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For a forced harmonic oscillator with constant viscous damping coefficient $m\gamma >0$, the average power divided by its maximum is ${\gamma }^2{\omega }^2/[({\omega }_0^2-{\omega }^2{)}^2+{\gamma }^2{\omega }^2]$. It peaks at ${\omega }_0=\sqrt{k/m}$. Its two positive half-power angular frequencies are ${\omega }_{1,2}=(\sqrt{4{\omega }_0^2+{\gamma }^2}\mp \gamma )/2$. Hence their separation is exactly $\gamma$, and their product is ${\omega }_0^2$. The narrow-resonance approximation ${\omega }_{1,2}\simeq {\omega }_0\mp \gamma /2$ is needed for symmetry about the peak, not for the exact bandwidth in angular-frequency units.

A time-dependent response can be represented by its Fourier transform in angular frequency. With $\hat{f}(\omega )=\int f(t)e^{i\omega t}dt$, a time derivative transforms to $-i\omega \hat{f}$ when boundary terms vanish. A time-translation-invariant response convolution becomes multiplication by a frequency-dependent response coefficient. This is why a dispersive electrical conductivity is naturally specified in the frequency domain. The inverse-transform modes use $e^{-i\omega t}$.

For equal masses $m>0$, positive end spring constants $k_1>k_2$ and coupling spring constant $\kappa >0$, the stiffness matrix is $K=((k_1+\kappa ,-\kappa ),(-\kappa ,k_2+\kappa ))$. Its eigenvalues divided by $m$ are the squared normal mode angular frequencies ${\omega }_{\pm }^2=(k_1+k_2+2\kappa \pm D)/(2m)$, where $D=\sqrt{(k_1-k_2{)}^2+4{\kappa }^2}$. Holding the opposite mass fixed gives ${\omega }_j^2=(k_j+\kappa )/m$. Since $D>k_1-k_2$, subtraction proves the title ordering. Holding a mass fixed retains the coupling stiffness; removing the coupling spring produces a different system.

Frequency modulation carries a message by changing the instantaneous frequency of a carrier wave. In the ideal model the wave amplitude stays constant: information changes the rate at which the wave phase advances. Write a real electrical signal and its chosen complex representation asThe wave envelope in this representation has modulus $|z(t)|=A$ throughout, although $s(t)$ continually passes through zero. A smaller instantaneous value of $s(t)$ therefore does not mean a smaller wave amplitude. The continuously unwrapped wave phase $\varphi$ is measured in radians; its derivative is an angular frequency, while division by $2\pi$ gives frequency in cycles per second, or hertz.

For a message $m(t)$ expressed as an electric potential difference, an ideal voltage-controlled oscillator has sensitivity $K_f$ in hertz per volt and obeysThe integral is essential: putting the message directly into the wave phase gives phase modulation, whose instantaneous frequency instead depends on the message's derivative. Amplitude modulation changes the multiplying wave amplitude. These are different ways to encode the same kind of message signal.

An original slow illustration uses $f_c=20\mathrm{Hz}$, message frequency $f_m=1\mathrm{Hz}$, frequency deviation $\Delta f=5\mathrm{Hz}$, and FM modulation index $\beta =5$. With zero initial wave phase,Here $t$ is in seconds. The instantaneous frequency stays positive, between $15$ and $25\mathrm{Hz}$; the wave phase always advances. The oscillations bunch up where $f_i$ is large and spread out where it is small, while the wave envelope remains at $A$.

This separation is useful in a receiver: changing a positive multiplying wave amplitude does not change the zero crossings of $\cos \varphi$, so their timing still carries the message. Noise that moves those crossings, a vanishing wave amplitude, and frequency response distortion can still damage it. The surprising beauty of FM is that a constant wave envelope can conceal a rich frequency spectrum: the components interfere to change timing instead of the envelope's height.

