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A physicist whose work connects special relativity, general relativity, Brownian motion and the photoelectric effect. He worked at the Institute for Advanced Study from 1933 until his death in 1955.^[ref]

In 1905, Albert Einstein used the photoelectric effect to explore quantized light, related Brownian motion to molecular agitation, formulated special relativity, and connected changes in mass and energy. These problems link visible measurements to otherwise inaccessible structure: emitted electron energies, suspended-particle displacements and comparisons of clocks.^[ref]

The Millikan photoelectric experiment follows one of those predictions into the laboratory. Its stopping-voltage measurements test a quantitative relation; the experiment's apparatus, surface preparation and electrical corrections matter to that inference. The Einstein field equations and general relativity lead to a different set of tests involving gravity and spacetime.

The Nobel Prize in Physics in 1921 recognized his theoretical work, particularly the law of the photoelectric effect; the award was presented in 1922.^[ref]

Primary text in English translation: en.wikisource.org/wiki/Translation:Attempt_of_a_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Bodies

Original-language facsimile record: archiv.ub.uni-marburg.de/eb/2021/0037/

Introduction and sections II and IV formulate moving-body electromagnetism relative to a stationary ether and introduce transformations of the equations. The 2021 date in the facsimile repository is the digitization date, not the publication date. The source is used to distinguish nineteenth-century ether dynamics from Albert Einstein's 1905 principle-based treatment.

Boris Podolsky coauthored the 1935 argument of Albert Einstein, Boris Podolsky and Nathan Rosen about the completeness of the wave function. Their example correlates two distant particles so that a measurement on one can predict either the other's position or its momentum. Combining their certainty-based criterion for an element of reality with the assumption that the remote measurement does not disturb the second particle leads to a tension with a complete wave function description. The argument states assumptions about reality and disturbance; it is distinct from the later experimentally testable Bell inequalities.^[ref]

By Albert Einstein, Boris Podolsky, Nathan Rosen.

Physical Review 47, 777–780. The reality criterion and position–momentum inference are distinct from later Bell inequalities.

In 1915, Albert Einstein and Wander Johannes de Haas suspended a soft-iron cylinder by glass fibre between coaxial coils. Alternating magnetization drove torsional resonance; a mirror reflected lamplight onto a scale. They compensated Earth's magnetic field, tilted the coils, and rotated the suspension attachment to test unwanted magnetic dipole torques. Added known moment of inertia calibrated the mechanical response.^[ref]

Their paper acknowledges a failed phase determination and claims only ten-percent precision for the revised measurement. Its agreement with the orbital gyromagnetic ratio prediction was not a direct discovery of spin.^[ref] Comparing that orbital gyromagnetic ratio prediction with the electron spin g-factor explains why a successful qualitative demonstration of angular momentum transfer can coexist with an inadequate quantitative interpretation. The stated precision is the authors' estimate; it does not establish that all systematic errors were bounded by that amount.

The general control problem follows directly from the model. Unwanted magnetic dipole torque can drive the same mechanical resonance as the desired reaction torque. In the ideal geometry $m_z$ and $B_z$ alone produce no axial cross product, but transverse magnetic dipole moments, stray transverse magnetic fields and imperfect alignment can. Nulling stray magnetic fields and deliberately varying alignment test this contribution. Merely observing a large resonant twist cannot identify its cause.

For a nearly instantaneous magnetic response, a sinusoidal magnetization makes $m_z$ follow the drive while $-{\dot{m}}_z/\gamma$ is in phase quadrature. An ordinary magnetic contribution proportional to $m_z$ is then distinguishable by phase, after removing the mechanical frequency response. Hysteresis or delayed magnetic response can spoil this simple separation. The measured magnetization, rather than the coil electric current alone, must set the reference. Mechanical vibration, magnetostriction coupled through imperfect geometry, and changing suspension properties require separate checks; none is eliminated simply by reversing the coil electric current.

By Albert Einstein. Paper 5, pp. 177–197; originally Annalen der Physik 17, 132–148 (1905).

Cited passages: §§7–8, pp. 191–196.

A discovery becomes more intelligible when its motivating question, proposed explanation and actual measurement can be followed together. The photoelectric effect connects Albert Einstein's light-quantum reasoning with the Millikan photoelectric experiment. The Schrödinger equation and the Davisson–Germer experiment connect mathematical wave mechanics with a laboratory test of matter waves.

