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For a uniform ideal gas sphere, set $\rho =mn$, $p=n{k}_{B}T$ and $M=4\pi \rho R^3/3$. Building the sphere by concentric shells gives gravitational potential energy $W=-3G{M}^2/(5R)$. During a small homologous contraction, released gravitational energy balances the work against gas pressure when $3G{M}^2|dR|/(5R^2)=4\pi R^{2}p|dR|$, or $GMm/R=5k_{B}T$. Eliminating $R$ gives the displayed mass. At the same initial mass density and temperature, larger masses have stronger gravitational contraction than pressure support in this estimate. Its numerical coefficient belongs to the uniform-sphere pressure-work model; the infinite-medium Jeans instability and a pressure-confined isothermal sphere use different equilibrium and boundary assumptions.

For the auxiliary-field convention with cost $M^2/(2U)$ and coupling $-M\bar{\psi }{\sigma }_z\psi$, the homogeneous saddle point equation equates M to U times spin polarization per site. M has energy units. It is not itself a magnetic moment, and its coefficient changes if the contact interaction or density-of-states spin convention is changed. The exact three-component decoupling must be distinguished from selecting one constant spin direction as a mean-field approximation.

For a nonadhesive obstacle, $g$ is the signed separation and $r$ its outward normal contact force. Nonpenetration gives $g\ge 0$, absence of adhesion gives $r\ge 0$, and reaction is possible only at contact, giving $gr=0$. In a small-slope taut-string model, $g=w-f$ and force balance gives $r=p-T{w}^{\prime \prime }$ for downward applied load $p$. The three conditions then become $(w-f)(T{w}^{\prime \prime }-p)=0$, $w\ge f$, $T{w}^{\prime \prime }-p\le 0$. Multiplication by admissible displacement differences and integration gives the associated variational inequality, connecting a local contact law to constrained energy minimization.

A time-independent Hamiltonian evolves a state by the unitary operator $U(t)=e^{-iHt/\hslash }$.

For a closed quantum system, unitary time evolution changes a quantum state through a unitary operator, preserving norms and inner products. For a time-independent self-adjoint Hamiltonian operator $H$, choose the initial time as zero:Here $\hslash =h/(2\pi )$ is the reduced Planck constant. The same $U(t)$ acts on every initial quantum state; it depends on the Hamiltonian operator and elapsed time, rather than the chosen preparation.^[ref]

The self-adjointness of $H$ makes the exponential a unitary operator. In finite dimension, conjugating its matrix exponential givesThe two exponents commute because both are multiples of $H$. Hence, for any two initial vectors,This short proof explains why normalization survives, why orthogonality of initial vectors survives, and why the transition probability between two states evolved by the same $U$ is unchanged. It does not require that a fixed measurement in quantum mechanics have time-independent probabilities: its measurement basis need not move with the quantum states.

For an unbounded self-adjoint operator, define the exponential through the spectral theorem, rather than assuming that its power series acts on every vector. It is a unitary operator on the entire Hilbert space; the strong Schrödinger equation holds for initial vectors in the operator domain $D(H)$. For a density operator, unitary time evolution reads $\rho (t)=U(t)\rho (0)U(t{)}^{†}$. It preserves eigenvalues and purity of a density operator, but can change its entries in a fixed basis.

The useful distinction is between conserving total probability and conserving each measurement's probabilities. A fixed energy spectrum supplies different ticking rates for different probability amplitudes. Their relative quantum phases can move the Bloch vector and create quantum interference even while every energy eigenstate population stays fixed. Qubit precession under a constant Hamiltonian makes this visible; Ramsey interferometry converts the accumulated relative quantum phase into a population difference that a simple readout can detect.

If a unitary $T$ satisfied $U(t)T=TU(-t)$, differentiation at zero would giveIt would map every energy $E$ to $-E$. A Hamiltonian unbounded above would consequently be unbounded below and have no stable ground state.

