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Exchange constraint. Exchange of two identical particles cannot change physical predictions. For the usual three-dimensional bosonic or fermionic state space, the exchange operator obeys $P_{12}^2=1$, so its allowed eigenvalues are $\pm 1$. bosons use the symmetric overall wavefunction, $P_{12}\Psi =+\Psi$; fermions use the antisymmetric overall wavefunction, $P_{12}\Psi =-\Psi$. Exchange includes the spatial coordinates and all internal labels. The identity $P_{12}^2=1$ alone does not prove which particle species has which statistics; that assignment is the Spin-statistics theorem.

Factored fermion states. If $\Psi ={\psi }_{\mathrm{space}}{\chi }_{\mathrm{spin}}$ and both factors have definite exchange symmetry, their exchange signs must multiply to $-1$. Thus a symmetric spatial factor requires an antisymmetric spin factor, and an antisymmetric spatial factor requires a symmetric spin factor. For two spin-one-half particles, the former spin factor is the singlet $(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)/\sqrt{2}$; the latter factors are the three triplet states. A more general entangled spin-space state need not factor, but must still be antisymmetric overall.

hydrogen spin isomers. The two protons are identical spin-one-half fermions. parahydrogen has total nuclear spin $I_{\mathrm{n}}=0$, the antisymmetric singlet; orthohydrogen has $I_{\mathrm{n}}=1$, the symmetric triplet. Here $I_{\mathrm{n}}$ is a spin quantum number, distinct from the moment of inertia used below.

Rotational restriction. In the electronic and vibrational ground state, these two factors are symmetric under nuclear exchange. Exchanging the protons reverses the internuclear axis, $(\theta ,\varphi )\mapsto (\pi -\theta ,\varphi +\pi )$. The spherical harmonic rotational state contributes the exchange sign $(-1{)}^{\ell }$. Overall antisymmetry therefore requiresHence parahydrogen has even $\ell$; orthohydrogen has odd $\ell$. The $\ell$ here labels molecular rotation, rather than the angular momentum obtained by also coupling the nuclear spins.

Conversion heat. The ortho-para conversion heat follows from the linear rigid rotor approximation, $E_{\ell }={\hslash }^2\ell (\ell +1)/(2I)$ with $I=\mu r^2\simeq m_p{r}^2/2$. Thus conversion of one molecule from $\ell =1$ to $\ell =0$ releasesNeglecting the small electron contribution to the molecular mass, a sample of mass $M$ contains $M/(2m_p)$ molecules. Only the initial orthohydrogen fraction releases this energy, givingThe estimate uses $r=74$ pm, $m_p\simeq 1.67262\times 10^{-27}$ kg and $\hslash \simeq 1.05457\times 10^{-34}$ J s. Handbook-level rounding and the rigid rotor approximation justify about three significant figures, not a high-precision thermochemical prediction.^[ref]

Storage consequence. The internal conversion is a delayed heat source even in a well insulated container, causing warming or additional evaporation of the liquid. Insulation reduces external heat transfer but does not remove this source. Converting to predominantly parahydrogen during liquefaction, while removing the released heat, reduces the subsequent storage heat load. No evaporation fraction can be calculated without a latent heat and a model for how the heat is removed.

With two coherent paths, the probability amplitude at a screen is ${\psi }_1(x)+e^{i\varphi }{\psi }_2(x)$. The Born rule givesThe final term shifts when the relative phase changes, producing interference fringes. For slit separation $d$ in the far field, the phase difference is approximately $2\pi d\sin \theta /\lambda$; constructive interference occurs at $d\sin \theta =n\lambda$. Finite slit widths add a diffraction envelope, so an ideal equally bright, infinitely narrow slit model does not predict the full observed envelope.

single-electron interference pattern buildup starts with isolated localized detector events at sufficiently low source intensity. Repeated events accumulate the interference probability distribution. Thus detection is discrete while propagation is described by coherent probability amplitudes. The effect cannot be explained by interference between different particles if the apparatus is occupied by at most one particle at a time. A motivating realization used a field-emission electron source, an electron biprism to recombine two paths, electron lenses and a position-sensitive electron counter; it recorded individual arrivals building up the fringes. This was a biprism realization of two-path interference, not a plate with two literal slits.^[ref]

The crucial comparison is between both coherent paths open and a preparation that supplies which-path information. If the paths become correlated with detector states $|D_1⟩,|D_2⟩$, the joint state is ${\psi }_1|D_1⟩+e^{i\varphi }{\psi }_2|D_2⟩$. Ignoring that detector multiplies the interference term by $⟨D_1|D_2⟩$. Orthogonal detector states remove the term completely, while identical detector states preserve it. This is a consequence of entanglement and the partial trace; a human need not read the detector.

In a real test, check the single-path distributions, detector background, source coherence and stability of the relative phase. Classical phase drift can wash out fringes without supplying which-path information. Conversely, localized detections alone do not establish coherent two-path propagation; the controlled phase dependence of their accumulated distribution provides the discriminating evidence.

shot noise arises from fluctuations in discrete transfer events. For equal charges $q$ arriving independently with a constant Poisson rate $\lambda$, the charge collected during $\tau$ is $Q=qN$, withThus the relative counting uncertainty is $1/\sqrt{\lambda \tau }$. Longer integration improves the precision of the mean current even when the microscopic arrivals remain irregular.