The frequency ratio of two nonzero frequencies is a dimensionless ratio. Ordinary and angular frequencies give the same result because both are converted by the same factor $2\pi$. Rational ratios imply a common period for their sinusoidal motions, while the special ratio two means that one mode completes two cycles during one cycle of the other.

The front velocity is the propagation speed of the earliest boundary of a newly introduced disturbance's support. It asks when an initially unaffected observer can first respond, whereas group velocity asks how a narrow wave envelope translates. A detector threshold defines a practical arrival time, but can miss arbitrarily small earlier responses; a packet peak is another distinct timing convention. In a causal dielectric model, a high-angular frequency contour argument gives exactly zero field for $t<x/c$, even when phase velocity exceeds $c$. This is a statement about that model's relativistic causality, not a universal identification of group velocity with front velocity.^[ref]

For the stipulated Klein-Gordon equation with compactly supported real Cauchy data, front velocity is bounded by $c$, and $c$ is the model's maximal speed of influence. A short energy method outline usesIntegration outside an expanding interval shows that zero initial energy there remains zero. Thus initial support in $[-a,a]$ cannot influence $x>a$ before $t=(x-a)/c$. The positive energy density includes the restoring term ${\omega }_0^2{u}^2/2$: this adds oscillation behind the front without extending its domain of dependence. A particular disturbance may fail to excite the leading boundary, so the maximal speed is not a promise of a detectable response for every source.

A Gaussian function is nonzero arbitrarily far away, so the Gaussian envelope in a Klein-Gordon medium has no sharp initial support boundary. Its small distant tails are already in the initial data. Likewise, keeping only a narrow interval of the frequency spectrum cannot reproduce an exact abrupt onset. Cutting off a Gaussian tail changes its Fourier transform and the accompanying initial velocity; it is a different initial-value problem. The exponential initial-velocity tail for the Klein-Gordon equation is another concrete example of a pre-existing tail outside the main disturbance.

The water-tank measurement in phase velocity records free surface elevation with calibrated resistance probes and uses a ruler and stopwatch for an independent check. The inspected protocol calls the front or back of a group a timing feature. That visible edge establishes an envelope transit estimate, not the first nonzero disturbance in the sense of front velocity.^[ref] A proposed additional control would repeat the transit fit at several detection thresholds and train durations, using matched probe calibration and the same sampling rate. Dependence on the threshold diagnoses wave envelope reshaping or limited sensitivity; records should be compared before reflections return. Such a test could expose limitations of a group velocity estimate, but would not establish an exact causal front or validate the stipulated Klein-Gordon equation for water waves.

For a dominant-pole voltage-feedback operational amplifier with $A(s)=A_0/(1+s/{\omega }_a)$ and $A_0\gg 1$, the unity-gain angular frequency is approximately ${\omega }_t=A_0{\omega }_a$. In the range where this pole dominates, a roughly constant closed-loop noise gain $G_N$ gives a bandwidth approximately $f_t/G_N$. This approximation depends on the amplifier model and feedback control circuit; it is not a universal statement about all amplifiers or all frequency-dependent feedback networks. ^[ref]

A fundamental Gaussian beam is an ideal beam of light whose transverse wave amplitude has a Gaussian function profile. Here consider a circularly symmetric beam in vacuum, with propagation along $z$, two transverse coordinates $x,y$, and radius $r=\sqrt{x^2+y^2}$ from the optical axis. With the beam waist at $z=0$, total power $P>0$, and vacuum wavelength $\lambda$, its time-averaged light intensity, meaning irradiance in $\mathrm{W}{\mathrm{m}}^{-2}$, isHere $w_0$ is the minimum beam radius, and $z_R$ is the Rayleigh range. Far from the beam waist, $w(z)\sim \theta |z|$ with beam divergence half-angle $\theta =\lambda /(\pi w_0)$. These are the fundamental circular Gaussian beam laws within the scalar paraxial approximation, appropriate when $w_0\gg \lambda$ and the beam divergence is small. The underlying envelope solves the parabolic wave equation exactly; this does not make it an exact vector solution of Maxwell equations. ^[ref].

