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Saccharomyces cerevisiae is a particular species of yeast, used in baking, brewing and laboratory research. During ordinary vegetative growth it reproduces by budding: a smaller daughter grows from the mother before separation. Laboratory strains can be maintained as either haploid or diploid biological cells. This makes it possible to investigate genes in a single eukaryote, then use controlled crosses and meiosis to connect a genotype to a phenotype. See Burgess, Powers and Mell, Budding Yeast Saccharomyces cerevisiae as a Model Genetic Organism (2017), pp. 1–2.

The practical value of Saccharomyces cerevisiae as a model organism is the combination of easy laboratory cultivation, accessible genetics, and cellular functions shared with other eukaryotes. A haploid strain has one set of chromosomes, so a changed allele is not accompanied by a second copy at that genetic locus. But a mutation need not produce an observable phenotype under every condition. The yeast is also a whole organism, not a miniature human: transferring a result requires evidence that the relevant function is shared in the target system. See Duina, Miller and Keeney, Budding Yeast for Budding Geneticists (2014): cultivation, genetic tools and conserved functions.

For a concrete example, Hartwell, Culotti and Reid's 1970 work followed temperature-sensitive Saccharomyces cerevisiae strains with successive microscope photographs. Failure to complete a particular part of the cell cycle helped identify required gene functions. Their distinction between when a function is required and where a cell finally stops is especially instructive: the visible endpoint need not be the time of the underlying defect. See Hartwell, Culotti and Reid, Genetic control of the cell-division cycle in yeast. I. Detection of mutants (1970): primary abstract. This is a useful way to turn an observable change in a biological cell into a question about genes and causal order.

In alcoholic fermentation, Saccharomyces cerevisiae converts sugars into ethanol and carbon dioxide. The carbon dioxide helps dough rise; the ethanol is useful in alcoholic drinks. The metabolism also supports production of ATP, a usable cellular energy carrier. This is one metabolic ability of one species, not the definition of every yeast. See Clark, Douglas and Choi, Biology 2e, 7.5: alcohol fermentation and the Burgess, Powers and Mell account above.

For glucose, the ideal net stoichiometry isOne molecule of glucose supplies six carbon chemical atoms: four end up in the two ethanol molecules and two in the two carbon dioxide molecules. The twelve hydrogen and six oxygen chemical atoms also balance. This is a net material balance across several enzyme-catalyzed steps, not a single chemical reaction step or the complete bookkeeping for cellular energy carriers. The standard pathway accounting yields a net two ATP per glucose: the initial breakdown produces the ATP, while the later conversion to ethanol regenerates a coenzyme needed to keep the breakdown going. See Clark, Douglas and Choi, Biology 2e, 7.2: net ATP accounting and the section 7.5 cited above.

At complete conversion, this toy gives $20\mathrm{mmol}$ of ethanol and $20\mathrm{mmol}$ of carbon dioxide. The carbon balance is $6\times 10=2\times 20+1\times 20=60\mathrm{mmol}$ of carbon chemical atoms. A real culture also builds cellular material, can use other metabolic routes, and may retain dissolved carbon dioxide; the toy is a reference balance rather than a prediction of collected gas or beverage yield.

Although alcoholic fermentation does not require molecular oxygen as a reactant, the presence of molecular oxygen does not force Saccharomyces cerevisiae to stop fermenting. De Deken's 1966 experiments reported predominantly fermentative glucose use during exponential growth in air. Consequently, bubbles do not establish either absence of molecular oxygen or an exclusively fermentative metabolism. See De Deken, The Crabtree Effect: A Regulatory System in Yeast (1966): primary abstract.

For a planet on a circular stellar orbit, a common convention is $\Theta =\frac{1}{2}(v_{\mathrm{esc}}/v_k{)}^2$. The escape-to-orbital-speed ratio measures the size of a strong gravitational assist relative to the stellar binding energy of a nearby comet. Large values tend to favour ejection over collision; small values favour repeated weak encounters with more collision opportunities. This is a statistical comparison, dependent on the initial orbit and encounter geometry, rather than a universal outcome threshold.

Compare two entrance channels feeding the same isolated resonance and the same selected exit channel, each at its own resonance energy. The exit partial decay width and the resonant denominator cancel. Different entrance momenta, spin weights and entrance widths remain. The formula assumes the same resonance, the stated unpolarized spin average and negligible nonresonant interference; a ratio at one shared laboratory energy would generally be different.

A sampling calorimeter alternates passive shower-forming material with active signal layers. Only part of the deposited energy is measured directly. Fluctuations of that fraction contribute a stochastic energy-resolution term. Fine segmentation and robust absorbers can be economical, while fine sampling and a larger active fraction generally improve resolution.

