isolated point; information does not abruptly vanish below it.

Boundary conditions quantize the transverse wavevector of an electromagnetic waveguide. For an ideal hollow rectangular conducting guide of sides $a,b$, filled with a nondispersive medium of wave speed $u$, a transverse electric mode of a rectangular waveguide or transverse magnetic mode of a rectangular waveguide obeysThe allowed integer pairs differ: TE permits one zero index, but not both; TM requires both positive. Here $\beta$ is the longitudinal propagation constant. Below cutoff $\beta$ is imaginary, so the mode is evanescent. Above cutoff,The positive branch approaches the straight line $\omega =u\beta$ far above cutoff, while near cutoff its slope tends to zero. Different frequencies acquire different group delays and a pulse can spread. A wave phase speed above $u$ does not convey superluminal information; in this ideal guide energy transport follows the group velocity. A single hollow conductor cannot support a TEM mode; two-conductor guides can have a zero-cutoff TEM branch. Material dispersion, dielectric fibres and lossy boundaries require their actual dispersion relation and need not preserve the product rule.

The populations giveTogether with $⟨S_y⟩=(\hslash /2)\sin (\Omega t)$, this describes Larmor precession of a spin initially perpendicular to the field. The positive angular frequency is $|\Omega |$; the sign specifies the rotation sense through $d⟨\mathbf{S}⟩/dt=\gamma ⟨\mathbf{S}⟩\times \mathbf{B}$. The factor of two between the spinor amplitudes and the measured spin is essential. The model with quantum coherence predicts no decay; spin relaxation or dephasing in a real solid can damp the oscillation.

In a classically forbidden region, the Schrödinger equation has evanescent solutions rather than forcing the wavefunction to vanish. Matching at both sides of a finite barrier permits nonzero transmission. For a broad barrier the WKB approximation giveswhere $C$ is a matching prefactor. Transmission is exponentially sensitive to barrier width and height. A single decaying exponential function carries no probability current; the matched scattering solution includes the combination needed for transmitted current. energy is conserved throughout.

In alpha decay, a quasibound alpha particle penetrates the nuclear Coulomb barrier. Multiplying the penetration probability by an assault frequency estimates the decay rate; the strong dependence on alpha energy explains the wide range of lifetimes. In a scanning tunnelling microscope, a voltage drives electron quantum tunnelling between a conducting tip and sample. Approximately $I\propto V{e}^{-2\kappa d}$, so constant-current feedback resolves changes in gap width, with electronic density of states also affecting the image. Piezoelectric motion, vibration isolation and stable voltage are necessary; a bright feature is not automatically a taller atom. Atomic-scale imaging was demonstrated with a vacuum-tunnelling microscope in the original surface study.^[ref] In a Josephson junction, Cooper pair quantum tunnelling through a weak link gives $I=I_c\sin \phi$, allowing supercurrent without an applied voltage and phase-sensitive interference. These are distinct applications of barrier penetration and coupling with quantum coherence, not particles borrowing energy to climb over a barrier.

Uniform samples at $t_n=n/f_s$ cannot distinguish frequencies separated by integer multiples of $f_s$:For a real cosine, opposite frequencies also coincide. Thus an unknown high-frequency component can appear in the sampled record at its folded frequency $f_{\mathrm{alias}}\in [0,f_s/2]$. For example, 2 Hz and 8 Hz cosines give identical samples at 10 samples per second. This ambiguity is aliasing, rather than uniform quantization error.

The Nyquist–Shannon sampling theorem permits reconstruction of an ideally bandlimited function of bandwidth $B$ when $f_s>2B$. Use an analogue anti-aliasing filter before conversion and allow a transition band between the desired passband and the frequencies that fold into it. Sampling faster relaxes the filter requirement. Digital filtering after undersampling cannot separate already identical sample sequences. Finite observation time, clock jitter, noise and imperfect filter rejection impose further practical limits.