The radius convention matters. Writing $U$ for the complex wave amplitude envelope, the same $w(z)$ is the wave amplitude $1/e$ radius and the light intensity $1/e^2$ radius:At $r=w(z)$ these ratios are $e^{-1}\approx 0.367879$ and $e^{-2}\approx 0.135335$, respectively. The light intensity $1/e$ radius is instead $w(z)/\sqrt{2}$. None of these radii is a sharp edge: the ideal Gaussian function has nonzero tails at every finite radius. Kogelnik and Li explicitly use the wave amplitude $1/e$ radius in their fundamental-mode treatment. ^[ref].

For a numerical example, take vacuum wavelength $\lambda =632.8\mathrm{nm}$ and beam waist radius $w_0=0.500\mathrm{mm}$. Substitution gives Rayleigh range $z_R\approx 1.24114754\mathrm{m}$ and far-field beam divergence half-angle $\theta \approx 0.402852992\mathrm{mrad}$. At $z=z_R$, the beam radius is $w=\sqrt{2}{w}_0\approx 0.707107\mathrm{mm}$ and the on-axis irradiance is half its waist value. At the same wavelength, halving the beam waist radius to $0.250\mathrm{mm}$ quarters the Rayleigh range to $0.310286885\mathrm{m}$ and doubles the beam divergence half-angle to $0.805705984\mathrm{mrad}$. At fixed power, that smaller beam waist also quadruples the peak irradiance.

The contours in the left panel are $r=\pm w(z)$ in a plane through the optical axis; they mark an light intensity level, not an outer boundary. The right panel compares the exact model radii with the far-field asymptotes $w=\theta z$. The horizontal and vertical axes use different length units, so the drawn slopes are not physical angles.

The normalization expresses conserved power during free propagation. In transverse polar coordinates, an annulus has area $2\pi rdr$, soA proof outline is the substitution $s=2r^2/w(z{)}^2$ in this integral. Spreading reduces peak irradiance as $w^{-2}$ while increasing transverse area as $w^2$, leaving power unchanged. In particular, the disk of radius $w(z)$ contains $1-e^{-2}\approx 0.8646647168$, or $86.4665\%$, of the total power; the local light intensity at its rim is only $13.5335\%$ of the on-axis light intensity. These are different quantities.

The left panel compares normalized wave amplitude magnitude and irradiance; the right panel integrates the transverse area to show enclosed power. For an ideal absorbing circular aperture of radius $a$ centered on the optical axis, the same fraction gives the power passing the aperture at that plane. An aperture of radius $w$ clips about $13.5335\%$ of the incident power; radius $2w$ clips only $e^{-8}\approx 0.000335463$, or $0.0335463\%$. After clipping, extra diffraction means the transmitted field no longer obeys the unmodified ideal Gaussian beam law. Every finite aperture cuts some ideal tail. Kogelnik and Li distinguish their large-aperture beam analysis from their finite-aperture diffraction analysis. ^[ref].

The useful tradeoff is that a small bright spot cannot remain narrow indefinitely. An optical lens can create a smaller beam waist, but diffraction then makes the beam spread faster. The particularly simple relation $w_0\theta =\lambda /\pi$ ties localization directly to beam divergence. A proof outline for the propagation law starts with $U(x,y,0)=U_0{e}^{-(x^2+y^2)/w_0^2}$. The parabolic wave equation separates into two transverse factors, and a Gaussian integral in each coordinate givesHere $q$ is dimensionless and $k$ is the wavenumber; the carrier convention is $e^{i(kz-\omega t)}$, with angular frequency $\omega$. Taking the squared complex modulus gives the stated $w(z)$ and its $w^{-2}$ light intensity scaling. Each transverse factor contributes $q^{-1/2}$, explaining the $q^{-1}$ prefactor here and the different prefactor in the preserved Gaussian beam with one transverse coordinate.