In the frequency range where independent arrivals can be treated as instantaneous pulses, the one-sided current power spectral density isFor an ideal unit-gain rectangular bandwidth $B$, this gives $i_{\mathrm{rms}}=\sqrt{2qIB}$. boxcar averaging of Poisson shot noise over duration $\tau$ has effective noise bandwidth $B_{\mathrm{n}}=1/(2\tau )$, reproducing the counting result $i_{\mathrm{rms}}=\sqrt{qI/\tau }$. The two-sided spectral convention instead assigns $qI$ to each sign of frequency. Stating that convention prevents a factor-of-two error.

A photodiode illuminated at a stable rate provides a useful measurement: subtract the mean current, amplify its fluctuations through a calibrated linear filter, and compare the variance with the mean photocurrent. Independent detected photoelectrons predict a variance proportional to mean current. Measure the dark and amplifier backgrounds separately, check the filter's effective noise bandwidth and vary illumination to distinguish this term from classical intensity noise, whose variance can scale quadratically with mean current. The shot noise model concerns counting statistics, whereas thermal electrical noise can exist in an unbiased resistor. Walter Schottky's motivating analysis treated amplifier limits and independent electron transfers in thermionic discharge tubes.^[ref]

The Poisson result is a model, not a universal consequence of charge quantization. Correlated arrivals, detector dead time, avalanche multiplication or a non-Poisson light source can change the fluctuations. Filtering reduces integrated noise at the cost of temporal resolution. Increasing the detected event rate reduces relative shot noise but raises its absolute variance; balanced photodetection cancels shared classical fluctuations, not the independent shot noise of two uncorrelated detectors.

The free electron model replaces the microscopic ionic potential and interacting electronic motion by independent mobile electrons in a uniform neutralizing background, with approximately parabolic dispersion $ϵ={\hslash }^2{k}^2/(2m^{\ast })$. Its quantum version fills states according to Fermi-Dirac statistics and the Pauli exclusion principle. Scattering is represented by a momentum relaxation time $\tau$, rather than derived from the model's ideal free trajectories. This model explains a Fermi energy much larger than room-temperature thermal energy, an electronic heat capacity proportional to temperature at low temperature, and a Drude electrical conductivity. With appropriate scattering assumptions it also yields the Wiedemann-Franz law. It cannot by itself explain band gaps, detailed band anisotropy, strong correlations, or the microscopic temperature dependence of $\tau$.

The inertial term is mass times the rate of change of mean drift velocity. The electric force is $eE$, taking the printed $e$ as the signed electron charge, and $-\gamma ⟨v⟩=-m^{\ast }⟨v⟩/\tau$ describes loss of directed momentum to scattering. electrons retain large microscopic velocities; this damping is of their mean directed motion. In a steady DC field,Both charge factors are negative for electrons, so the conductivity is positive. If $e$ instead denotes a positive charge magnitude, the force and current equations each require a minus sign and give the same result.

For AC response choose physical fields as real parts of phasors proportional to $e^{-i\omega t}$. Substitution givesThusThe real part decreases from ${\sigma }_0$ to zero. The imaginary part starts at zero, reaches ${\sigma }_0/2$ at $\omega \tau =1$, and falls as $1/\omega$. The real part falls as $1/{\omega }^2$ at high frequency. With an $e^{+i\omega t}$ convention, only the sign of the imaginary part changes. Positive $\mathrm{Re}\sigma$ gives positive cycle-averaged dissipation $⟨jE⟩=\mathrm{Re}\sigma |\hat{E}{|}^2/2$.

The supplied DC conductivity determines the transport time:Inertia matters when $\omega \tau \sim 1$, giving ${\omega }_c\simeq 4.02\times 10^{13}\mathrm{rad}{\mathrm{s}}^{-1}$, or ${\nu }_c={\omega }_c/(2\pi )\simeq 6.40\times 10^{12}\mathrm{Hz}$. Quoting the former number as hertz would introduce an erroneous factor of $2\pi$.

A completely filled valence band carries no net current: its occupied crystal-momentum states have velocities that cancel. Near a valence-band maximum, an almost full band is conveniently described by the few missing electrons, or electron holes. Removing an electron of charge $-e_0$, with $e_0>0$, changes the charge relative to the filled band by $+e_0$. This is a quasiparticle description of electronic motion, not a mobile positive ion or a Positron.

For an isotropic parabolic maximum, ${ϵ}_v(\mathbf{k})=E_v-{\hslash }^2{k}^2/(2m_h^{\ast })$. Assign the missing-electron state momentum ${\mathbf{k}}_h=-{\mathbf{k}}_e$ and hole energyIt has positive effective mass $m_h^{\ast }$. Its velocity equals the velocity of the missing electron state, and its positive-charge current replaces the negative of that state's electronic current. Semiclassical motion is thus $\hslash {\dot{\mathbf{k}}}_h=+e_0(\mathbf{E}+{\mathbf{v}}_h\times \mathbf{B})$. With momentum relaxation time ${\tau }_h$, the low-field mobility is ${\mu }_h=e_0{\tau }_h/m_h^{\ast }$ andIf electrons also contribute, $\sigma =e_0(p{\mu }_h+n{\mu }_e)$: the two species' longitudinal conductivities add.

