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.

A systematic error is a reproducible bias in the measurement model or procedure, such as a zero offset, incorrect gain, background contamination, misalignment, loading or temperature-dependent calibration. Repeating the same biased procedure more often reduces independent random scatter but does not remove the bias. An unknown correction also has a measurement uncertainty, even when the mechanism is systematic.

A simple instrument model is $y=(1+g)x+a+d(t)+ϵ$, with gain error $g$, offset $a$, drift $d(t)$ and random noise $ϵ$. Comparison with calibrated standards at several input values separates an offset from a scale error and can test nonlinearity. Use standards that cover the operating range, verify units and reference values, and propagate uncertainty in the standards into the corrected result. Calibration changes the model; it does not make the corrected measurement exact.

A blank or source-off measurement estimates additive background. For independent source-plus-background count $K_s$ in time $t_s$ and background count $K_b$ in time $t_b$, the net-rate estimate and its counting variance areSubtracting a measured background removes its estimated mean, not its fluctuations or its uncertainty. A long background exposure can reduce the latter, while drifting backgrounds require monitoring or a joint model.

Reversal can distinguish a wanted signal from an unchanged offset. For example, $V(+I)=IR+V_{\mathrm{th}}$ and $V(-I)=-IR+V_{\mathrm{th}}$ give $R=[V(+I)-V(-I)]/(2I)$ when the thermoelectric offset is stable between readings. Interchanging sample and reference, reversing field direction or reversing an apparatus orientation can similarly expose sign-dependent biases, but only if the wanted and unwanted contributions transform differently.

Track slow instrumental drift using frequent references, interleaved readings or an ABBA measurement sequence. Equal-time readings in the order A, B, B, A cancel a common linear drift in the difference estimator $(y_1+y_2-y_0-y_3)/2$. Nonlinear drift or unequal sample conditions can survive. Randomizing run order helps prevent a time drift from imitating dependence on a controlled experimental variable.

Control physical couplings directly: stabilize temperature, minimize gradients, shield unwanted fields, align geometry, and use high input impedance when loading is the concern. Differential and bridge measurements suppress common changes when their channels are well matched. Varying an apparatus parameter or using an independent method can reveal an unsuspected bias. Report residual corrections, their uncertainties and correlations; adding more identical readings cannot justify discarding a shared calibration uncertainty.

Bayes' theorem updates prior knowledge using the likelihood of new observations. For alternative hypotheses $H_j$,The prior $P(H_j)$ describes information before the data $D$; $P(D\mid H_j)$ is the sampling likelihood, and the posterior $P(H_j\mid D)$ describes the updated probabilities. Posterior odds equal prior odds times the likelihood ratio. A likelihood is a probability model for data at specified parameters, not automatically a normalized probability distribution over those parameters.

For a continuous parameter $\theta$, the same rule uses densities: $\pi (\theta \mid D)=L(D\mid \theta )\pi (\theta )/\int L(D\mid u)\pi (u)du$. Priors can encode positivity, physical limits, previous independent measurements or calibration knowledge. Their parameterization matters: a density uniform in a rate is not uniform in its logarithm. Proper priors make model evidences well defined; an arbitrary normalization in an improper prior is especially problematic for comparing models.

A counting example uses Poisson-gamma conjugacy. If a detector registers $K$ events in exposure $t$ with constant rate $\lambda$, $L\propto {\lambda }^K{e}^{-t\lambda }$. A gamma prior with shape $\alpha$ and rate parameter $\beta$, $\pi (\lambda )\propto {\lambda }^{\alpha -1}{e}^{-\beta \lambda }$, givesThe posterior mean combines the prior rate $\alpha /\beta$ with the observed rate $K/t$, weighted by the prior and actual exposures. The parameters must use a consistent shape-rate convention. Small counts can leave substantial prior sensitivity; abundant informative data usually reduce it for regular identifiable models.