The approximate fifth-power scaling of an allowed beta decay rate with available energy when electron mass and Coulomb distortion are negligible and the nuclear matrix element is held fixed. It follows from ${\int }_0^Q{p}^2(Q-p{)}^{2}dp=Q^5/30$. Different nuclear matrix elements or nonrelativistic electron kinematics spoil an exact universal law.

A satellite test of greenhouse spectral change compares outgoing radiance with radiative transfer predictions. The 2001 study compared IRIS on Nimbus 4 (1970) and IMG on ADEOS (1997). Matching April–June ocean regions, masking land, screening clouds and correcting resolution and wavenumber controlled comparability. Spectral radiance became brightness temperature readouts. Regional comparisons and independent calculations tested interpretation. Carbon dioxide and methane features supported changed absorption; residual clouds, calibration and interannual variability limited background inference.^[ref]

This observational comparison tests radiative transfer; it does not by itself measure equilibrium climate sensitivity or validate a complete climate model.

To understand the readout, write an instrument's spectral channel schematically asHere $I$ is spectral radiance in the viewing direction $\hat{\mathbf{n}}$, $K_j$ is a normalized channel response, $b_j$ is calibration offset and $e_j$ is random error. Subtracting two channel readings isolates an atmospheric difference only after their response functions and sampling are comparable. Matching kernels reduces a false difference caused merely by observing a narrow spectral line at different resolutions; it cannot remove every calibration offset.

Outgoing longwave radiation in an energy budget also integrates radiance over upward directions and over the thermal spectrum, with the appropriate projected-angle factor. A brightness temperature converts one spectral radiance to the equivalent blackbody value through Planck's law; it is not automatically surface temperature. This explains why a spectral difference, a broadband radiative flux difference and surface warming are related observables with different units and different inferences.

With the ground-state energy set to zero, the chemical potential of an ideal Bose gas cannot exceed zero. The limiting population of excited states as it approaches zero is the saturation particle number, provided the integral converges. Additional particles occupy the ground state macroscopically. For a three-dimensional uniform box, $N_c=\zeta (3/2)V/{\lambda }^3$ with thermal wavelength $\lambda =h/\sqrt{2\pi m{k}_{B}T}$.

An Ising model on a chain of corner-sharing triangles, with one base spin and one apex spin per cell. If the apex interacts only with the two base corners, summing each apex independently gives a two-by-two transfer matrix on base spins. For symmetric Ising spins and energy $Js{s}^{\prime }+\bar{J}\bar{s}(s+s^{\prime })$, its entries are $2e^{-\beta Js{s}^{\prime }}\cosh [\beta \bar{J}(s+s^{\prime })]$.

With ${\Delta }_F=⟨0|T\{\varphi (x)\varphi (y)\}|0⟩$, for a mode of positive energy $E>0$ the Feynman i-epsilon prescription puts poles $+E-i0$ and $-E+i0$ in opposite half-planes. The energy contour integral closes clockwise below for positive time separation and counterclockwise above for negative separation. The residue theorem gives $e^{-iE|t|}/(2E)$ in either case. Thus the four-dimensional Fourier integral has numerator $i$ and denominator $p^2-m^2+i0$. The integral is understood as a distribution limit, not an ordinary absolutely convergent integral; changing the definition to $i{\Delta }_F$ would change the displayed normalization.

For $V(r)=c/r^2$ with $c>0$, nonzero physical angular momentum $h$ and positive energy $E$, put $\beta =\sqrt{1+2mc/h^2}>1$ and $r_{\min }=\sqrt{(c+h^2/(2m))/E}$. Direct integration of the effective potential radial equation gives the displayed orbit. It has one radial minimum and two straight-line asymptotes separated by polar angle $\pi /\beta$. The repulsive scattering deflection is $\pi -\pi /\beta$; no bounded or circular orbit exists at finite positive energy.

The Schrödinger equation governs the evolution of a quantum state between instances of measurement in quantum mechanics. For a particle of mass $m>0$ moving in a prescribed real potential energy $V(\mathbf{r},t)$, its position wavefunction satisfiesHere $h$ is the Planck constant, ${\nabla }^2$ is the Laplace operator, and the Hamiltonian operator combines kinetic energy with potential energy. Specify an initial wavefunction, an operator domain and any boundary conditions: the differential expression alone does not completely specify a physical model. The Born rule then turns the evolving probability amplitude into position probabilities through $\rho =|\psi {|}^2$ and integration over a region.