A Hall bar separates carrier concentration from mobility. Apply current density $j_x$, magnetic field $B_z$, and measure transverse electric field $E_y$. For one isotropic hole population, transverse force balance gives $E_y=v_x{B}_z$, soFor thickness $t$, the Hall-voltage magnitude is $|V_H|=IB/(e_{0}pt)$. A longitudinal four-terminal resistance measurement supplies $\sigma$ independently. Use ohmic contacts, known conducting thickness, controlled temperature and low enough current to avoid self-heating. Reversing magnetic field removes even-in-field longitudinal offsets from contact misalignment; reversing current also helps reject thermoelectric offsets. These are central controls, rather than merely taking one transverse voltage reading.^[ref]

The simple inference assumes one carrier type and a known Hall scattering factor. More generally $R_H=r_H/(e_{0}p)$, so $\sigma R_H$ measures Hall mobility rather than automatically the drift mobility. With two carrier populations at weak field,Its sign alone need not identify the more numerous carrier. A Hall experiment also cannot determine $m_h^{\ast }$ and ${\tau }_h$ separately: their ratio sets mobility. cyclotron resonance supplies an independent mass measurement by sweeping frequency or magnetic field through $\omega =e_{0}B/m_c$ and detecting resonant absorption. The measured cyclotron effective mass can depend on band anisotropy and need not equal every curvature mass; the isotropic parabolic model identifies them. Its linewidth provides a scattering-time estimate only with a specified damping model.

semiconductor doping generates holes by adding acceptor impurities. In silicon, a substitutional group-III atom such as boron provides one fewer valence electron than silicon. A shallow acceptor level lies slightly above the valence-band edge: accepting a valence electron makes the dopant negatively ionized and leaves a mobile hole in the valence band. The impurity stays fixed in the lattice. charge neutrality and the nondegenerate equilibrium mass-action law giveFor fully ionized, uncompensated acceptors in the extrinsic regime, $p\simeq N_A$ and $n\simeq n_i^2/N_A$; with fully ionized donor compensation, $p\simeq N_A-N_D$ while holes remain dominant. The Fermi level moves towards the valence band. At low temperature, semiconductor carrier freeze-out suppresses mobile electron holes; at high temperature intrinsic electron-hole generation can dominate. Increasing dopant density also changes scattering, so a conductivity increase alone cannot distinguish increased carrier density from changed mobility. Temperature-dependent Hall and conductivity measurements separate these effects, subject to the carrier and scattering qualifications above.

The effective mass follows from the curvature of the electronic band structure. Differentiating twice givesAt either first Brillouin zone boundary $k=\pm \pi /a$,The boundary is a band maximum, with negative curvature and zero group velocity. In the semiclassical single-band description, an applied force changes $\hslash k$; the resulting acceleration is opposite the force near that maximum. This is a band-response coefficient, not a negative microscopic electron rest mass. A nearly filled band is often described more conveniently through positively charged electron holes near its maximum.

Take the electron's signed charge to be $e=-q$, where $q>0$ is the elementary-charge magnitude. Conventional electric current flows along $+x$, while the mean electron drift velocity is negative. With $\mathbf{B}=B_z\hat{\mathbf{z}}$, the magnetic Lorentz force separates charge across the width until a transverse electric field balances it. The accumulated edge charge then sustains a Hall voltage without a transverse steady current. In a one-carrier model, the Hall coefficient identifies the sign and number density of carriers; real multiband materials require a more careful interpretation.

The acceleration term contains the effective mass describing the local band response. The relaxation term removes drift momentum on the mean momentum-relaxation time $\tau$. The remaining two terms are the electric and magnetic Lorentz force on signed charge $e$; the magnetic force is perpendicular to velocity and does no instantaneous work. The relaxation-time approximation replaces individual scattering events by an averaged drift equation.

Let $n$ be the electron number density, distinct from the negative charge $e$. In transverse open circuit $⟨\mathbf{v}⟩=(v_x,0,0)$ and $j_x=ne{v}_x=I_x/(wt)$. The steady component equations givehenceConsequentlyThere is no sign ambiguity in this field-defined $R_H$ once axes and carrier charge are fixed.

A measured voltage ratio additionally needs probe polarities, which the geometric bar diagram does not label. To reproduce the positive ratio specified for electrons, define the longitudinal voltage drop $V_x=\phi (0)-\phi (l)=E_{x}l$ and the transverse rise $V_y=\phi (w)-\phi (0)=-E_{y}w$. ThenReversing either voltage's probes reverses the measured ratio, without changing the physical carrier density. With both voltages instead defined as drops along their positive axes, the algebraic ratio would have the opposite sign. The figure records the convention used here.

For the density, geometry gives $\rho =Rwt/l=Rt/(l/w)$ and $|R_H|=t|V_y|/(I_x|B_z|)=tR|V_y/V_x|/|B_z|$. ThusandThe mobility is $\mu =|R_H|/\rho =0.0130{\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$. With $m^{\ast }=m_e$,

A monovalent copper model assigns roughly one delocalized conduction electron per atom to a broad, approximately free-electron-like band. Using $m_e$ is therefore a sensible first estimate for its transport scale. Monovalence alone does not prove that every band curvature or orbit has exactly the free mass: periodic-potential mixing, the nonspherical Fermi surface and direction-dependent scattering limit the model. The calculated $\tau$ is conditional on the stipulated effective-mass approximation.

Experimentally, reverse both current and magnetic field and extract the part of the transverse voltage odd in each. This rejects fixed contact offsets and much longitudinal pickup; use four-terminal longitudinal sensing, measure dimensions independently and check heating before assigning material parameters. These controls are implemented in a reusable Hall/conductivity teaching experiment, although that experiment uses germanium rather than the copper sample here.^[ref]

The free electron model replaces the detailed periodic potential by a constant background, treating conduction electrons as mobile particles with approximately parabolic dispersion $E={\hslash }^2{k}^2/(2m)$. It is useful to distinguish the classical Drude model from the quantum free electron gas. They share a simple transport picture, but differ fundamentally in occupation statistics.