The Nobel Prize in Physics provides one route through these subjects. Following the original papers and the people who built the apparatus also brings collaborators into view: Lester Halbert Germer appears alongside Clinton J. Davisson in the Davisson–Germer experiment. The history belongs in the articles on those people and concepts as well as in award records.

A theoretical physicist who coauthored the 1935 Einstein–Podolsky–Rosen paradox argument with Albert Einstein and Boris Podolsky. The paper asks whether predicting a distant system's measurement result with certainty, without disturbing it, establishes an element of reality absent from a complete wave function description. Its concrete tension between correlations, locality and completeness helped motivate later Bell tests.^[ref]

Awarded to Albert Einstein for theoretical physics, with particular recognition of the law of the photoelectric effect. He received the award in 1922.^[ref]

The Millikan photoelectric experiment makes the stopping-voltage test and its experimental limitations explicit.

It is implausible to deny every case of revolutionary theory change, but reasonable to reject the claim that such changes are instantaneous, complete or wholly disconnected from earlier science. Much turns on the strength of the proposed notion of revolution.

Thomas Kuhn challenges a picture of scientific development as the steady addition of true propositions. His contrast between normal science and revolutionary episodes concerns changes in accepted problems, methods and exemplary achievements as well as changes in propositions. A community can revise the framework within which it decides what needs explaining.^[ref] This is a historical proposal, not the claim that every important discovery is a revolution.

The transition from Newtonian mechanics to special relativity supplies a clear physical example of a deep conceptual change. Consider two spatially separated events, and let their differences in one inertial frame be $\Delta t$ and $\Delta x$. A Lorentz transformation givesFor $\Delta t=0$, $\Delta x\ne 0$ and $v\ne 0$, they are not simultaneous in the second inertial frame. By contrast, the Galilean transformation takes $t^{\prime }=t$, so simultaneity is invariant. The change concerns how time coordinates assigned by separated clocks enter physical measurement, not merely one corrected numerical constant. Albert Einstein begins by defining simultaneity through a clock procedure and then derives the transformation from his postulates.^[ref]

One reason to deny revolutions is precisely that the new account retains old achievements. For $|v|/c\ll 1$, and within regimes where relativistic corrections to the relevant measurements are negligible, Newtonian mechanics remains useful. Experimental records, instruments, mathematical methods and many successful predictions cross the transition. That continuity is real. It does not show that the physical interpretation of simultaneity remained unchanged. Numerical approximation in a restricted domain is weaker than identity of conceptual commitments.

A second objection is that major changes emerge gradually. Earlier theories and experiments prepare the transition; different researchers adopt it at different times; its consequences take decades to develop. This undermines a picture of one decisive moment transforming all science. It does not undermine a notion of revolutionary theory change as an extended reorganization of a field. Political revolutions are themselves extended events; the metaphor need not require an instantaneous rupture.

A third objection concerns rational comparison. If a revolution changes standards and meanings, perhaps rival theories cannot be compared at all. But this extreme conclusion is unnecessary. Rival accounts can share instrument records, limited predictions and mathematical limiting relations while differing about their interpretation. Explaining why those records agree in one domain and discriminate theories in another is itself a rational comparison. Conceptual change can make translation partial and difficult without making all evidence inaccessible across the divide.

These qualifications also constrain the historical claim. A local change in one field need not create a universal new worldview. Nor should every anomaly be described as a crisis: normal science routinely works on discrepancies without replacing its framework. Whether an episode is revolutionary should depend on evidence of the extent and character of reorganization, not on a dramatic label attached afterwards.

The defensible conclusion is therefore selective. Denying total, unprepared and irrational breaks may be historically plausible. Denying substantial transformations of fundamental concepts and research standards is much less so. The example of simultaneity shows how continuity of successful calculation can coexist with a change deep enough to merit the name revolution.

Theory change defeats an inference from scientific success to final, unrestricted truth. It does not show that successive theories make no progress in accuracy, explanation or warranted knowledge. Calling two theories false says too little about how their claims compare.

The strongest pessimistic argument is historical: theories once regarded as successful and well supported have later lost central commitments. If successful predecessors could misidentify what exists, the success of a present theory cannot by itself guarantee its full ontology. Larry Laudan uses the historical record to challenge a simple connection between success, reference and approximate truth.^[ref] The challenge is serious for scientific realism, particularly when it treats an entire accepted framework as a single package of warranted beliefs.