Identify the physical coupling that causes a systematic error and change the apparatus so that it can be measured or cancelled. measurement calibration against known reference standards at several points reveals zero and gain errors; blanks and background runs measure additive contributions. Carry uncertainties of the standards and fitted corrections into the final result.

Reversal provides a particularly transparent test. If $D_{+}=S+b$ and $D_{-}=-S+b$, the combination $(D_{+}-D_{-})/2$ isolates $S$ while $(D_{+}+D_{-})/2$ measures the measurement zero offset. Interleave and repeat the reversals so drift can be detected. The cancellation assumes a stable measurement zero offset and a genuinely odd signal; a common gain error remains. null measurements and bridge comparisons can reduce dependence on detector gain. Swapping sensors or leads exposes asymmetry; checking alignment, shielding, grounding and temperature control tests specific unwanted couplings. Varying signal amplitude or measurement frequency can reveal heating, nonlinear response or pickup.

Use independent measurement principles where possible, and test the predicted response to each deliberate change. Repeated readings alone do not remove a fixed systematic error. Correct identified significant effects and retain uncertainty for imperfectly known corrections; complete elimination cannot generally be demonstrated. Type A and Type B classify how uncertainty is evaluated, rather than simply meaning random measurement errors and systematic errors.^[ref]

A representative bipolar-input 741 amplifier has $R_{\mathrm{in}}$ of order $10^6\Omega$; the LM741 specification gives a typical $2\mathrm{M\Omega }$ at $25^{\circ }\mathrm{C}$. This is an illustrative device specification, not an identification of an unspecified practical-class part. Other input architectures can have much larger electrical resistance, and finite input capacitance makes the input impedance frequency dependent.^[ref]

The open-loop gain is large and roughly constant at low frequency, then falls as internal poles become important. A dominant-pole approximation isAbove $f_p$, but before higher poles dominate, $|A|\propto f^{-1}$, a slope of $-20\mathrm{dB}$ per decade, with increasing phase lag. The approximate unity-gain frequency is $f_T=A_0{f}_p$. A 741-class amplifier has typical low-frequency gain about $2\times 10^5$; this is the open-loop gain, distinct from the feedback-set signal gain. The finite-gain formula at the parent level explains why the closed-loop response eventually rolls off.^[ref]

An ideal operational amplifier draws no current at either input. With stable operational-amplifier negative feedback in its unsaturated linear range, infinite open-loop gain also gives a virtual short, $V_{+}=V_{-}$. The amplifier adjusts its output to enforce that equality; it is not a conducting connection between the inputs. Here $V_{+}=0$, so the ideal summing node is a virtual ground. Kirchhoff's current law gives $V_{\mathrm{in}}/R_1=-V_{\mathrm{out}}/R_2$, hence the ideal gain is $-R_2/R_1$.

For the finite-input-resistance inverting amplifier let the inverting-node voltage be $v$, and continue to neglect output impedance. The controlled output and node-current balance areEliminating $v$ givesDividing the denominator by $R_2$ gives the requested resistance-sum form. The ideal result follows whenever $|A|\gg 1+R_2/R_1+R_2/R_{\mathrm{in}}$; taking $R_{\mathrm{in}}\to \infty$ alone is insufficient if $A$ is small. At finite frequency $A$ and the gain are phasor ratios, subject to feedback stability.

With equal resistors and sufficiently large loop gain, $V_{\mathrm{out}}\simeq -V_{\mathrm{in}}$. This gives a controlled sign reversal or 180-degree phase inversion, useful for subtraction and opposite-phase signal paths, with amplitude set by a resistor ratio rather than uncertain open-loop gain. The source still sees approximately $R_1$, so this is not a high-input-impedance voltage follower. For the ideal-input, single-pole model, the noise gain of an operational amplifier is two and the closed-loop bandwidth is approximately half the gain-bandwidth product. Output swing, slew rate and load constraints must also permit linear operation.