With a momentum-relaxation time $\tau$, both lead in a simple isotropic one-band approximation toThis explains approximately Ohmic longitudinal conduction and a carrier-density scale from the Hall effect. The relaxation time is phenomenological: introducing it does not itself predict scattering from defects or phonons. A four-probe resistance measurement separates the sample's longitudinal response from current-contact voltage drops; field and current reversals isolate the transverse signal.

The classical model incorrectly assigns roughly $3N{k}_B/2$ of electronic heat capacity, far too large in a degenerate metal. Fermi-Dirac statistics fixes this: most occupied states cannot change occupation after a small thermal perturbation because nearby low-energy states are already filled. Only a fraction of order $k_{B}T/E_F$ near the Fermi energy participates. For $T\ll T_F$,The quantum model also gives a finite Fermi velocity even at zero temperature and Pauli paramagnetism. Under predominantly elastic scattering, its electronic heat and charge conductivities obey the Wiedemann-Franz law,This is a quantum success; classical equipartition does not give the correct small heat capacity and Lorenz number together.

Its limitations are equally instructive. Without the periodic potential it has no band gaps, so it cannot distinguish a filled-band insulator from a metal or explain semiconductor doping. The simplest single-electron Hall formula cannot explain positive Hall coefficients; band curvature and electron holes, or a multiband transport treatment, are needed. It neglects much Fermi-surface anisotropy and detailed optical interband absorption. Noninteracting electrons alone do not explain ferromagnetism or superconductivity, and lattice thermal behavior needs phonons. The nearly-free electron model, which allows reciprocal-lattice Fourier components of the potential to mix plane waves, provides a controlled first correction near Bragg planes.

Its main success is a simple quantitative baseline for degenerate conduction electrons; its main failure is treating the crystal and interactions as too featureless. A fit of conductivity alone is not a test of the whole model: compare heat capacity, Hall sign, optical response and temperature dependence using independently constrained parameters.

A p-n junction brings p-type and n-type regions into contact. Majority-carrier diffusion initially leaves exposed ionized acceptors and donors, producing a depletion region and a built-in potential. At thermal equilibrium the electrochemical potential is constant: drift and diffusion currents cancel. The internal barrier is not a freely extractable battery voltage.

For a nondegenerate abrupt homojunction,Here $V>0$ denotes forward bias and the depletion approximation is assumed. forward bias lowers the barrier and injects minority carriers; reverse bias widens the depleted region and suppresses ordinary majority injection. In the ideal diffusion regime,Recombination can change the ideality factor; series resistance, high-level injection, leakage and breakdown limit this simple Shockley diode equation. The original junction theory explicitly treats both electron and hole currents, which is essential to carrier injection and transistor action.^[ref]

As a rectifying diode, the junction conducts strongly in one bias direction and weakly in the other before breakdown. This permits power rectification, signal detection and electronic switching. Stored minority charge affects turn-off time, so a static exponential current-voltage relation alone does not specify switching speed. A junction capacitance $C_j\simeq {ϵ}_{s}A/W$ varies with reverse bias and can be used in a voltage-controlled capacitor; heavily doped or high-field junctions can instead exploit controlled tunnelling or avalanche breakdown.

In a photodiode, absorbed photons generate electron-hole pairs and the depleted-region field assists collection. reverse bias widens the collecting region and can improve speed, but adds dark-current noise and breakdown constraints. In a solar cell, the same carrier generation supplies current against an external voltage. An ideal illuminated characteristic is $I=I_s(e^{qV/(k_{B}T)}-1)-I_{\mathrm{ph}}$: at open circuit the forward diode current balances photocurrent, while useful power is taken at an intermediate operating point. Neither short circuit nor open circuit delivers power to a load.

A forward-biased light-emitting diode injects electrons and holes that can recombine radiatively. Efficient emission favors suitable direct-gap materials and carrier confinement; the characteristic photon energy is near the semiconductor band gap. nonradiative recombination, escape losses, series resistance and heating limit efficiency. Two coupled junctions also form a bipolar junction transistor: an emitter injects minority carriers through a thin base, and a reverse-biased collector collects them. Useful current gain depends on efficient injection and limited base recombination, not simply on joining any three arbitrary regions.

These functions are experimentally distinguished by measuring dark and illuminated current-voltage curves, bias-dependent capacitance, transient switching or optical emission, with controlled temperature and current limits. The common mechanism is bias-controlled carrier injection or collection at a depletion barrier; the device geometry and operating mode determine which function is useful.

Operators, quantization and rotations. Angular momentum is the generator of spatial rotations. In quantum mechanics, its components are Hermitian operators on their appropriate domains and obeyThese angular momentum commutation relations prevent all three components from having simultaneous sharp values. For example, the Robertson uncertainty principle gives $\Delta J_x\Delta J_y\ge \hslash |⟨J_z⟩|/2$. One conventionally diagonalizes ${\hat{J}}^2$ and a selected component ${\hat{J}}_z$:The allowed labels are $j=0,\frac{1}{2},1,\frac{3}{2},\dots$ and $m=-j,-j+1,\dots ,j$. An angular-momentum multiplet has $2j+1$ states. The magnitude $\hslash \sqrt{j(j+1)}$ differs from the largest possible component $\hslash j$; the remaining components fluctuate even in a state with maximal $m$.