But the slogan conflates several claims: that an unrestricted theory contains a false assertion, that every one of its assertions is false, and that its successes were accidental. The first does not entail either of the others. A theory can be wrong outside its proper domain while accurately representing relationships within it. Establishing that domain is itself a scientific achievement.

The velocity-addition formula makes the difference precise. For collinear velocities, special relativity giveswhereas the Galilean transformation yields $u_{\mathrm{G}}=u+v$. Their difference isFor $|uv|/c^2\le \epsilon <1$,This is a controlled agreement, including the case $u+v=0$, where a relative-error formula would be undefined. At $u=v=0.6c$, the classical value $1.2c$ differs dramatically from $15c/17\simeq 0.882c$. At small velocities the same comparison explains why ordinary classical calculations work so well. The relativistic rule and its low-speed relation to the classical rule appear in Albert Einstein's treatment of velocity composition.^[ref]

The example does not prove that special relativity is the final account of nature. It shows that a successor can explain an earlier theory's restricted success and predict when it fails. One obtains a stronger account of the evidence than either blanket rejection of the older theory or unqualified belief in it. The surviving mathematical relation also need not preserve the older interpretation of absolute time.

Structural realism develops this possibility by locating continuity in interpreted relations rather than every proposed entity. John Worrall uses continuity between Augustin Fresnel's optics and later electromagnetism to motivate such a response.^[ref] Its attraction is discrimination: some tested structure can survive an ontological change. Its difficulty is specifying which structure deserves belief before the next change, rather than selecting the survivors afterwards. An equation without a physical interpretation and domain is too weak a candidate.

An empiricist can accept the improvement in prediction and systematic representation while withholding belief about unobservables. That shows that progress does not logically require scientific realism. The realist can reply that independent tests, novel successful predictions and quantitatively explained limiting relations support cautious belief beyond the immediately observed. Neither side should treat historical examples as a deductive proof of its complete position.

The original claim is therefore defensible only in a thin sense: successive unrestricted theories may each contain falsehoods. Used to deny progress, it is misleading. Theories can become more accurate, discriminate more possibilities and explain why earlier approximations succeeded. Theory change should make scientific confidence selective and revisable, rather than erase the distinction between a useful approximation and an arbitrary mistake.

One can derive a family containing the Lorentz transformations without assuming anything about light, but the Einstein's principle of relativity alone does not select that family member over the Galilean transformation. Extra assumptions must be stated: spatial and temporal translational symmetry, isotropy of SPACE, continuous transformations between equivalent inertial frames, reciprocity, and closure under composition. These are substantive restrictions, rather than consequences of the bare assertion that physical laws agree between inertial frames.

Here is the algebraic content. Translational symmetry makes the transformation between coincident origins a linear transformation. In one spatial direction the moving origin $x=vt$ fixes $x^{\prime }=a(v)(x-vt)$. Isotropy and the inverse transformation at $-v$ allow the standard calibrated formTo see why $K$ is universal, temporarily write the upper-right entry as $-b(v)$, leaving the lower-left entry $-v$. Equality of the two diagonal entries in a composite boost gives $b(u)v=ub(v)$, so $b(v)/v=K$ is constant for nonzero boosts. Inversion fixes $a(v{)}^2(1-K{v}^2)=1$, with the positive branch continuous at zero. This is the universality of an invariant speed under boost composition.

The resulting velocity-addition formula law and invariant quadratic expression areFor $K>0$, $C=K^{-1/2}$ is an invariant limiting speed; setting the empirically determined $C$ equal to the vacuum speed of light gives the standard Lorentz transformation. For $K=0$, $t^{\prime }=t$ and $x^{\prime }=x-vt$: an entirely consistent Galilean transformation satisfying the stated Einstein's principle of relativity. Thus the general assumptions do not establish that an invariant finite speed exists, much less identify it with light.

There is also a formal negative-$K$ branch, which has a rotation-like geometry in the time–space plane. Standard derivations exclude it using additional global assumptions, such as a preserved time orientation and unrestricted composition of admissible boosts. Without identifying those assumptions it is misleading to claim a two-way classification from the Einstein's principle of relativity alone. In three spatial dimensions isotropy requires the same constant in every direction.