In a finite potential barrier with $E<V$, the stationary Schrödinger equation has exponentially decaying and growing solutions, with local decay constant $\kappa =\sqrt{2m(V-E)}/\hslash$. Matching the wavefunction and its derivative at finite steps produces a transmitted probability current even though a classical particle at that energy could not cross. energy remains conserved. For a smooth thick barrier, the leading WKB transmission factor isFor a rectangular barrier of height $V_0$, width $d$ and $0<E<V_0$, exact matching gives $T=[1+V_0^2{\sinh }^2(\kappa d)/(4E(V_0-E)){]}^{-1}$. Thickness and energy therefore affect transmission exponentially in the opaque limit; prefactors and near-turning-point corrections require more than the leading exponent.

alpha decay illustrates escape from a nuclear well through a Coulomb barrier. A rough decay rate is an internal encounter frequency times barrier penetrability, multiplied by the probability of forming the outgoing cluster; penetrability alone is not a complete nuclear lifetime calculation. In a scanning tunnelling microscope, electron transmission through the vacuum gap gives an approximately exponential distance dependence $I\propto e^{-2\kappa d}$ at fixed conditions. A feedback loop adjusts tip height to maintain current, providing surface information. Bias, tip electronic structure and local electronic density of states also affect the current, so constant-current height is not purely geometric topography. Both applications exploit the same sensitivity of transmission to a classically forbidden region.

The Time-dependent Schrödinger equation $i\hslash {\partial }_t|\psi ⟩=H(t)|\psi ⟩$ governs unitary evolution of a closed system. If $H$ is independent of time, expand in its energy eigenstates:An individual energy eigenstate changes only by a phase, so its probability density and time-independent-observable expectations are stationary. Superpositions can have interference terms oscillating at $(E_n-E_m)/\hslash$; fixed energy populations do not imply that every observable is constant.

For general $H(t)$, $U(t,t_0)=\mathcal{T}\exp [-(i/\hslash )\int H(t^{\prime })d{t}^{\prime }]$ requires time ordering unless Quantum Hamiltonians at different times commute. The equivalent Heisenberg description evolves observables rather than the state, giving $d⟨A⟩/dt=(i/\hslash )⟨[H,A]⟩+⟨{\partial }_{t}A⟩$. Explicit time dependence can transfer energy from an applied perturbation.

For a weak perturbation $V(t)$ to a known $H_0$, the first-order transition amplitude from $n$ to $m\ne n$ isIts squared modulus gives the leading transition probability while depletion remains small. A nearly monochromatic perturbation emphasizes energy differences near its quantum of frequency; a continuum and suitable long-time limit yield Fermi's golden rule. These approximations have stated weak-coupling and time-resolution limits; they do not replace exact unitary dynamics in every regime.

For an experimental-noise interpretation of Nyquist's theorem, a resistor $R$ in thermal equilibrium at temperature $T$ has one-sided open-circuit voltage-noise spectral densityin the classical range $hf\ll k_{B}T$, over an approximately rectangular bandwidth $\Delta f$. Across a matched load, half the open-circuit voltage appears, giving available noise power $k_{B}T\Delta f$. This is thermal agitation, present without a mean current, and is distinct from shot noise. A real readout weights the spectrum with its transfer function: $⟨v_{\mathrm{out}}^2⟩={\int }_0^{\infty }{S}_V(f)|H(f){|}^{2}df$. Cooling, reduced bandwidth and suitable impedance/noise matching reduce measured noise; the resistor's noise does not simply disappear in an ideal amplifier model.^[ref]

The bare name also has a sampling interpretation. The Nyquist–Shannon sampling theorem permits exact reconstruction of a bandlimited signal from sufficiently fast uniform samples. The practical condition is $f_s>2f_{\max }$, with an anti-aliasing filter before sampling; otherwise distinct continuous frequencies produce identical samples. Ideal reconstruction uses shifted sinc functions. Critical-rate equality needs an appropriate signal class and can fail for unrestricted edge-frequency sinusoids with unknown phase. Sampling fast enough prevents aliasing, but does not remove thermal noise. This distinction covers the source's unspecified use of “Nyquist's theorem”.^[ref]

A closed conducting enclosure redistributes surface charge so that an empty interior is protected from an external static electric field. Grounding fixes a reference and drains deposited charge, but an ideal isolated closed conductor also provides inward electrostatic shielding. Real seams, apertures and cables need careful bonding, shielding and feedthrough filtering. For time-varying fields the relevant wavelength, conductivity, geometry and penetration depth matter; a static argument alone is insufficient. Differential wiring and small pickup-loop areas can complement shielding, while careless multiple ground connections create unwanted current loops.