This scalar Schrödinger operator describes nonrelativistic quantum mechanics, with relevant momenta much smaller than $mc$, where $c$ is the speed of light. It omits the constant rest energy $m{c}^2$, assumes fixed particle number, and suppresses active spin dynamics and coupling to a magnetic vector potential. A free electron with a fixed, uncoupled spin state can still use this scalar model; the Pauli equation supplies the additional spin components and magnetic coupling when needed. A prescribed real $V$ describes a closed evolution model or an approximation in which the source of that potential energy is treated as external. Dissipation, particle creation and a general measurement in quantum mechanics need further modelling.

For a time-independent self-adjoint Hamiltonian operator, the spectral theorem defines unitary time evolution byInitial states in the operator domain satisfy the differential equation in the Hilbert space sense; the unitary operator also evolves all other states in that Hilbert space. Time-dependent Hamiltonian operators need suitable regularity and domain hypotheses; their evolution is not generally the exponential of $-itH(t)/\hslash$.

The local reason for conservation is especially tangible. Multiply the Schrödinger equation by the complex conjugate ${\psi }^{\ast }$, subtract its conjugate multiplied by $\psi$, and use the fact that the real potential energy terms cancel. For smooth wavefunctions this gives the probability continuity equationThe divergence theorem identifies the change in probability inside a region with minus the outward probability current. Decay at infinity, or appropriate probability-conserving boundary conditions, preserves wavefunction normalization. Conservation of total probability does not require a stationary probability density.

Its usefulness comes from being a linear partial differential equation: quantum superpositions of solutions are solutions, while different quantum phases can produce interference. For a fixed Hamiltonian operator, the time-independent Schrödinger equation finds the energy eigenstates; each then acquires just its own time-dependent quantum phase. One eigenvalue problem therefore organizes both allowed energies and the evolution of arbitrary quantum superpositions, with a continuous spectrum included when required. This is the attractive bridge between a spatial wave problem and a clock: energy differences determine the rate at which relative quantum phases change. It makes atomic spectral lines, quantum tunnelling and quantum scattering calculable from a specified model.

For an explicit example without confinement, take a free particle on the whole line, $V=0$, initially in a Gaussian wave packet with position standard deviation ${\sigma }_0$ and mean momentum $\hslash k_0$:Unlike a single plane wave, this wavefunction satisfies wavefunction normalization. A Fourier transform writes it as a quantum superposition of plane waves. Their dispersion relation is $\omega (k)=\hslash k^2/(2m)$, so each Fourier component gains the factor $e^{-i\hslash k^{2}t/(2m)}$. Completing the square in the resulting Gaussian integral giveswith the square root chosen continuously from $t=0$. Its position probability density and standard deviation areThe centre moves with the group velocity ${\omega }^{\prime }(k_0)=v_0$ and the wave packet spreads because different wavenumbers have different group velocities. This is unitary wave dispersion, without a random force or loss of probability. Independently, the initial momentum standard deviation is $\Delta p=\hslash /(2{\sigma }_0)$ and the initial position-momentum covariance vanishes. Free evolution gives $x(t)=x(0)+p(0)t/m$ in the Heisenberg picture, henceThis independently checks the spreading law. The Gaussian integral of $\rho$ is one at every time; the decreasing peak is exactly offset by increasing width.

As hypothetical preparation parameters, choose a free electron with ${\sigma }_0=1\mathrm{nm}$ and $k_0=2{\mathrm{nm}}^{-1}$. Using $\hslash \simeq 1.054571817\times 10^{-34}\mathrm{J}\mathrm{s}$ and $m\simeq 9.1093837\times 10^{-31}\mathrm{kg}$ givesThe rounded electron mass uses the 2022 adjustment.^[ref] At $t=t_s$, the centre is $4\mathrm{nm}$ and ${\sigma }_t=\sqrt{2}\mathrm{nm}$; at $t=2t_s$, they are $8\mathrm{nm}$ and $\sqrt{5}\mathrm{nm}$. The mean kinetic energy staysIt includes both mean motion and the momentum spread. The mean speed is about $7.72\times 10^{-4}c$ and the momentum spread corresponds to $1.93\times 10^{-4}c$, making the nonrelativistic model accurate for the appreciably occupied part of this ideal Gaussian distribution. Its mathematically unbounded tails are an idealization.

The complex width also explains where the probability goes. Write $\psi =\sqrt{\rho }{e}^{i\theta }$, where $\theta$ is the dimensionless quantum phase. Differentiating the exact solution givesRelative to the moving centre, the probability current $j_{\mathrm{rel}}=j-v_0\rho$ points outward on both sides for $t>0$. This is a flux statement, not an additional assumption about individual particle trajectories. At $t=t_s$ and $x-v_{0}t=\pm {\sigma }_0$, the relative flow speeds $j_{\mathrm{rel}}/\rho$ are $\pm {\sigma }_0/(2t_s)$, about $\pm 2.8942\times 10^4\mathrm{m}{\mathrm{s}}^{-1}$. The probability within a moving interval of half-width ${\sigma }_0$ falls from about $0.6827$ initially to $0.5205$ at $t_s$, exactly because of this outward probability current.