Historically, Gaussian beam theory developed alongside early laser resonator theory and lens-guided transmission. In their 1966 review, Kogelnik and Li credit Boyd and Gordon, and Goubau and Schwering, with early beam treatments; they summarize earlier work rather than claim the whole construction as a new discovery. Their paper brings beam propagation, focusing and finite-aperture diffraction into a common practical account. ^[ref].

Use the quantum harmonic oscillator in units with reduced Planck constant $\hslash =1$, mass $m=1$, and angular frequency $\omega =1$. Its Hamiltonian operator and trial Gaussian wave packet areThis is the same Gaussian wave packet used for the Heisenberg uncertainty relation: $|{\psi }_{\sigma }{|}^2$ is a normal distribution with mean zero and variance ${\sigma }^2$. The Gaussian integral gives wavefunction normalization and $⟨x^2⟩={\sigma }^2$. These smooth, rapidly decaying trials lie in the one-dimensional harmonic oscillator form domain and even in the operator domain.

Since ${\psi }_{\sigma }^{\prime }=-x{\psi }_{\sigma }/(2{\sigma }^2)$, integration by parts with vanishing boundary terms gives the kinetic energy and potential energy:Narrow trials cost kinetic energy, while broad trials cost potential energy. The exact differenceshows that the unique optimal width is ${\sigma }^2=1/2$, with $E=1/2$. At $\sigma =1/2$ and $\sigma =1$, the trial energies are both $5/8=0.625$; the kinetic energy and potential energy contributions exchange places.

Minimizing within this trial family alone establishes $E_0\le 1/2$. To establish exactness, use the Heisenberg uncertainty relation to obtain $E_{\psi }\ge 1/2$ for every admissible quantum state, as in its existing oscillator example. Together the bounds give $E_0=1/2$. Direct substitution also yields $H{\psi }_{1/\sqrt{2}}={\psi }_{1/\sqrt{2}}/2$, agreeing with the Gaussian eigenstate for a quadratic potential. This is a useful distinction: the variational method supplies the upper bound, while the Heisenberg uncertainty relation supplies the complementary lower bound.

A gravity-capillary wave has restoring contributions from both gravity and surface tension. Consider an initially resting, horizontally unbounded liquid layer of constant depth $h$, with uniform mass density $\rho$, incompressible flow, inviscid flow and irrotational flow. The upper gas exerts constant pressure and has negligible inertia. Assume a clean interface with constant surface tension $\gamma$: no surfactant stresses, imposed current or wind. The wave amplitude $a$ must satisfy $ka\ll 1$ and $a\ll h$, so linearized boundary conditions apply. Deep water means $kh\gg 1$, rather than a particular depth in metres. ^[ref]

Let $z=0$ be the undisturbed free surface, with liquid below it. Potential flow gives a velocity potential $\varphi$ with $\mathbf{u}=\nabla \varphi$, and incompressible flow gives the Laplace equation. The linearized kinematic boundary condition for a free-surface graph and the Unsteady Bernoulli equation, including gravity and the Young–Laplace equation, giveHere $\eta (x,t)$ is the interface displacement; the signed small-slope pressure jump is $p_{\mathrm{liquid}}-p_{\mathrm{gas}}=-\gamma {\eta }_{xx}$. A wave crest has ${\eta }_{xx}<0$, so its capillary pressure is positive. Choose a right-travelling Fourier mode of positive wavenumber $k$ and angular frequency $\omega$:The factor $e^{kz}$ decays into the deep liquid, satisfies the Laplace equation, and makes ${\varphi }_z={\eta }_t$ at $z=0$. Substituting in the dynamic boundary condition leavesThis derivation displays the two restoring contributions with the same sign. Dimensional analysis also checks the result: $\gamma /\rho$ has units ${\mathrm{m}}^3{\mathrm{s}}^{-2}$, so both $gk$ and $(\gamma /\rho )k^3$ have units ${\mathrm{s}}^{-2}$. ^[ref]