The angular momentum lowering operator and its raising counterpart, ${\hat{J}}_{\pm }={\hat{J}}_x\pm i{\hat{J}}_y$, satisfyThe ladder terminates at $m=\pm j$. A rotation through angle $\theta$ about unit vector $\hat{\mathbf{n}}$ acts through the unitary operator $\exp [-i\theta \hat{\mathbf{n}}\cdot \hat{\mathbf{J}}/\hslash ]$. Thus quantized components and the continuous rotation of a state are compatible.

Orbital angular momentum. The orbital contribution is associated with motion in space:For a spinless spatial wavefunction, the simultaneous angular eigenfunctions are the spherical harmonics $Y_{\ell m}(\theta ,\varphi )$, with $\ell =0,1,2,\dots$ and $m=-\ell ,\dots ,\ell$. Single-valued ordinary scalar spatial wavefunctions give integer orbital labels. In a central potential energy, $[\hat{H},{\hat{L}}^2]=[\hat{H},{\hat{L}}_z]=0$, so stationary states may be labelled by these angular quantum numbers. The angular kinetic contribution appears in the radial equation as $\ell (\ell +1){\hslash }^2/(2m{r}^2)$, the centrifugal potential. Noncentral interactions need not conserve these orbital labels.

Intrinsic spin and addition. Spin angular momentum is an intrinsic internal degree of freedom with the same commutation algebra. It is not the angular momentum of a small classical charged sphere literally rotating about its centre. For an electron, $s=1/2$ and $m_s=\pm 1/2$, so ${\hat{S}}_i=(\hslash /2){\sigma }_i$ in terms of the Pauli matrices. Its squared spin is $3{\hslash }^2/4$. Other particles can have integer or half-integer spin, including spin zero.

For one particle with orbital and spin degrees of freedom, $\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$; the two component sets commute with each other. Addition of angular momentum permits $j=|\ell -s|,|\ell -s|+1,\dots ,\ell +s$. The coupled states $|j,m_j⟩$ are linear combinations of the uncoupled $|\ell ,m_{\ell }⟩|s,m_s⟩$ with $m_j=m_{\ell }+m_s$, determined by Clebsch-Gordan coefficients. A rotationally invariant spin-orbit interaction can conserve $J^2,J_z$ while mixing the separate component labels.

Angular momentum also carries a magnetic moment. For an electron's orbital motion and intrinsic spin,The minus signs reflect negative electric charge. With the Bohr magneton ${\mu }_B=e\hslash /(2m_e)$, a spin projection $m_s$ has ${\mu }_z=-g_s{\mu }_B{m}_s$. A magnetic field can therefore distinguish angular-momentum states through their interaction energy $-\mathbf{\mu }\cdot \mathbf{B}$.

Stern–Gerlach measurement. In the Stern-Gerlach experiment, atoms emerge from a heated source, pass through collimating apertures, traverse a strongly inhomogeneous magnetic field, and strike a position-sensitive screen. The beam is neutral, so its separation is interpreted through the atomic magnetic moment rather than the ordinary charged-particle Lorentz force. Near a chosen quantization direction, the interaction givesA uniform field can exert a torque but would not separate the beam by this translational force; the gradient is essential. Different allowed magnetic-moment components become correlated with different trajectories. A classical ensemble with continuously varying orientations would yield a continuous spread, whereas discrete quantum components yield separated channels.

In the modern electronic description of ground-state silver, filled shells cancel their electronic angular momenta and the unpaired $5s$ electron has $L=0$, $S=1/2$, $J=1/2$. The effective electronic measurement therefore has two components, approximately ${\mu }_z=\pm {\mu }_B$, rather than an orbital $\ell =1/2$, which is not allowed. For positive field gradient the electronic $m_J=+1/2$ component has negative ${\mu }_z$ and deflects downward in this coordinate convention. Nuclear spin and hyperfine structure are neglected in this simplified two-channel account. Historically the original splitting preceded the electron-spin interpretation, so the modern explanation should not be confused with the original investigators' interpretation.

More generally a prepared spin state $a|+z⟩+b|-z⟩$ gives the two outcomes with probabilities $|a{|}^2,|b{|}^2$. Selecting one channel prepares a state for further measurements. An ideal second analyser along the same axis then gives a definite result, whereas one along a perpendicular axis gives two equally likely outcomes for a spin-one-half eigenstate. This connects the apparatus directly to incompatible spin components and the Born rule. Angular momentum is quantized through its operator algebra; the Stern–Gerlach gradient turns that discrete internal information into spatially distinguishable outcomes.

For fixed electron number $N$, the specified electron degeneracy pressure scales as $P\propto V^{-5/3}$. Differentiating at fixed low temperature gives the degeneracy-pressure bulk modulus,A monovalent sample has one conduction electron per atom. Its carrier density isusing the Avogadro constant $N_A=6.02214076\times 10^{23}\mathrm{mo}{\mathrm{l}}^{-1}$. Substitution with reduced Planck constant $\hslash =1.05457\times 10^{-34}\mathrm{J}\mathrm{s}$ and electron mass $m_e=9.10938\times 10^{-31}\mathrm{kg}$ givesEquivalently, $P=2n{E}_F/5$ and $K_T=2n{E}_F/3$ for this free electron model. This is the assumed electronic contribution, not a calculation of the complete equation of state of real potassium: ionic cohesion and interaction corrections have been excluded as instructed. The supplied pressure is the zero-temperature form, appropriate to leading order for $T\ll T_F$.