The light postulate is therefore dispensable as the starting point for a conditional kinematical derivation; empirical input remains indispensable in identifying our world's value and sign of $K$. Albert Einstein's original construction uses the light postulate and Einstein synchronisation, making that empirical connection explicit.^[ref] Replacing the postulate by an independently established finite invariant speed changes the route, rather than deriving all of special relativity from relativity alone.

Historically, electromagnetism was central to Albert Einstein's 1905 argument. His opening example compares a moving magnet and a moving conductor. The same relative motion produces the observed induction, but a description using electric fields in one frame and the magnetic part of the Lorentz force in another seemed to divide the explanation according to which object was called stationary. The unsuccessful search for an observable preferred state of rest also motivated treating the laws of mechanics, optics and electromagnetism on a common footing. That establishes a strong historical connection, not a deduction that relativity applies only to electromagnetic systems.^[ref]

The logical starting points of the kinematical argument are Einstein's principle of relativity and an invariant speed of light in vacuum, together with the assumptions needed to compare homogeneous inertial frames and physical measuring standards. Einstein synchronisation assigns distant clock readings by light exchange. In standard coordinates, the resulting Lorentz transformation for a boost along $x$ isNeither the derivation of this transformation nor the deductions of relativity of simultaneity and time dilation require solving Maxwell equations for every particular material system. They constrain the relation between measurements made in different inertial frames.^[ref]

There is nevertheless an empirical commitment in identifying $c$ with the speed of vacuum light. It is not a freely chosen numerical label: the theory says different inertial observers, using the prescribed standards, obtain the same measured value. More generally, $c$ specifies the causal structure of Minkowski spacetime. A theory could contain a massive scalar field with a Lorentz invariant equation and no electromagnetic field at all. Its particles, proper times and causal propagation would still be governed by relativistic geometry. This gives a direct sense in which the framework is logically more general than electromagnetism.

Conversely, the presence of Maxwell equations alone would not establish the full relativity principle for all matter. One could append a law involving a physically preferred timelike vector field, giving a nonelectromagnetic sector a preferred frame while retaining the usual electromagnetic equations. The covariant equation with a preferred timelike vector is an explicit model of this distinction. Requiring every relevant interaction to respect Lorentz invariance adds content; writing its components in four-dimensional notation does not supply that content automatically.

The magnet–conductor puzzle is resolved because electric fields and magnetic fields are related parts of one electromagnetic field, with the Lorentz force transforming consistently. The measured current must be compared for the same physical arrangement, including the conductor's state of motion, rather than by demanding the same electric and magnetic components separately. This is an application of the general frame transformations to a particular interaction, not the definition of every possible relativistic interaction.

Albert Einstein's later distinction between a principle theory and a constructive theory clarifies what the 1905 argument provides. It places universal restrictions on admissible laws without constructing every rod and clock from a particular microscopic model. A constructive explanation can additionally investigate why actual material standards approximate the required behavior. Harvey R. Brown and Oliver Pooley emphasize this dynamical question; it does not make the kinematical transformations meaningless or confine them to electromagnetic phenomena.^[ref][ref]

The strongest account therefore distinguishes historical motivation, the operational use of light, and the theory's universal scope. The 1905 paper addresses an electromagnetic problem through a framework whose restrictions extend to every physical process claimed to obey special relativity.

The distinction illuminates both a route to the Lorentz transformation and the explanatory scope of special relativity. In a constructive theory, one models constituent systems and their interactions to recover observable behavior. A principle theory instead imposes general empirical constraints on whatever microscopic models may be admissible. Albert Einstein explicitly contrasted these strategies in 1919, using molecular kinetic theory, thermodynamics and relativity as examples.^[ref]

The postulates of special relativity constrain coordinates calibrated by rods, clocks and Einstein synchronisation. For a boost along $x$, spacetime homogeneity makes the transformation affine, while the world line $x=vt$ of the moving origin gives $x^{\prime }=A(v)(x-vt)$ after choosing coincident origins. Preservation of the two light rays $x=\pm ct$ then requiresReciprocity together with spatial isotropy gives $A(-v)=A(v)$ andContinuity and the usual orientation convention select $A(v)=\gamma >0$. Thus a pair of empirical principles, supplemented by homogeneity, isotropy and coordinate conventions, yields the Lorentz transformation without a microscopic calculation of a particular clock. Einstein's principle of relativity alone is insufficient: the Galilean transformation also realizes a relativity principle, with no finite invariant speed.^[ref]

The distinction also tells us what has not been derived. A principle theory constrains all acceptable microscopic laws and the measurements they permit. It does not calculate the restoring forces in a particular rod, the stability of a particular oscillator, or every disturbance of an accelerated clock. Those are questions for detailed dynamics. Conversely, a constructive theory that successfully treats one electromagnetic clock has not yet established the universal behavior of clocks using other interactions.