At finite depth, a flat impermeable bottom at $z=-h$ requires ${\varphi }_z=0$. Replacing $e^{kz}$ by the normalized vertical factor $\cosh (k(z+h))/\cosh (kh)$ makes the surface ratio ${\varphi }_z/\varphi =k\tanh (kh)$. Eliminating the mode amplitudes then givesThus the same model supplies its own depth check: use the deep-water formula only when $\tanh (kh)$ is sufficiently close to one for the required accuracy. For example, $kh=3$ reduces $\omega$ by about $0.248\%$ relative to deep water, while $kh=1$ reduces it by about $12.7\%$. These percentages are calculations from this factor. ^[ref]

In deep water the phase velocity magnitude follows a wave crest, while the group velocity magnitude follows the wave envelope of a narrow wave packet. Differentiating the positive-frequency dispersion relation givesFor $k{\ell }_c\ll 1$, the surface gravity wave limit has $c_g\simeq c_p/2$. For $k{\ell }_c\gg 1$, the deep-water capillary-wave dispersion relation has $c_g\simeq 3c_p/2$: a wave crest slips backward relative to the faster wave envelope, even while both advance. Neither speed is the oscillating fluid element's own velocity. ^[ref]

There is a particularly neat coincidence at the crossover. Differentiate $c_p^2=g/k+\gamma k/\rho$: its unique minimum is at $k_c=\sqrt{\rho g/\gamma }=1/{\ell }_c$, exactly where the restoring terms are equal. ThereforeFor the stipulated parameters in capillary length, the minimum is $0.231\mathrm{m}{\mathrm{s}}^{-1}$. At $\lambda =3\mathrm{mm}$, direct substitution gives $f=\omega /(2\pi )\simeq 131\mathrm{Hz}$, $c_p\simeq 0.394\mathrm{m}{\mathrm{s}}^{-1}$ and $c_g\simeq 0.580\mathrm{m}{\mathrm{s}}^{-1}$. This short wavelength is capillary dominated, but its group velocity is still slightly below the pure-capillary ratio $3c_p/2$ because gravity has not vanished.

For total ball mass $m$, arm length $l$ and imposed constant shaft speed $\Omega$, the reduced effective potential is $V_{\mathrm{eff}}/(m{l}^2)=-{\Omega }^2{\sin }^2\theta /2-{\omega }_0^2\cos \theta$. Its equilibrium equation is $\sin \theta ({\omega }_0^2-{\Omega }^2\cos \theta )=0$. The downward state is stable for $|\Omega |<{\omega }_0$ and loses linear stability above it; nonzero stable openings then satisfy the displayed relation. Extending the angle to signed values gives a supercritical pitchfork bifurcation. Squared angular frequencies of small oscillations are ${\omega }_0^2-{\Omega }^2$ at the downward state and ${\Omega }^2-{\omega }_0^4/{\Omega }^2$ at the open states. At threshold the quadratic term vanishes but the quartic potential is positive, so zero linear frequency does not imply finite-amplitude free motion.

The second derivative of spectral phase with respect to angular frequency; it measures variation of group delay with angular frequency.

Variation of inverse group velocity with angular frequency; for a uniform medium of length $L$, the accumulated group delay dispersion is $L{d}^{2}k/d{\omega }^2$.

For a rapidly spinning gyrocompass with axle constrained horizontally, take azimuth $\theta$ positive from north towards west and positive spin angular momentum $J_s=I_1\omega$. In the terrestrial rotating reference frame, the vertical component of the torque equation, with zero external vertical torque, is $I_2\ddot{\theta }+J_s{\Omega }_E\cos \lambda \sin \theta \simeq 0$. Here $I_2$ is the vertical moment of inertia and terms of order ${\Omega }_E^2$ have been neglected. The small-angle approximation about $\theta =0$ gives simple harmonic motion with the displayed squared angular frequency. Linearizing about $\theta =\pi$ reverses its sign and gives an unstable equilibrium. At the poles this restoring effect vanishes. For a thin rotationally symmetric flywheel, the perpendicular axis theorem gives $I_1=2I_2$.