Historically, the 1926 Quantisierung als Eigenwertproblem (Erste Mitteilung) by Erwin Schrödinger began with the stationary hydrogen atom problem. It distinguishes discrete negative energies from a positive continuous spectrum; it is not yet the general evolution equation displayed above.^[ref] The opening of quantum mechanics discusses its original vibrating-string analogy.

In the 12 December 1933 lecture The fundamental idea of wave mechanics by Erwin Schrödinger, Erwin Schrödinger explains the move from independent rays to waves through diffraction, then applies that analogy to atomic physics. His closing pages explicitly acknowledge the difficulty of reconciling wave language with observed particle tracks. This retrospective physical argument motivates the model; it does not derive the modern Born rule from classical mechanics.^[ref]

The dates separate proposal from measurement: the de Broglie wavelength belongs to the 1924 thesis, the Schrödinger equation to 1926, and the Davisson–Germer experiment's electron diffraction beams to 1927.^[ref] That linked experiment describes the prepared nickel crystal, evacuated electron gun, angular collector scans, retarding-potential selection and heating controls. Its measured collector electric current tests wave-dependent scattering probabilities; it neither measures a wavefunction directly nor establishes this equation uniquely.

A brief flash of light produced when an energetic particle deposits energy in a material. Counting resolved flashes can count individual incident particles; this differs from atmospheric scintillation.

A material that emits optical photons after an ionizing particle deposits energy. Coupling it to photomultipliers or other light sensors converts deposited energy into an electrical pulse.

JHEP 09 (2014), 176. Full title writes the collision energy as $\sqrt{s}=8\mathrm{TeV}$. Inclusive zero-lepton jets-plus-missing-momentum search with control-region background inference.

During continued compression of a first hydrostatic core, molecular hydrogen dissociation absorbs energy and softens the effective equation of state. This can trigger a second rapid contraction before a compact stellar core becomes pressure-supported. It is distinct from the initial nearly isothermal collapse and from later Kelvin-Helmholtz contraction at roughly fixed mass.

Overlap delayed replicas in a sufficiently thin phase-matched second-order crystal and collect their cross-generated sum-frequency signal. Its integrated energy gives the displayed autocorrelation under undepleted, broadband-response assumptions. Temporal resolution depends on gate overlap, group-velocity mismatch and spectral acceptance. Duration inference requires a pulse-shape assumption, and arbitrary temporal phase is not recovered.

For an isolated system, the second law requires $\Delta S\ge 0$. More generally, include all affected surroundings to obtainEquality describes the reversible limit; irreversible macroscopic processes produce entropy. A closed system can lose entropy by exporting heat, so nondecrease does not apply to that system alone. The Clausius statement of the second law forbids a cyclic device whose sole effect is transferring heat from cold to hot. The Kelvin-Planck statement of the second law forbids a cyclic device whose sole effect is converting heat from one thermal reservoir completely into work. "Sole effect" matters: a refrigerator moves heat from cold to hot by consuming work. ^[ref]

For direct transfer of $Q=1000\mathrm{J}$ from a thermal reservoir at $600\mathrm{K}$ to one at $300\mathrm{K}$, with no work extracted,The reverse transfer alone would have negative total entropy change. Both directions conserve energy, but only the hot-to-cold direction is spontaneous. The finite-body example under entropy replaces these constant temperatures by changing ones.

Friction, free expansion and heat transfer across a finite temperature difference all introduce irreversibility. Moving arbitrarily slowly is insufficient if friction remains. A reversible thermodynamic process must permit restoration of both the system and surroundings. This gives engineers a useful ideal benchmark: irreversibility spends an opportunity to obtain work, even though energy itself is conserved. ^[ref]

At a material interface, incident P or SV energy can scatter into both P and SV modes, subject to conserved horizontal slowness and traction/displacement conditions. SH is decoupled in a planar isotropic model; some conversions vanish at symmetry directions such as normal incidence.

For uniform slip on a shear fault, scalar seismic moment is $M_0=\mu Ad$, with shear modulus $\mu$, fault area $A$ and slip magnitude $d$. More generally it integrates rigidity-weighted slip. It measures source strength rather than radiated energy alone.

A seismic shadow range is an interval of receiver distances missing a specified family of geometrical rays. A velocity decrease can separate shallow-turning and deep-turning arrivals; diffraction or other wave families can still deliver energy there.