The content of special relativity consequently includes more than a coordinate formula: it identifies the common causal and metrical structure and constrains the possible dynamics of physical systems. Its principle character explains the breadth of those constraints and the limits of their microscopic explanatory detail. A constructive realization can complement them. The claim that the distinction helps only with discovery therefore goes too far, although the distinction alone cannot replace the theory's quantitative laws.

The photoelectric effect, black-body radiation and the Davisson–Germer experiment challenge different parts of a fully classical picture. Classical electromagnetic waves successfully describe propagation, interference and diffraction, while classical particles follow trajectories determined by forces. The difficulty is combining that picture with the observed exchange of energy between radiation and matter, the equilibrium distribution of radiation, and electron scattering. The three experiments expose the roles of discrete energy exchange and coherent matter waves.

In a photoelectric effect experiment, monochromatic light illuminates a metal cathode in vacuum and a second electrode collects emitted electrons. Varying the collecting voltage measures a photocurrent; reversing it until collection ceases determines the stopping potential. The observations distinguish two controls: at fixed frequency above threshold, increasing light intensity increases the emission rate, whereas the largest emitted kinetic energy is determined mainly by frequency. There is a material-dependent photoelectric threshold frequency, and emission above threshold does not require the long charging-up delay predicted by a simple picture of gradual classical energy accumulation at low intensity.

A classical wave carries continuously distributed energy, with intensity proportional to squared field amplitude. On a fully classical absorption picture, increasing intensity would be expected to increase the energy delivered to an electron, and sufficiently long exposure could accumulate the escape energy even below the observed threshold. It does not naturally produce the measured universal straight-line relation between stopping potential and light frequency. In 1905, Albert Einstein instead treated light-energy transfer as localized quanta. His argument distinguished the fastest escaping electrons from electrons losing energy before escape.^[ref]

In the single-photon regime, conservation of energy gives the Einstein photoelectric equationwhere $\Phi$ is the surface work function. The photon supplies energy $h\nu$; the escape cost is paid first, and the remainder sets the upper electron-energy edge. Increasing intensity at fixed frequency increases the number of available photons, so it raises current without raising this edge. Below ${\nu }_0$, one photon cannot pay the escape cost. In an actual measurement, stable surface preparation, suppression of unwanted higher-frequency light, and the photoelectric contact-potential correction are needed to test the slope $h/e$ reliably. Robert Andrews Millikan's experiment used selected mercury spectral lines, freshly exposed metal surfaces and an electrometer to make such a quantitative test.^[ref] The interpretation is for weak single-photon illumination; multiphoton photoemission at sufficiently intense fields is a different regime.

Black-body radiation tests equilibrium physics rather than the escape of individual electrons. A small aperture in an isothermal absorbing cavity approximates an ideal blackbody: radiation entering the hole undergoes repeated interactions with the walls, and the equilibrium spectrum depends on temperature rather than wall material. A spectrally resolving detector outside the aperture measures how emitted power varies with frequency or wavelength. Experimentally the spectrum has a finite peak, is strongly suppressed at short wavelengths, shifts toward shorter wavelengths when the cavity is heated, and has integrated emission proportional to $T^4$.

For a large cavity, the electromagnetic mode density in a box, including two independent transverse polarizations, is $g(\nu )=8\pi {\nu }^2/c^3$ per volume per unit frequency. The classical equipartition theorem assigns mean energy $k_{B}T$ to each mode, leading to the Rayleigh-Jeans lawThis agrees at low frequencies but diverges as frequency increases; integrating it over all frequencies gives infinite energy density at any nonzero temperature. This is the ultraviolet catastrophe, a failure of classical equipartition applied to the electromagnetic modes.