A half-wave rectifier with one diode and a resistor load passes one half of each input cycle.^[ref]

For an original ideal example, let an ideal voltage source supply the sinusoid $v_S(t)=V_p\sin (\omega t)$, with $V_p=5\mathrm{V}$ and angular frequency $\omega >0$. Connect its positive reference terminal to the diode anode, the diode cathode to the upper terminal of a load resistor with electrical resistance $R_L=1\mathrm{k\Omega }$, and the lower terminal to the source return. Let $v_L$ be the upper-minus-lower load electric potential difference, and $i_L$ the downward load electric current. Assume the ideal diode model, zero source electrical resistance, and no filter capacitor, inductor or switching delay.

The Kirchhoff cycle law gives $v_S=V_D+v_L$, and Ohm's law gives $v_L=R_L{i}_L$. If $v_S>0$, the blocking assumption would give $V_D=v_S>0$, contradicting the ideal diode model, so the diode conducts with $V_D=0$. If $v_S<0$, conduction would give $i_L=v_S/R_L<0$, also forbidden, so the diode blocks. At the crossings both values vanish. ThusThe peak load electric current is $5\mathrm{mA}$. The output is unidirectional but still varies in time, reaching zero throughout every negative half-cycle. Over a period of a function $2\pi /\omega$, write the wave phase as $\theta =\omega t$. Its average value of a function and root mean square follow from two integrals:Consequently the average load power is $v_{L,\mathrm{rms}}^2/R_L=6.25\mathrm{mW}$, not ${\overline{v}}_L^2/R_L$. The ideal diode has zero loss; the resistor converts supplied energy into Joule heating. The same switching argument predicts $v_L=\min (0,v_S)$ when the diode is reversed. Directional selection changes the waveform and its average value of a function; it does not by itself produce a constant output.

Near a stable equilibrium $x_0$ with $V^{\prime }(x_0)=0$ and $V^{\prime \prime }(x_0)>0$, the harmonic approximation truncates the Taylor series of the potential energy at quadratic order. With constant mass $m$, Newton's second law then gives a harmonic oscillator with angular frequency $\omega =\sqrt{V^{\prime \prime }(x_0)/m}$. The approximation requires displacements small enough that the omitted terms are negligible; it does not describe a flat minimum where $V^{\prime \prime }(x_0)=0$.

A quadratic potential energy about a stable equilibrium. In one dimension $k>0$ is the restoring stiffness and the force is $-dV/dx=-kx$. A mass $m$ undergoing undamped motion has angular frequency $\sqrt{k/m}$. A smooth potential with a nondegenerate local minimum is approximately harmonic for sufficiently small displacements; higher-order terms describe departures from simple harmonic motion.

Keep the same bond force constant $k$ and approximate the nuclear masses of ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, and ${}^{35}\mathrm{Cl}$ by $1u$, $2u$, and $35u$, respectively. Here $u$ is the unified atomic mass unit, and $\mathrm{D}$ denotes deuterium. These are chosen mass number approximations, not measured atomic masses or natural-abundance averages. With the same chlorine isotope in both molecules,This is the isotope scaling of diatomic vibration prediction. Doubling the light partner's mass nearly doubles the reduced mass, lowering the frequency by about $28.3\%$. The factor differs from $1/\sqrt{2}$ because the chlorine partner also moves.

For the same small initial stretch $A$ and zero velocity, both models start with the same internal mechanical energy $k{A}^2/2$, but the DCl oscillation period is longer by $\sqrt{72/37}\approx 1.394972$. In the centre of mass frame, $u_{\mathrm{H}}=-35q/36$ and $u_{\mathrm{Cl}}=q/36$ for HCl; $u_{\mathrm{D}}=-35q/37$ and $u_{\mathrm{Cl}}=2q/37$ for DCl. These ratios explain which partner moves and are separate from the change in frequency.