Ordinary copper or aluminium does not generally exclude a static magnetic field. High-permeability shells instead guide magnetic flux around the protected volume. Their effectiveness depends on shape, continuity, openings and saturation; magnetic history and mechanical stress can worsen performance. Nested shields and degaussing can reduce residual fields. Superconducting shielding can exploit the Meissner effect, but trapped flux, openings and critical-field limits require attention.

At nonzero frequency, eddy currents in conducting shields oppose changing magnetic flux. The skin depth isfor a good conductor in the usual diffusion approximation. Thickness of several skin depths suppresses penetration, but the very-low-frequency limit is weak for a normal metal. High-permeability and conducting layers can therefore complement each other. Active compensation coils are an additional method, not passive shielding. Test performance by applying known fields along relevant axes and measuring the residual field or induced signal over the apparatus bandwidth; room-temperature shielding materials and electronics can themselves add noise.

The ideal operational-amplifier rules are zero input currents and, when stable negative feedback maintains unsaturated linear operation, equal input voltages. Infinite open-loop gain motivates the latter; it must not be assumed to guarantee asymptotic stability for every external network.

For the numerical sketch and usefulness discussion, use the transfer function derived in part (c). The chosen values give $K=5$ and $\tau =RC=10^{-4}\mathrm{s}$. With $x=\omega \tau$,The formal amplitude falls from 5 to zero, with high-frequency slope $-40$ dB per decade. Its unwrapped phase rises from zero through $+90^{\circ }$ at $\omega =10^4{\mathrm{s}}^{-1}$ to $+180^{\circ }$; the corresponding characteristic frequency is 1.59 kHz. At that frequency the amplitude is 2.5.

These are formal sinusoidal particular-solution curves, not a stable settled filter response. For the formal sinusoidal response of an unstable linear system, replacing $i\omega$ by the Laplace variable $s$ gives $G(s)=5/(1-s\tau {)}^2$, with a repeated pole at $s=+1/\tau$. A homogeneous output contains $(A+Bt)e^{t/\tau }$ and grows, so real perturbations lead to supply-limited nonlinear behaviour. With equal positive resistors and capacitors in the frequency-setting network, the general denominator is $1+(3-K)s\tau +s^2{\tau }^2$: the Sallen-Key equal-component stability bound is $K<3$. For example $K=3-\sqrt{2}$ gives a Butterworth filter response. Thus the Sallen-Key low-pass filter architecture is useful for active filtering with suitable gain and damping, but the stated $K=5$ component set is unsuitable for linear filtering. The component values are retained in this diagnosis.^[ref]

For elliptical polarization from circular and linear components, use coherent waves of the same frequency, propagation direction and fixed relative phase. Let the circular component be $A(\hat{\mathbf{x}}\cos \psi +\hat{\mathbf{y}}\sin \psi )$ and the linear component $B\hat{\mathbf{x}}\cos \psi$, with $A,B>0$. Their sum has $E_x=(A+B)\cos \psi$ and $E_y=A\sin \psi$, soThe tip of the electric field traces an ellipse with unequal semiaxes $A+B$ and $A$. Coherence is necessary: averaging unrelated phases does not create this fixed polarization ellipse.