Independent old data can become the prior for a new experiment, giving the same result as joint updating under a consistent model. The new data must not also have been used to construct that prior, or the evidence is counted twice. A credible interval has a specified posterior probability under the chosen model and prior; this interpretation differs from repeated-sampling coverage of a confidence interval. Predictions for a future observation average its sampling distribution over the posterior, incorporating parameter uncertainty. Prior sensitivity checks and a defensible likelihood are therefore part of the inference, rather than Bayes' theorem itself supplying missing physical knowledge.

Introduce an additive background rate $B$, from ambient radiation, detector dark counts or another weakly attenuated contribution. If $S_0$ denotes the unattenuated signal alone, the attenuation count model with an additive background isIt approaches a nonzero constant as the transmitted signal becomes small. The source-off background should ideally be measured in a separate long exposure and its uncertainty included in the fit; otherwise estimate $B\ge 0$ jointly using a Poisson count likelihood.

There is a normalization detail: the given $N_0$ was itself obtained from an $n=0$ count measurement. If that measurement included the same background, then it measures $S_0+B$, and the attenuation fit normalized to a total reference rate isWriting $N_0{e}^{-bn}+B$ instead treats $N_0$ as a background-subtracted signal calibration. The difference is small here because the background is a few counts per second while $N_0$ is about two thousand, but the distinction prevents double-counting background. The last three observations alone do not give an exact value of $B$, because residual transmitted signal and sizeable counting fluctuations remain.

For a candidate attenuation coefficient, the one-second expected count is ${\mu }_n=2018.5e^{-bn}$. The model predictions areUse first the conservative signal-dominated range $n=1,\dots ,4$, where the count approximation is reliable and background is a small correction. Using the estimated errors from part (c), the weighted residual statistic isIt gives $\boxed{Q_{\mathrm{obs}}(1)=4.45,Q_{\mathrm{obs}}(1.1)=12.74}$. Since both $b$ candidates are fixed and $N_0$ was calibrated independently, neither was fitted to these four counts: the approximate reference has $\nu =4$, not three. The upper-tail probabilities are approximately $0.349$ and $0.0126$, respectively. Thus $b=1$ is compatible with the scatter, whereas $b=1.1$ is disfavored at the usual $5\%$ level in this approximate test.

A Pearson chi-squared statistic, using model variances instead of estimated observed variances, gives the same preference:Including the fifth point also leaves the decision unchanged: the estimated-error statistics are $8.37$ and $12.74$ on five degrees of freedom, with approximate upper-tail probabilities $0.137$ and $0.0259$; the Pearson statistics are $6.34$ and $15.63$. HenceThe small uncertainty in $N_0$ contributes correlated prediction uncertainty, as in chi-squared comparison of a calibrated attenuation model, but is negligible compared with the one-second counting errors here. Choosing the test range after inspecting the observations limits a strict interpretation of its nominal significance; the numerical comparison and its stated range are the relevant evidence.

If all eight observations are inserted into the estimated-error formula, the totals are $14.99$ and $22.48$. Their nominal eight-degree-of-freedom calibration is unreliable because the final expected counts are much smaller than one for some predictions. In fact the full-range Pearson statistics are $43.34$ and $118.25$, exposing the late excess much more strongly. The full Poisson deviance values are $24.09$ and $47.03$, again favoring $b=1$ but signaling that a background-free exponential is inadequate over the complete range. These sparse-count comparisons should not be used to claim that either unmodified model explains all eight measurements.

Assume independent radioactive detections with negligible dead-time loss, so accumulated counts have a Poisson distribution. The reference measurement has total count $K_0=N_0{t}_0=2018500$ over $t_0=1000\mathrm{s}$. Since a Poisson count has variance equal to its mean, the rate uncertainty isIts relative uncertainty is $7.04\times 10^{-4}$, or $0.0704\%$. For each one-second measurement, ${\sigma }_y\simeq \sqrt{y}$, giving the estimated count and rate uncertaintiesin counts, numerically the same in counts per second for these one-second exposures. Their fractional uncertainties range from about $3.7\%$ at the first sheet to $35$–$45\%$ for the smallest counts. The long, high-rate reference is therefore much more precise. These are statistical errors; a longer exposure does not automatically remove calibration errors, source drift or background bias.