The atomic number density isOne fully ionised single-electron donor per million atoms givesThe usual simplified answer treats conductivity as proportional to electron concentration and assumes unchanged electron mobility, giving about $1.46\times 10^3$ times the undoped conductivity. There is a qualification: the conductivity ratio of doped and intrinsic semiconductors is not fixed by electron counts alone. Intrinsic material also has $p=n_i$, whereas the heavily n-type sample has $n\simeq N_D$ and $p\simeq n_i^2/N_D$. ThereforeWith unchanged electron mobility and negligible intrinsic hole conductivity this reduces to $1.46\times 10^3$; with equal, unchanged electron and hole mobilities it is about $7.29\times 10^2$. The problem supplies no mobilities, so the density-ratio estimate requires its stated approximation. Ionised impurities may additionally reduce mobility.

For a hydrogenic donor state, replace the bare electron mass by $m^{\ast }=0.1m_e$ and the Coulomb permittivity by ${ϵ}_r{ϵ}_0$ with ${ϵ}_r=16$. Rescaling the Bohr model gives the positive ground-state binding magnitudeThe radius isusing the Bohr radius $a_0=5.29177\times 10^{-11}\mathrm{m}$. The supplied positive hydrogen “energy” formula should be interpreted as a binding magnitude; the bound level's energy relative to the conduction-band edge is $-E_B$.

At $300\mathrm{K}$, $k_{B}T=25.9\mathrm{meV}$, about five times the binding energy. Thus a substantial fraction, ordinarily most of the shallow donors in the dilute nondegenerate regime, will be ionised. An exact fraction also depends on the Fermi level, donor concentration and degeneracies; comparing $E_B$ with $k_{B}T$ alone does not determine it exactly. The large radius explains why a weakly bound donor extends across many unit cells.

Semiconductor doping introduces impurities with levels close to the band edges. A donor impurity can release an electron to the conduction band, leaving a fixed positive ion; an n-type semiconductor has electrons as majority carriers. An acceptor impurity can take an electron from the valence band, leaving a mobile hole and fixed negative ion; a p-type semiconductor has holes as majority carriers. The opposite species remains as a minority carrier population, and a neutral bulk sample balances the ionised dopants with mobile carriers.

At a p-n junction, concentration gradients initially drive electrons and holes across the join. Their removal near the join exposes negative acceptor ions on the p side and positive donor ions on the n side, forming a depletion region. Its electric field opposes further majority-carrier diffusion. Equilibrium has a flat Fermi level, no net current and spatially bent band edges. With p on the left and n on the right, the electron band edges fall towards the n side; the difference is the equilibrium barrier in electron energy. A flat Fermi level does not mean flat conduction and valence edges.

A semiconductor has a filled valence band and an empty conduction band at zero temperature in the intrinsic idealisation, separated by a modest band gap. Thermal excitation or deliberately introduced impurities can create mobile electrons and electron holes. Its conductivity is consequently very sensitive to temperature and doping. A metal has available electronic states at the occupied energy boundary, through a partly filled band or overlapping bands, and need not thermally excite carriers across a gap.

For a direct band gap, the conduction-band minimum and valence-band maximum occur at the same crystal momentum. An electron can recombine with a hole near these extrema and emit a photon while conserving crystal momentum, since optical photon momentum is small on the lattice scale. For an indirect band gap, the extrema lie at different crystal momenta. A band-edge radiative transition then ordinarily needs a phonon as well, making it less probable and allowing nonradiative processes to compete more effectively.

gallium arsenide has a direct fundamental gap, while silicon has an indirect one. This makes efficient band-edge emission and hence a light-emitting diode easier to obtain from gallium arsenide. It does not mean silicon has no optical transitions or that every direct-gap sample emits efficiently: defects, surfaces and carrier confinement also matter.

A band effective mass describes the acceleration of an electronic wavepacket in a periodic lattice through the local band curvature. It incorporates the periodic lattice potential into a useful transport parameter without changing the bare electron mass. For a Bloch state in an isolated band,Differentiation givesIn one dimension this becomes $\boxed{m^{\ast }={\hslash }^2/(d^2ϵ/d{k}^2)}$. A locally parabolic isotropic band minimum has $ϵ\simeq {ϵ}_c+{\hslash }^2|\mathbf{k}-{\mathbf{k}}_c{|}^2/(2m^{\ast })$ with positive $m^{\ast }$. A flatter band has a larger effective mass. At a band maximum the electron curvature is negative, so the electron accelerates oppositely to the applied force. Describing the missing electron as a positive-charge electron hole with positive mass is often simpler.

In the scalar relaxation-time approximation, a conduction electron in an electric field reaches ${\mathbf{v}}_d=-e\tau \mathbf{E}/m^{\ast }$. ThusFor a crystal with different directional curvatures, conductivity is a tensor and involves the occupied states and their scattering. Contributions from electrons and holes add: $\sigma =e(n{\mu }_e+p{\mu }_h)$ in an isotropic two-carrier description. A filled band contributes no ordinary longitudinal current because opposite group velocities cancel; a partially filled band or holes permit a current response.

Effective mass therefore connects measured electrical transport to electronic band structure, but carrier density and momentum relaxation time matter as much as curvature. Ionised impurities and phonons alter $\tau$. A smaller mass does not guarantee a higher conductivity if carrier population or scattering changes. Nor are all reported masses identical: density-of-states, cyclotron and conductivity masses weight anisotropic bands differently. The simple semiclassical formula assumes an isolated band, slowly varying weak fields and negligible interband transitions; near degeneracies or strong fields a single constant scalar mass is inadequate.