Reading the time axis, the rapid oscillation has period about $2\mathrm{s}$, while successive envelope maxima are about $10\mathrm{s}$ apart. The envelope stays nonzero and has a slowly sinusoidal modulation about a positive mean. The corresponding amplitude-modulation model is $y=[A+B\cos (2\pi 0.1t)]\cos (2\pi 0.5t)$, with $A>|B|$. Product-to-sum givesThus the intended positive-frequency components are $0.4$, $0.5$ and $0.6\mathrm{Hz}$. The envelope frequency $0.1\mathrm{Hz}$ is not itself a separate additive displacement component. These values use the illustrated sinusoidal-modulation interpretation; a finite qualitative trace alone does not uniquely specify every possible spectral component.

At long wavelength, the acoustic phonon branch moves the two atoms in a cell nearly in phase: $u_M\simeq u_m$. In the $q\to 0$ limit this is rigid translation and $\omega \to 0$. The optical phonon branch moves them oppositely, with stationary cell centre of mass: $M{u}_M+m{u}_m=0$. Its $q=0$ frequency is finite because adjacent unlike atoms change their separation. For unequal masses the lighter atom has the larger optical displacement. Three-dimensional crystals have three acoustic branches and, with two atoms per primitive cell, three optical branches; longitudinal and transverse directions describe displacement relative to the wavevector. The name optical does not mean that every optical phonon branch is infrared-active: coupling also depends on effective charges and symmetry.

For a narrow packet, phase velocity is $u_p=\omega /k$ and group velocity is $u_g=d\omega /dk$. Set $b=\sigma /{\rho }_1$. The uncovered surface hasEquality occurs at $k_{\ast }^2=g/b$, also the minimum of $u_p$. Numerically,The long-wave gravity limit has $u_g/u_p\to 1/2$; the short-wave capillary limit has $u_g/u_p\to 3/2$.

A compact packet subsequently spreads because different wavenumbers have different group speeds. In the gravity regime $\omega \simeq \sqrt{gk}$, so smaller $k$ and longer waves move ahead. Individual crests move twice as fast as the envelope, appearing at its rear and disappearing at its front. In the capillary regime $\omega \simeq \sqrt{b}{k}^{3/2}$, shorter waves move ahead; the envelope overtakes its slower individual crests, which drift backward relative to it. Thus both the wavelength ordering within the spread packet and the crest motion relative to its envelope reverse.

For the sheet define $c=a{\rho }_2/{\rho }_1=8.0\times 10^{-5}\mathrm{m}$. The elastic-cover frequency ratio for water waves isAt small $k$, $F=1-ck/2+O(k^2)$: sheet inertia lowers the frequency, while bending first enters at higher order. At sufficiently large $k$, bending dominates the restoring force and sheet inertia dominates the extra mass, giving $F\sim \sqrt{B/(\sigma c)}{k}^{1/2}$. The curve initially dips below one, reaches a minimum, then rises and crosses one. This asymptote describes the supplied ideal dispersion law; arbitrarily large $k$ eventually violates a thin-sheet continuum model.

For a pinned-square surface displacement mode, $k_{nm}=\pi \sqrt{n^2+m^2}/L$ with $n,m\ge 1$. The fundamental has $k_{11}=74.05{\mathrm{m}}^{-1}$. The uncovered frequency is $4.3729\mathrm{Hz}$ and $F(k_{11})=0.997051$, soThis uses a square two-dimensional surface mode, not the one-dimensional $\pi /L$ estimate. A clamped plate or additional hydraulic boundary conditions would require a different mode calculation.