An exponential count rate appears as a straight line on a semilogarithmic plot:The early points decrease by roughly a factor $e$ per added sheet. Keeping the supplied $N_0$ fixed, a simple unweighted straight-line estimate from the first four logarithms isso the early trend is consistent with $b$ near one. This logarithmic estimate is illustrative; the final statistical comparison should use the count uncertainties rather than treating the logarithms as equally precise.

The pure exponential gives a useful approximate description through about the first four or five sheets. The first four observations are the clean high-count range; the fifth remains compatible with $b=1$ within its counting scatter. At $n=6,7,8$ the measurements are roughly constant at a few counts per second, so a single straight line no longer describes the whole pattern. Agreement of the sixth point with one exponential prediction by itself does not remove the systematic late flattening. There is no sharply determined physical cutoff from eight noisy measurements; the range used for a formal test should be stated explicitly.

A chi-squared test compares residuals with the random scatter predicted by a model. For independent measurements $y_i$ with known Gaussian standard deviations ${\sigma }_i$ and predictions ${\mu }_i(\theta )$, formIf the mean model and error assumptions are correct, a full-rank linear fit of $p$ parameters gives a chi-squared distribution with $\nu =N-p$ statistical degrees of freedom. With a fully specified prediction and no fitted parameters, $\nu =N$. Nonlinear fits usually make this an approximation. Independent external calibration is not the same as fitting a parameter to these test data.

Under the reference distribution, the mean statistic is $\nu$ and its variance is $2\nu$. The upper-tail p-value $\mathrm{Pr}({\chi }_{\nu }^2\ge Q_{\mathrm{obs}})$ quantifies how unusual a discrepancy at least this large would be under the assumed model. A small value suggests model failure or incorrect uncertainties; it is not the probability that the model is true. An unusually small statistic can also reveal inflated uncertainties or correlations. A rejection threshold should be specified in advance.

For independent Poisson distribution counts, the model variance is its expected count ${\mu }_i$. The Pearson chi-squared statistic is $\sum (y_i-{\mu }_i{)}^2/{\mu }_i$. The estimated-error alternative ${\sigma }_i\simeq \sqrt{y_i}$ is common for large counts but can differ substantially when counts are small. In that regime the Poisson deviance or a direct likelihood calculation is preferable, with simulation if an accurately calibrated goodness-of-fit probability is required.

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.

A diatomic molecule has one internal bond-stretching vibrational coordinate. Near its equilibrium separation, approximate the bond by a spring of stiffness $K$ and use the reduced mass $\mu =m_1{m}_2/(m_1+m_2)$. Its frequency is $\omega =\sqrt{K/\mu }$ and its quantum harmonic oscillator levels are ${ϵ}_r=(r+1/2)\hslash \omega$.

For independent molecules in thermal equilibrium, set $x=\hslash \omega /(k_{B}T)={\theta }_v/T$. The single-molecule canonical partition function is a geometric series:Using $U=-\partial \log (z_v^N)/\partial \beta$, where $\beta =1/(k_{B}T)$, givesPer mole replace $N{k}_B$ by $R$. The zero-point energy is present in $U_v$ but is temperature-independent and contributes nothing to the heat capacity.

For $T\ll {\theta }_v$, occupation of the first excited vibrational state is exponentially suppressed: $C_{V,v}\simeq N{k}_B{x}^2{e}^{-x}\to 0$. The vibrational degree of freedom is frozen out. For $T\gg {\theta }_v$, $C_{V,v}\to N{k}_B$. This recovers classical equipartition: the quadratic kinetic term and quadratic elastic term each contribute $k_B/2$ per molecule.

The vibrational heat capacity of a diatomic gas must be distinguished from the total heat capacity. In a dilute gas with rotations already active and vibration still frozen, translation contributes $3R/2$ and rotation $R$, so molar $C_V\simeq 5R/2$. Once vibration is active it adds $R$, giving $C_V\simeq 7R/2$. For an ideal gas of a fixed number of intact molecules, $C_P=C_V+R$, so the corresponding values are $7R/2$ and $9R/2$. Very low temperatures also freeze rotations, while sufficiently high temperatures bring anharmonicity, dissociation or electronic excitations; the harmonic plateau is restricted to the range where those effects are negligible.