When the converted P-wave is evanescent, its amplitude decays away from the boundary and the reflected SV amplitude has modulus one. The P field stores reactive energy near the boundary but carries zero mean normal power.

For the O3 nonlinear sigma model with energy $\int |\nabla \mathbf{n}{|}^2/2$, the rational map $w(z)=\rho /(z-z_0)$, $\rho \ne 0$, is a degree-one sigma-model lump. Its energy density is $4|\rho {|}^2/(|z-z_0{|}^2+|\rho {|}^2{)}^2$, whose integral is $4\pi$. The complex center $z_0$ gives its position, while $|\rho |$ gives its size and $\arg \rho$ its internal phase. Varying the size preserves energy. The limit $\rho \to 0$ is singular at the center, so conserved degree under smooth deformations does not prevent concentration.

A coherent disordered mesoscopic conductor has reproducible, sample-dependent conductance variations when magnetic field, carrier energy or disorder configuration changes. Their typical scale is of order $e^2/h$ in a fully coherent diffusive sample, with numerical factors determined by symmetry, dimensionality and geometry. Thermal averaging and finite phase coherence length reduce the measured amplitude.

The up quark $u$ is the first-generation up-type quark, with electric charge $+2e/3$, spin $1/2$ and baryon number $1/3$. It is a light quark, and the up quark and down quark together account for the net valence flavor content of ordinary nuclei.^[ref]

The minimal proton content $uud$ has total charge $2(2e/3)-e/3=e$ and baryon number one. The proton's mass is not simply the sum of three free-quark masses: its interacting quark and gluon fields carry most of its energy. The quark model organizes the flavor content, while quantum chromodynamics describes the dynamics needed to compute hadron properties.

In a uniform U-tube containing incompressible liquid of density $\rho$, cross section $A$ and total moving length $l$, let one surface fall by $y$ while the other rises by $y$. The height difference is $2y$, and potential energy relative to equilibrium is $U=\rho gA{y}^2$. The whole moving mass is $\rho Al$, so $K=\rho Al{\dot{y}}^2/2$. Newton's second law or energy conservation gives $\ddot{y}+(2g/l)y=0$ and ${\omega }_0=\sqrt{2g/l}$. This assumes small excursions that keep the free surfaces in the uniform vertical limbs.

Vacuum energy is the energy assigned to a theory's vacuum state. In a free bosonic field description it includes the sum of oscillator zero-point energies, formally $\frac{1}{2}{\sum }_{\mathbf{k}}\hslash {\omega }_{\mathbf{k}}$. The continuum limit of that expression is generally divergent and requires a prescription, rather than direct identification with a finite measured energy density.

Adding a constant to a nongravitational Hamiltonian changes an overall evolution phase but not transition probabilities. Energy differences between boundary configurations can nevertheless have physical effects, as in the Casimir effect. When gravity is included, a vacuum energy density contributes to the gravitational source and is related to the cosmological constant. Normal ordering removes a reference-theory constant by convention; it does not solve every interacting or gravitational vacuum-energy problem.

The vacuum energy density is the bulk limit ${\rho }_{\mathrm{vac}}={\lim }_{\mathcal{V}\to \infty }{E}_0(\mathcal{V})/\mathcal{V}$, with boundary effects removed and a fixed convention for the Hamiltonian. In an effective field theory at a homogeneous vacuum it is the value of the renormalized scalar potential. A global supersymmetry algebra expresses the regulated vacuum energy as a sum of supercharge norms, giving nonnegative density; the scalar potential of supergravity instead includes a negative gravitational term.

In $c=\hslash =1$ units, vacuum energy density has mass dimension four, Newton constant dimension minus two, and ${\Lambda }_{\mathrm{geom}}=8\pi G{\rho }_{\mathrm{vac}}$ dimension two. A value quoted in $\mathrm{Ge}{\mathrm{V}}^4$ is an energy density, not the geometric cosmological constant.