The postulates of quantum mechanics give a framework for predicting observations from a prepared state. The experiments below motivate different features of that framework; no one experiment proves the whole set of postulates or specifies every Hamiltonian operator.

A pure quantum state is a normalized vector in a complex Hilbert space, with vectors differing by an overall quantum phase representing the same physical preparation. In position representation, $\int |\psi (\mathbf{r}){|}^2{d}^{3}r=1$. The linear state space permits quantum superposition, while the Born rule assigns probability ${\int }_{\Omega }|\psi {|}^2{d}^{3}r$ to a position outcome in a region $\Omega$. The wavefunction is a probability amplitude; it is not a classical distribution of definite particle positions and momenta.

An observable is represented by a self-adjoint operator. If $Q$ has spectral projectors $P_q$, its possible measurement outcomes are its spectral values and $\mathrm{Pr}(q)=⟨\psi |P_q|\psi ⟩$. The mean is $⟨Q⟩=⟨\psi |Q|\psi ⟩$. For an ideal repeatable projective measurement, obtaining $q$ leaves the normalized conditional state $P_q\psi /\sqrt{\mathrm{Pr}(q)}$. This projects onto an entire eigenspace if the outcome is degenerate. General detectors need more general measurement in quantum mechanics models; the projective update is an idealization. Noncommuting observables need not have simultaneous sharp values, as the canonical commutation relation and Heisenberg uncertainty relation illustrate.

Between measurements, an isolated system has unitary time evolution generated by its Hamiltonian operator:This preserves normalization and superpositions. Composite systems use a tensor product of state spaces, allowing entangled states as well as product states; identical particles additionally require the bosonic or fermionic exchange symmetry appropriate to their species. Statistical mixtures are described by a density matrix, with $\mathrm{Pr}(q)=\mathrm{tr}(\rho P_q)$. These rules separate preparation, coherent evolution and measurement.

The photoelectric effect exposes discrete energy transfer by light. Illuminate a clean metal cathode in an evacuated tube and measure the collected electron current while varying the light frequency, intensity and a retarding voltage. The stopping potential $V_s$ probes the largest emitted electron kinetic energy, whereas the photocurrent measures the rate at which charge is collected. In the ordinary single-photon regime there is a threshold frequency; above it the maximum energy grows with frequency, rather than with intensity. Changing intensity at fixed frequency mainly changes the number of emitted electrons. These observations are summarized by the Einstein photoelectric equationHere $\Phi$ is the surface work function, and the stopping voltage is understood after any photoelectric contact-potential correction. A photon delivers $h\nu$ in an absorption event; emission is allowed only when that energy can pay the escape cost. This accounts for the frequency threshold and the lack of the classical waiting time associated with slowly accumulating energy in a single electron. The experiment supports energy quantization of light; it does not imply that every incident photon emits an electron or that every emitted electron has the maximum energy. Surface losses and collection geometry affect the full current-voltage curve.

The Davisson–Germer experiment demonstrates a complementary wave property of matter. An electron gun uses a heated emitter, accelerating voltage and apertures to send a narrow beam onto a nickel single crystal in vacuum. A movable collector measures scattered electric current versus direction; rotating the crystal also tests its azimuthal symmetry. The crystal gives sharp angular beams whose positions depend on voltage and crystal orientation. A prominent feature occurs near $54\mathrm{V}$ accelerating voltage and roughly $50^{\circ }$ detector colatitude. The nonrelativistic de Broglie wavelength predicted at that voltage isDiffraction wavelengths inferred using the known crystal structure agree with this voltage dependence. These angular peaks arise from coherent addition of scattering amplitudes from the crystal, not from classical specular reflection of independent point projectiles. Bragg's law is $2d\sin {\theta }_B=r\lambda$ for lattice-plane spacing $d$ and angle to those planes; a detector colatitude is a different angle and must not be substituted without its geometry. The historical readout was current, not a photograph of individual electron trajectories.^[ref]

The double-slit experiment makes the role of amplitudes especially direct. A coherent source illuminates two narrow slits; an observation screen a distance $D$ away records the pattern. For slit spacing $d$ and small screen displacement $y$, the path difference is approximately $dy/D$. The two amplitudes add before probabilities are formed:For equal coherent illumination the interference term varies as $\cos (2\pi dy/(\lambda D))$, giving fringe spacing $\lambda D/d$ beneath the single-slit diffraction envelope. Blocking one slit removes that cross term. Young's optical experiment established wave interference; later particle-resolved realizations with photons or electrons show localized detection events whose accumulated distribution has the interference pattern. Low arrival rates demonstrate that this pattern does not require different incident particles to collide with one another.

If the paths become correlated with distinguishable detector states $|D_1⟩$ and $|D_2⟩$, the cross term acquires their overlap $⟨D_2|D_1⟩$. Perfectly distinguishable path records make that overlap zero, suppressing interference even if the spatial beams still overlap. This connects quantum entanglement, measurement and loss of observable coherence. Without a path measurement, the two alternatives are amplitudes for one preparation, rather than classical mutually exclusive events whose probabilities may simply be added.

Together these experiments motivate a description in which energy transfer and detector outcomes can be discrete, yet propagation retains complex quantum phases and interference. The Born rule connects those amplitudes to repeatable frequencies of outcomes. The Schrödinger equation supplies the dynamics once a Hamiltonian operator is specified, while observable operators, state preparation and the measurement model specify what the apparatus can read.