Indistinguishable particles have the same intrinsic properties and cannot be assigned physically observable permanent labels. Exchanging their mathematical coordinates leaves all physical predictions unchanged. In ordinary three-dimensional quantum mechanics the total state, including spin, obeysThe Spin-statistics theorem associates integer spin with bosons and half-integer spin with fermions. Exchange does not mean physically transporting two particles along some prescribed trajectories; it is the symmetry of the many-particle state under permutation of its arguments.

For orthonormal one-particle states $u,v$, normalized two-particle spatial combinations are $[u(1)v(2)\pm v(1)u(2)]/\sqrt{2}$. They illustrate coherent exchange amplitudes. If both fermions have the same complete one-particle state, the antisymmetric combination vanishes. This is the Pauli exclusion principle: a fermionic mode has occupation zero or one. Bosonic modes can have arbitrarily large occupation. A many-fermion product basis is antisymmetrized by a Slater determinant.

It is the full spin-spatial state that must have the appropriate symmetry. For two electrons, an antisymmetric spin singlet state can accompany a symmetric spatial orbital, so opposite-spin electrons can occupy the same spatial orbital. A symmetric spin triplet instead needs an antisymmetric spatial state. Thus exclusion concerns identical complete states, not an unconditional prohibition on sharing a position or spatial orbital.

Counting allowed occupations gives Bose-Einstein statistics and Fermi-Dirac statistics. For an ideal gas in thermal equilibrium,Bosonic accumulation permits Bose-Einstein condensation; exclusion gives a filled Fermi sea and degeneracy pressure even at zero temperature. In the dilute classical limit the occupations are small and both expressions approach the Maxwell-Boltzmann distribution. Exchange effects become important when wavepackets overlap appreciably, often characterized by number density times the cube of the thermal wavelength.

Composite particles follow their total exchange statistics while remaining intact. The composite statistics of neutral atoms depend on electrons as well as nucleons, as in the two lithium isotopes above. Symmetrization can produce bunching or antibunching in joint detection probabilities even in the absence of an interaction potential; it is not an additional classical force.

The Hamiltonian operator isSince multiplication by $x$ commutes with multiplication by $V(x)$, part (b) gives $[x,H]=i\hslash p_x/m$. Similarly $[p_x,H]=-i\hslash V^{\prime }(x)$. Substituting into part (c) gives Ehrenfest theorem:Thus the mean momentum changes according to the mean force, and the derivative of mean position is mean velocity. These exact quantum equations resemble Hamilton's classical equations, but $⟨V^{\prime }(x)⟩$ generally differs from $V^{\prime }(⟨x⟩)$. The classical point-particle equation for the center is exact for potentials at most quadratic and is an approximation for a localized packet when the force varies little across it. A spatially broad packet in a nonlinear potential need not follow a classical trajectory. Even where its center follows classical motion, its width and measurement uncertainties remain quantum properties.

For a normalized state, the expectation value isFor a time-independent operator, differentiation givesThe Schrödinger equation says $|\dot{\psi }⟩=-iH|\psi ⟩/\hslash$. The self-adjointness of $H$ gives $⟨\dot{\psi }|=i⟨\psi |H/\hslash$, soIn the integral derivation, the same self-adjointness step is integration by parts with vanishing boundary terms. An explicitly time-dependent observable would also contribute $⟨\partial Q/\partial t⟩$. These are the expectation-value equations underlying the Heisenberg equation of motion.

Expand the proposed product rule without interchanging any operator factors:Applying this commutator identity with $A=x$ and $B=C=p_x$ givesFor a differentiable multiplication potential, apply the other commutator to a test function:ThereforeThese operator equalities are understood on a common domain where the indicated products and derivatives are defined, with the boundary conditions appropriate to the state.