The string speed is $c=\sqrt{T/\rho }$. The characteristic impedance is the ratio of the forward-wave transverse force to transverse velocity, $Z=T/c=\sqrt{T\rho }$. With peak displacement amplitude $A$, the mean power isDisplacement and transverse force are continuous at a massless junction. Since $Z_2=2Z_1$, the displacement amplitudes are $r=(Z_1-Z_2)/(Z_1+Z_2)=-1/3$ and $t=2Z_1/(Z_1+Z_2)=2/3$. Taking the incident amplitude real, the light-string motion isIn this standing-and-travelling decomposition of a reflected string wave, the first term travels and the second stands; the decomposition is not a separation into independent incident-power fractions because these fields interfere. The reflected wave has a phase reversal. power coefficients and standing-wave ratio are

For a uniform inserted matching section, unequal exterior impedances require a quarter-wave impedance transformer: $Z_M=\sqrt{Z_1{Z}_2}$ and $k_{M}a=(2j+1)\pi /2$. At equal tension this means middle density $2{\rho }_0$ and an odd-quarter-wavelength length in that section. Reflections at its two ends then cancel at the design frequency. An arbitrary density or an arbitrary half-wavelength segment cannot perfectly match these unequal exterior strings; the ideal match is frequency-selective.

amplitude scaling along a slowly tapered string instead follows from matching over an adiabatic frequency range. The source fixes frequency, while conserved power and the final characteristic impedance determine amplitude. In the heavy string,Its wave speed halves. The no-reflection approximation needs small fractional characteristic impedance change over each local wavelength, rather than length alone for a taper containing abrupt features.

For $m\ddot{x}+b\dot{x}+kx=F_0\cos \omega t$, define ${\omega }_0=\sqrt{k/m}$ and $\beta =b/(2m)$. The parameter relating quality factor and viscous damping regimes isAt weak damping it also equals $2\pi$ times stored energy divided by energy lost per cycle. The stored-energy convention and frequency matter: this cycle-based interpretation should not be extended to free motions that never complete a cycle.

For $\beta <{\omega }_0$ the free motion is $e^{-\beta t}[A\cos {\omega }_{d}t+B\sin {\omega }_{d}t]$, where ${\omega }_d=\sqrt{{\omega }_0^2-{\beta }^2}$. Thus underdamping means $Q>1/2$. High $Q$ gives many oscillations per amplitude-decay time, narrow resonance and strong frequency selectivity; a lightly loaded tuning fork, optical cavity or low-loss electrical resonator provides examples. The amplitude decay time is ${\tau }_A=1/\beta =2Q/{\omega }_0$, while the slowly averaged energy decays with ${\tau }_E=1/(2\beta )=Q/{\omega }_0$. These times differ by a factor of two.

At critical damping, $\beta ={\omega }_0$ and $Q=1/2$, with $x=(A+Bt)e^{-{\omega }_{0}t}$. There is no ringing; a simple critically damped position indicator can settle without oscillation. For $\beta >{\omega }_0$ and $Q<1/2$, there are two real negative decay rates $-\beta \pm \sqrt{{\beta }^2-{\omega }_0^2}$. Increasing very strong drag slows the long-time return, whose rate approaches ${\omega }_0^2/(2\beta )$. A strongly viscously damped mechanical indicator illustrates this regime. These classifications concern the linear homogeneous equation, not every possible nonlinear damping law.

The driven displacement amplitude isThe displacement-resonance threshold of a viscous oscillator requires $\beta <{\omega }_0/\sqrt{2}$, or $Q>1/\sqrt{2}$; it lies at $\sqrt{{\omega }_0^2-2{\beta }^2}$. Some underdamped systems therefore have no such maximum. Average absorbed power instead peaks at ${\omega }_0$ for this viscous model. Its half-power frequencies are $\sqrt{{\omega }_0^2+{\beta }^2}\pm \beta$, so their separation is $2\beta$ and $Q={\omega }_0/\Delta \omega$. For broad response this ratio remains a useful parameter but not a claim of many free oscillations. The familiar displacement-linewidth version is a high-$Q$ approximation.

A series RLC circuit has ${\omega }_0=1/\sqrt{LC}$ and $Q={\omega }_{0}L/R$ for the charge oscillation. Resistance dissipates energy, inductance provides the inertial analogue and capacitance the restoring analogue. Ringdown and a calibrated half-power scan independently estimate loss; parasitic resistance, loading and radiation change the measured value. A large $Q$ increases frequency selectivity but also lengthens settling time, so it is not universally the best instrumental choice.