Long imaginary-time evolution multiplies an energy eigenstate by $e^{-T{E}_n}$. After normalization it projects onto the lowest energy component with nonzero initial overlap, or onto the lowest-energy subspace if the vacuum is degenerate. This provides vacuum boundary conditions for a path integral and is related to the Feynman i-epsilon prescription.

The rest mass of a stable soliton is the lowest energy in its topological sector minus the vacuum energy, in the infinite-volume limit. If Euclidean kernels $K_Q(\tau )$ have nonzero overlap with the sector's lowest-energy states, then $M=-{\lim }_{\tau \to \infty }{\tau }^{-1}\log [K_1(\tau )/K_0(\tau )]$, followed by the infinite-volume limit. A Euclidean path integral represents these kernels, with boundary wavefunctionals included if necessary.

For a massive spin-zero one-particle state with covariant normalization and a Lorentz scalar field, Lorentz invariance makes $F=⟨\Omega |\varphi (0)|p⟩$ independent of the on-shell four-momentum within a fixed species. A Lorentz boost to the rest frame, invariance of the vacuum state and scalar transformation at the origin prove this. Translation covariance of a quantum field then supplies the plane-wave factor. With noncovariant normalization, the boost law and the displayed momentum independence acquire energy factors.

Vadim Kaplunovsky calculated quantum corrections to gauge-coupling relations in string theory. His work shows how the first corrections in perturbation theory modify simple classical relations among the coupling constants of low-energy gauge theories. The calculation connects an assumed string vacuum with quantities such as the weak mixing angle and the scale of quantum chromodynamics. His corrected complete paper also distinguishes the revised calculation from a separately published erratum, which matters when using its numerical normalization.^[ref]

A physicist.^[ref]

The VSEPR model predicts approximate molecular geometry by arranging the electron domains around a central chemical atom to reduce their mutual repulsion. Start with a Lewis structure, count the domains, and then distinguish the resulting electron pair geometry from the positions of the chemical atoms themselves. A lone pair counts as a domain even though it is not another chemical atom.^[ref]

Three worked counts have the same total of eight valence electrons. In methane, carbon's four plus the contributions of four hydrogen chemical atoms give $4+4=8$: four bond pairs, no lone pairs, and tetrahedral molecular geometry. In Ammonia, nitrogen's five plus three from hydrogen give $5+3=8$: three bond pairs and one lone pair, producing trigonal pyramidal molecular geometry. In water, oxygen's six plus two from hydrogen give $6+2=8$: two bond pairs and two lone pairs, producing bent molecular geometry. All three have four electron domains; their molecular geometries differ.

The regular tetrahedral directions can be represented by the unit vectorsEach pair has dot product $-1/3$, so the ideal bond angle is $\theta =\arccos (-1/3)\approx 109.47^{\circ }$. This is a geometric calculation, rather than a prediction that every four-domain molecule has that exact bond angle. Lone pairs and different kinds of chemical bonds distort the ideal arrangement; water's H–O–H angle is about $104.5^{\circ }$.^[ref]

The model is attractive because small-number bookkeeping produces a spatial prediction without first solving for every molecular orbital. Shape then helps explain chemical polarity and which hydrogen bond donor or hydrogen bond acceptor sites are accessible. It is a qualitative model, not a full calculation of the electronic energy or a universal solution for every chemical compound. Gillespie and Nyholm's 1957 review ^[ref] is a historical anchor for its development.

An experimental nuclear physicist who coauthored measurements of neutrons in quantized vertical states in the Earth's gravitational field. A horizontal mirror below and an absorber above formed a height-selective guide. Transmission was suppressed when the gap was too small to accommodate the lowest state, turning a tiny gravitational energy scale into a measurable transmission threshold.^[ref]

The number of symmetry-related equivalent electronic valleys contributing to a specified low-energy energy band. It multiplies the single-valley density of states, independently of spin degeneracy. Equal electron and hole effective masses alone do not ensure equal total density of states values unless their valley degeneracies and spin degeneracies also agree.