Max Planck's essential modification was that energy at a frequency is exchanged in units $h\nu$. In a modern oscillator description, the thermal excitation levels are $nh\nu$, $n=0,1,\dots$. Using the Boltzmann distribution and summing a geometric series givesMultiplying by the mode density yields Planck's law for the spectral energy density,When $h\nu \ll k_{B}T$, expansion of the denominator recovers the classical law. When $h\nu \gg k_{B}T$, the large excitation cost suppresses occupancy exponentially, removing the divergence. Its integral gives the Stefan–Boltzmann law, while its peak gives Wien's displacement law. Thus quantization repairs the spectral shape, total emission and temperature dependence together. The displayed thermal energy excludes the oscillator's temperature-independent zero-point energy.

The Davisson–Germer experiment supplies the complementary evidence that matter cannot always be treated as classical particles with definite scattering trajectories. In 1927, Clinton J. Davisson and Lester Halbert Germer sent a narrow, accelerated electron beam normally onto a single nickel crystal in vacuum. A movable collector measured the current of scattered electrons as direction and accelerating voltage were varied; a retarding potential selected electrons retaining approximately the incident energy. Sharp angular beams depended on crystal orientation and beam energy, with a prominent feature near $54\mathrm{V}$ and detector colatitude $50^{\circ }$. These are scattering directions and currents, not photographs of electron paths.^[ref]

The de Broglie wavelength predictsAt $U=54\mathrm{V}$, this gives $\lambda \simeq 0.167\mathrm{nm}$, comparable to atomic spacings. A crystalline arrangement supplies regularly spaced scattering sites, so scattered probability amplitudes can add constructively in selected directions. For a suitable family of crystal planes, Bragg's law expresses the condition as $2d\sin {\theta }_B=n\lambda$, with ${\theta }_B$ measured relative to those planes. The detector colatitude is not automatically this angle. The measured angular structure and its variation with electron momentum are described by electron diffraction with the de Broglie wavelength. A classical point-particle deflection model has no phase wavelength $h/p$ and does not supply that interference condition.

The common quantum account preserves the successes of wave propagation and particle detection while changing what is quantized. Photons exchange energy in frequency-dependent units; a thermal distribution of those excitations gives the blackbody spectrum. Electrons have a wavefunction whose amplitudes interfere on scattering from a crystal, while detection transfers localized charge. The Born rule turns squared amplitudes into event probabilities. These results invalidate a fully classical description of matter and radiation; the photoelectric energy law alone does not establish every property of a quantized electromagnetic field, but the three phenomena together provide complementary evidence for the quantum description.

An eigenvalue of an observable in a known corresponding eigenstate is a simple example satisfying the Einstein–Podolsky–Rosen criterion of reality: its value is predictable with certainty. For example, a spin angular momentum component in a prepared spin-up eigenstate has a definite predicted value. For a distant system, Albert Einstein, Boris Podolsky and Nathan Rosen invoked locality: a measurement performed on a sufficiently separated, noninteracting partner should not disturb that system. In the relativistic version here, the measurement events have spacelike separation.

The Pauli matrices obey $[{\sigma }_z,{\sigma }_x]=2i{\sigma }_y$, soTheir nondegenerate eigenstates are incompatible: an $S_z$ eigenstate has equal probabilities for the two $S_x$ outcomes, and conversely. Thus no single quantum state has both components sharply determined. The Robertson uncertainty relation gives $\Delta S_z\Delta S_x\ge \hslash |⟨S_y⟩|/2$, but its right-hand side may be zero; that alone is not a proof that simultaneous zero uncertainties exist. The absence of a common eigenstate supplies the stronger conclusion for this spin-half pair.

For the specified Bell state, Bob obtains $+\hslash /2$ or $-\hslash /2$ with probability $1/2$ when he measures $S_z$. Conditional on his result, Alice obtains the same sign with certainty if she measures $S_z$. Under the EPR locality premise, this certainty supplies an element of reality for her $S_z$; it is a conditional prediction after Bob learns his own random result, not a premeasurement prediction of its sign.

The normalized $S_x$ eigenstates are $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$ and $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$. Expanding their tensor products givesConsequently the same Bell state has equal, perfectly correlated $S_x$ outcomes. Bob's two outcomes again each have probability $1/2$, and the same locality-and-certainty argument applies separately to Alice's $S_x$.