Quantum tunnelling is transmission through a finite classically forbidden region, where $E<V(x)$. The stationary Schrödinger equation permits an evanescent wavefunction inside such a barrier rather than forcing it to vanish at its entrance. Matching to propagating states beyond a barrier of finite width gives a nonzero transmitted probability current.

For a rectangular barrier of height $V_0$ and width $a$, with equal exterior potentials and $0<E<V_0$, write $\kappa =\sqrt{2m(V_0-E)}/\hslash$. The exact rectangular-barrier tunnelling probability isFor an opaque barrier its dominant dependence is $e^{-2\kappa a}$. For a slowly varying barrier, the WKB approximation generalizes this to $T\sim \exp [-(2/\hslash ){\int }_{x_1}^{x_2}\sqrt{2m(V(x)-E)}dx]$, apart from matching prefactors. Barrier width, particle mass and energy therefore have a strong influence. Energy is conserved in a static barrier: tunnelling does not require the particle to borrow energy or acquire a classical path with negative kinetic energy.

In alpha decay, a nuclear alpha-particle configuration is confined by the short-range nuclear interaction but separated from the exterior by a Coulomb barrier. Its energy can be positive relative to infinity while still lying below the barrier maximum. Escape by tunnelling produces alpha particles at that conserved energy, with an exponentially small decay probability per nuclear encounter. The pronounced dependence of half-life on alpha energy is a characteristic consequence of the barrier exponent, unlike a simple classical escape threshold.

In scanning tunneling microscopy, a conducting tip is brought within a nanometer-scale vacuum gap of a conducting surface and biased weakly. Electron tunnelling gives a current that depends approximately exponentially on separation, $I\propto V{e}^{-2\kappa s}$ in a simple low-bias model. Feedback adjusts the tip height to keep current fixed while scanning, producing a surface map sensitive to both geometry and electronic states. Finite current through the non-contact gap, and its steep dependence on separation, are the direct tunnelling signatures; current is not a direct measurement of topographic height alone.

A Josephson junction provides a collective superconducting example. Two superconductors separated by a thin barrier can carry a coherent pair current $I=I_c\sin \phi$ even at zero voltage. At constant voltage the AC Josephson effect gives $d\phi /dt=2eV/\hslash$ and oscillation frequency $2eV/h$. The coupled condensate quantum phases and pair transfer distinguish this from independent normal-electron transmission. These examples involve different microscopic barriers, but all require quantum coupling across a region that suppresses classical passage. Comparing weak-temperature tunnelling behavior with thermally activated escape helps distinguish the mechanisms experimentally.

Johnson-Nyquist noise is equilibrium electrical fluctuation caused by thermal agitation of charge carriers. A resistor can generate a fluctuating terminal voltage even when its mean current is zero. In the classical frequency range $hf\ll k_{B}T$, the fluctuation-dissipation theorem relates this noise to dissipative resistance. The one-sided voltage power spectral density of an unloaded resistor isEquivalently its short-circuit current spectrum is $4k_{B}T/R$. Over a flat bandwidth $B$ the rms voltage is $\sqrt{4k_{B}TRB}$. A real measurement instead integrates $S_V(f)|H(f){|}^2$ over positive frequency, with loading and the filter's effective noise bandwidth included. A flat spectrum does not imply an infinite physically measurable variance; the circuit and the quantum high-frequency regime limit the ideal white-noise model.

Cooling or narrowing the accepted bandwidth reduces thermal noise. Reducing resistance lowers its open-circuit voltage noise but raises its short-circuit current noise, so the appropriate choice depends on source impedance and amplifier loading. Thermal agitation is not a fluctuating externally applied voltage that can be removed by perfect shielding alone.

Shot noise arises from discrete transfer events. In the simplest shot noise counting model, equal charges $q$ arrive independently at constant rate $r$. A gate of duration $t$ contains a Poisson number $K$, with mean and variance $rt$. Therefore mean current is $I=qr$, and the averaged-current variance is $qI/t$. The corresponding one-sided low-frequency spectrum isfor a flat accepted bandwidth. Electron current has $q=e$. Vacuum emission or photodetection can approximate independent arrivals; correlations in a conductor can suppress or enhance the fluctuations. Discreteness by itself does not guarantee Poisson statistics.

For a fixed mean current the ideal shot-noise formula is not proportional to temperature, whereas Johnson noise is. Increasing current raises absolute shot noise but lowers its fractional size, since $\sqrt{2qIB}/I\propto I^{-1/2}$. Both can appear approximately white within a useful frequency band, and independent noise powers add after referring them to a common measurement point. Their spectra must use the same convention: two-sided densities are half these one-sided values. A real low-current experiment can also contain amplifier noise, low-frequency drift and correlated transfer noise, so a mean current or resistance alone does not specify its complete noise budget.

Under the one-carrier Drude model, the magnitude of the Hall coefficient gives an effective carrier densityConvert the material mass density to $1850\mathrm{kg}{\mathrm{m}}^{-3}$ and its molar mass to $0.009\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$. The atomic number density isHere $N_A=6.02214076\times 10^{23}\mathrm{mo}{\mathrm{l}}^{-1}$ is the Avogadro constant. The positive Hall coefficient gives a hole-like response in a one-carrier interpretation. The Hall carrier count versus valence distinction matters: beryllium has two valence electrons, but the result is neither two free electrons nor a literal fractional atom donating charge. Its energy bands and Fermi surface invalidate the simple isotropic one-carrier inference. Electron-like and hole-like contributions with different charge carrier mobilities can yield this sign and effective count.