The commutator is $\boxed{[A,B]=AB-BA}$. To evaluate it for differential operators, apply it to a smooth test wavefunction $f$:Thus the canonical commutation relation is $\boxed{[x,p_x]=i\hslash I}$. The associated Heisenberg uncertainty relation is $\Delta x\Delta p_x\ge \hslash /2$. Position and momentum cannot both have arbitrarily sharp outcome distributions in the same prepared state, even with ideal measuring apparatus. This concerns quantum-state preparation, not merely a calibration error in a particular instrument.

Taking the modulus squared gives the three-mode revival of a planar quantum rotor:The angular density has period $\pi$, while both the state and its density revive after $\boxed{T=2\pi /\Omega =\pi I/\hslash }$. Its integral over $0\le \varphi <2\pi$ is one because $\int \cos 2\varphi d\varphi =0$ and $\int {\cos }^{2}2\varphi d\varphi =\pi$.

At $t=0$, the density is $4{\cos }^4\varphi /(3\pi )$, with maxima at $0,\pi$ and zeros at $\pi /2,3\pi /2$. At $t=T/2$ it becomes $4{\sin }^4\varphi /(3\pi )$, interchanging those maxima and zeros. At $t=T/4$ and $3T/4$, it is $[1+{\cos }^{2}2\varphi ]/(3\pi )$, with four equal maxima along the coordinate axes and positive minima halfway between them. At $t=T$ the original profile returns.

The changing profile is interference between the stationary $m=0$ mode and the degenerate counter-rotating $m=\pm 2$ modes. It does not translate rigidly around the circle; the standing-wave weights alternate between two orientations.

Each energy eigenstate evolves by its own quantum phase $e^{-i{E}_{m}t/\hslash }$. The two $m=\pm 2$ components have the same energy, while the $m=0$ quantum phase stays one. ThusIn angle representation this isAt $t=0$, $1+\cos 2\varphi =2{\cos }^2\varphi$ recovers the initial state. The fixed coefficient magnitudes keep the angular momentum and energy probabilities constant even though the density in angle changes.

Expand the initial wavefunction in the angular momentum eigenbasis using ${\cos }^2\varphi =1/2+(e^{2i\varphi }+e^{-2i\varphi })/4$. With the normalized $u_m$ from part (a), this givesThe coefficient squares sum to one, also verifying the stated normalization. The Born rule therefore gives angular momentum outcomesFor an energy measurement the two distinct $m=\pm 2$ states belong to the same energy eigenspace. Sum their probabilities, rather than adding their amplitudes, to obtainTheir relative probabilities are $4:1:1$ for the angular momenta and $2:1$ for the energies. In particular, zero mean angular momentum does not imply zero rotational energy.

The quantum rotor has orientation angle $\varphi$ of period $2\pi$, so physical wavefunctions satisfy $\psi (\varphi +2\pi )=\psi (\varphi )$. The exponential solutions of $Hu=Eu$ are thereforeThey are normalized and orthogonal on $[0,2\pi )$. The corresponding angular momentum operator is $L_z=-i\hslash {\partial }_{\varphi }$, with eigenvalue $m\hslash$. The $m=0$ ground level is nondegenerate, and every $|m|>0$ energy level has the twofold degeneracy $m\leftrightarrow -m$. This uses ordinary periodic boundary conditions, with no flux or twisted quantum phase.

For removal of a half-oscillator wall, the sudden approximation keeps the state unchanged during the change of potential. Its probability in the new full-oscillator ground state ${\Psi }_0={\pi }^{-1/4}{\ell }^{-1/2}{e}^{-x^2/(2{\ell }^2)}$ is the squared overlap:where ${\int }_0^{\infty }x{e}^{-x^2/{\ell }^2}dx={\ell }^2/2$. ThereforeThe integral runs only over the initially occupied half-line. The initial state is not the odd full-line eigenstate: it is zero, rather than negative, on the left. Indeed its overlap with that final first excited state is $1/\sqrt{2}$, giving probability $1/2$, and the remaining probability lies in higher states. Subsequent evolution under the new fixed Hamiltonian operator changes the quantum phases of its energy components without changing these energy-measurement probabilities.