The disputed step is to combine predictions conditional on two incompatible choices into two simultaneous, measurement-independent pre-existing values. Quantum theory provides the two conditional ensembles, not a joint sharp assignment of $S_x$ and $S_z$. This is quantum steering. Bob cannot measure both incompatible observables sharply on the same member of the pair. Nor does conditional steering allow signalling: ignoring his result leaves Alice's reduced density matrix equal to $I/2$ for either measurement choice, by the no-communication theorem.

To test the stronger claim of a local hidden-variable theory, Alice needs several correlation settings, not just perfect agreement along the two common axes. For the CHSH inequality, choose observables with outcomes $\pm 1$:In the given Bell state, $⟨{\sigma }_z\otimes {\sigma }_z⟩=⟨{\sigma }_x\otimes {\sigma }_x⟩=1$ and the crossed correlations vanish. Henceexceeding the local bound $|\mathcal{S}|\le 2$. The bound assumes Bell local causality and measurement independence, with the appropriate treatment of sampling and settings.

Experimental Bell tests support violations rather than that local assignment. The Aspect switching experiment changed analyzer settings during separated photon flights, while leaving sampling and switching limitations. By the time of this examination, the Delft electron-spin Bell test had used electron spins at stations $1.3\mathrm{km}$ apart, efficient spin readout and rapid random basis choices. The combination addressed both detection and locality loopholes, providing evidence against the tested local model class.^[ref]

This does not prove that every conceivable notion of reality is false, or that noncommutation alone rules out every hidden-variable theory. A nonlocal theory can evade the local premise. What Alice can challenge is Bob's claim that locality, measurement-independent simultaneous values and all observed correlations can be retained together. An ordinary measurement disturbance at Bob's lab is not by itself a local explanation of Alice's conditional ensemble.

Paul Langevin described Brownian motion through a force balance containing drag and rapidly fluctuating molecular forces. In modern notation, the Langevin equation $m\dot{v}=-\gamma v+\xi (t)$ combines damping with a random force. At thermal equilibrium, the relation between fluctuations and drag gives the long-time diffusion coefficient $D=k_{B}T/\gamma$, where $k_B$ is the Boltzmann constant, $T$ is temperature, and $\gamma$ is the linear drag coefficient. His 1908 note recovers the displacement law developed by Albert Einstein through this mechanical route, connecting irregular microscopic forces with a predictable statistical diffusion law.^[ref]

The photoelectric effect is the release of electrons from matter following photon absorption. Here the focus is emission from a metal surface into vacuum: a photon supplies energy, the work function pays the escape cost, and the remainder can become the emitted electron's kinetic energy. A photocurrent counts collected electric charge; a stopping potential probes the upper kinetic energy limit. These are different measurements.

In 1905, Albert Einstein proposed that an individual light quantum could transfer its whole energy to one electron. His §8 allowed losses before escape, distinguished the fastest emitted electrons from the rest, and predicted a straight stopping potential versus frequency relation with a material-independent slope. He separately predicted that changing light intensity changes the number emitted without changing their velocity distribution under otherwise identical conditions. He also discussed incomplete transfer and limits to the account.^[ref]

The useful modern starting model is a clean, homogeneous metal with independent single-photon events, negligible initial thermal broadening, negligible lowering of the surface barrier by an applied electric field, and negligible electric fields from the emitted electrons themselves. Specify the surface and its work function; it is not enough to name the bulk material. The Einstein photoelectric equation describes the upper edge of the emission energy distribution. It neither predicts that every absorbed photon liberates an electron nor supplies the whole photocurrent curve. Its appeal is that a complicated surface process still exposes the universal Planck constant through a simple slope.

A physical theory organized around general empirical constraints on admissible processes, rather than a specified microscopic mechanism. Albert Einstein used thermodynamics and special relativity as examples. This classification concerns the explanatory strategy; a principle theory can constrain possible constructive theories.^[ref]

Wander Johannes de Haas and Albert Einstein investigated mechanical motion produced by changing the magnetization of a suspended body. Magnetic reversal changes microscopic angular momentum; a compensating rotation of the body tests angular-momentum conservation. Their experiment connected an apparently internal magnetic property with a visible mechanical response. Interpreting a gyromagnetic measurement requires the relation between magnetic moment and angular momentum, rather than treating magnetization as a force acting independently of its microscopic carriers.^[ref]

A physicist who researched magnetism and became a professor at Leiden in 1924.^[ref]