The common-$k$ assumption belongs to the leading Born-Oppenheimer approximation: fixed nuclear positions and electric charges determine the electronic potential energy, while nuclear masses govern motion on that curve. Quantitative comparison requires actual atomic masses, departures from that approximation, and separation of anharmonicity and molecular rotation from the measured band. Even with common $k$, the quantum harmonic oscillator predicts a different zero-point energy for each isotope.^[ref]

The Heisenberg uncertainty relation constrains the standard deviations of position and momentum in one preparation:On the real line, the position operator $x$ acts by multiplication, and the momentum operator is $p=-i\hslash d/dx$, with $\hslash$ the reduced Planck constant. Their canonical commutation relation $[x,p]=i\hslash I$ and the Robertson uncertainty principle give the bound on normalized quantum states satisfying the moment and product-domain assumptions above; Schwartz functions are a convenient sufficient class. Thus making a preparation more sharply localized in position forces a wider momentum distribution, even for ideal measurement in quantum mechanics.

An explicit saturating example is the normalized Gaussian wave packet, for $\sigma >0$,The wavefunction is a probability amplitude; its squared magnitude is the probability density function, by the Born rule. That density is a normal distribution of mean zero and standard deviation $\Delta x=\sigma$, rather than a density whose width is read directly from the amplitude's exponent. With the unitary, negative-exponent convention for the Fourier transform, its momentum amplitude isThe Gaussian integral evaluates this Fourier transform. Consequently $|{\tilde{\psi }}_{\sigma }(p){|}^2$ is another normal distribution, with mean zero and standard deviationFor example, in dimensionless units with $\hslash =1$, $\sigma =1/2,1,2$ give $(\Delta x,\Delta p)=(1/2,1),(1,1/2),(2,1/4)$. The Gaussian wave packets all saturate the same bound: narrowing one density broadens the other. More precisely, choose a reference length $L$ and set $X=x/L$, $P=pL/\hslash$; then $[X,P]=iI$, and the plotted widths are $\sigma /L$ and $L\Delta p/\hslash$.

The Heisenberg uncertainty relation connects localization to the cost in kinetic energy. In a quantum harmonic oscillator with mass $m>0$ and angular frequency $\omega >0$, the variance contribution to the expectation value of the energy is bounded using the arithmetic-geometric mean inequality:The mean-dependent contributions are nonnegative. The Gaussian eigenstate for a quadratic potential with ${\sigma }^2=\hslash /(2m\omega )$ and zero means reaches this ground state energy. This is why the tradeoff is more than a limitation on preparation: it predicts an unavoidable positive minimum energy.

Heisenberg's original article appeared in March 1927 as "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", Zeitschrift für Physik 43, pp. 172–198. Its abstract discusses simultaneous imprecision of canonically conjugate quantities and explanatory thought experiments. The precise standard deviation formulation here distinguishes preparation spreads from a slogan about measurement disturbance. ^[ref].

A point mass $M$ attached to a massless rotating hoop of radius $R$ is connected by a taut string wrapping around the hoop to a vertically suspended mass $m$. Measure $\theta$ from the downward radius toward the string's tangency side. The wrapped length decreases by $R\theta$, so the suspended mass descends by the same amount. Therefore $T=(M+m)R^2{\dot{\theta }}^2/2$ and $V=-MgR\cos \theta -mgR\theta$ up to a constant. The Euler-Lagrange equation gives the displayed equation. For $0<m<M$, the stable equilibrium has $\sin {\theta }_0=m/M$, and the small-oscillation angular frequency is $\omega =\sqrt{g/R}[(M-m)/(M+m){]}^{1/4}$. When $M\gg m$, the system approaches a simple pendulum of length $R$.