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If a weak interaction changes $|0,0⟩$ to a normalized state proportional to $|0,0⟩-g|1,1⟩$, tracing out either oscillator gives eigenvalues $1/(1+g^2)$ and $g^2/(1+g^2)$.

The purity of a density operator is $\gamma =\mathrm{Tr}({\rho }^2)$. It equals one exactly for a pure state; in finite dimension $d$ it lies between $1/d$ and $1$. The minimum occurs exactly at the maximally mixed state $I/d$.

A worked qubit family interpolates between the mixed state $I/2$ and the pure state ${\rho }_{+}=|+⟩⟨+|$, where $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$:The eigenvectors are $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$, with eigenvalues ${\lambda }_{\pm }=(1\pm p)/2$. Squaring and summing these eigenvalues givesHere $P_X(+)$ is the Born rule probability of the positive outcome in an $X$ projective measurement. The Bloch vector is $(p,0,0)$, since ${\rho }_p=(I+p{\sigma }_x)/2$ with ${\sigma }_x$ one of the Pauli matrices. At $p=0$ the purity is $1/2$; at $p=1/2$ it is $5/8$, with $P_X(+)=3/4$; at $p=1$ it is one. Within this family, $p=1$ is a pure state and every $p<1$ is a mixed state. Extending the formula to $p>1$ produces a negative eigenvalue $(1-p)/2$, so it ceases to be a density operator despite remaining Hermitian with trace one.

The Bloch vector length measures polarization: for any qubit density operator, $\gamma =(1+|\mathbf{r}{|}^2)/2$. This connects the interior of the Bloch ball with mixed states and its boundary, the Bloch sphere, with pure states. ^[ref]

For a short proof of the general bound, use the spectral theorem: the nonnegative eigenvalues ${\lambda }_j$ sum to one, so $\gamma ={\sum }_j{\lambda }_j^2$. ThenThe lower bound is attained only when all eigenvalues equal $1/d$; the upper bound is attained only when exactly one eigenvalue is nonzero, and hence equals one. Purity is therefore a simple quantitative test of mixedness, invariant under a unitary operator. It does not specify the entire density operator: even distinct pure states all have purity one. ^[ref]

The Von Neumann entropy of a density operator isIt vanishes for a pure state and measures mixedness and quantum entanglement in reduced states.

For $V(x)=V_0\delta (x)$, $E={\hslash }^2{k}^2/(2m)$, and $\gamma =m{V}_0/({\hslash }^{2}k)$, continuity and the derivative jump giveThey obey $|r{|}^2+|t{|}^2=1$ and $r^{\ast }t+t^{\ast }r=0$.

For delta interactions at points $x_j$, a Green-function solution reduces the Schrodinger equation to a finite linear system for the values $\psi (x_j)$.

An attractive one-dimensional delta potential has one even exponentially decaying bound state.

When a perturbation preserves each unperturbed degenerate block and has no inter-block coupling, diagonalizing each block can give the exact spectrum.

If the perturbation is already diagonal in a basis of a degenerate eigenspace, those basis vectors receive its diagonal entries as first-order shifts.

For the normalizationa dark soliton of speed $0\le U<1/\sqrt{2}$ is

For $\psi =e^{-iEt}f(x)$ with real $f$, the defocusing equation $i{\psi }_t+{\psi }_{xx}-2|\psi {|}^2\psi =0$ has first integralA nonzero smooth function tending to zero at both infinities cannot satisfy this identity: a nonzero extremum requires $f^2=E>0$, but then the right side is negative for sufficiently small nonzero $f$.

A forced harmonic oscillator responds at both its natural and forcing frequencies.

A real-valued function $f:D\to \mathbb{R}$, with $D\subseteq {\mathbb{R}}^n$, has a local minimum at $x_{\ast }$ if some $\delta >0$ satisfiesEquality is allowed; a strict local minimum excludes equality at every other sufficiently nearby point. The comparison uses a neighbourhood relative to the domain of a function. On an open set, this is an ordinary small ball around $x_{\ast }$. At an endpoint, only points belonging to the domain of a function are tested. Some elementary texts reserve "local" for interior points; here endpoints are allowed under the relative convention, but the zero-gradient condition still requires an interior point.

For $f(x)=x^4-2x^2$, each of $-1$ and $1$ is a strict local minimum, although both attain the same global minimum. For $h(x)=|x|$, zero is a strict local minimum despite the missing derivative. Thus the definition compares values and does not require differentiability.

A local minimum describes a valley that cannot be improved by a sufficiently small move; mathematical optimization also asks whether a better value lies farther away. At a smooth interior local minimum, the gradient is zero. If the function is twice continuously differentiable nearby, its Hessian matrix is positive semidefinite: restrict to a line through the point and its second derivative is nonnegative. The converse fails for a degenerate Hessian matrix, as $-x^4$ at zero shows.

The critical set of a differentiable map consists of the points where its derivative fails to have full rank.

Each broken continuous internal-symmetry generator normally produces a massless Goldstone mode; for a symmetry breaking pattern $G\to H$, their number is $\dim G-\dim H$ in the standard relativistic setting.

Homogeneity at a fixed point relates thermodynamic critical exponents to the correlation-length exponent $\nu$ and anomalous dimension $\eta$.

A renormalization group compares models at different observation lengths. A Wilsonian coarse-grained statistical Hamiltonian sums over short-distance fluctuations; a renormalization-group transformation rescales coordinates and fields for comparison. Kenneth Geddes Wilson's 1982 lecture treats fluctuations across successive length scales to explain critical phenomena.^[ref]

The experimental motivation is Andrews's carbon dioxide experiment. Dried carbon dioxide occupied a capillary tube above mercury; a screw compressed the transmitting water. A second air-filled tube gauged pressure. Flowing water controlled temperature, and a cathetometer read column positions for volume. The liquid meniscus vanished near the liquid-gas critical point. Residual air and imperfect air-gauge corrections limited accuracy.^[ref] These observations motivate critical-point theory without determining a correlation-length critical exponent. The renormalization group explains how critical measurements can become insensitive to many molecular details.

In 1972, Kenneth Geddes Wilson and Michael Ellis Fisher used an epsilon expansion about $d=4$ to compute nonclassical critical exponents. Their paper distinguishes a Gaussian fixed point from an interacting renormalization-group fixed point, tunes the thermal parameter, and extracts powers from a linearized recursion. Its Ising-like result is $\gamma =1+ϵ/6+O({ϵ}^2)$ for $d=4-ϵ$.^[ref] The worked model below isolates this mechanism with chosen coefficients; it is not a precision calculation for the three-dimensional Ising model.

The upper critical dimension is the dimension above which the Gaussian or mean-field fixed point controls critical exponents. It is four for the ordinary Ising critical point and three for its tricritical point.

Landau theory expands an effective free energy analytically in powers of an order parameter, constrained by symmetry.

An order parameter vanishes in a symmetric phase and becomes nonzero when that symmetry is spontaneously broken.

The density parameter is $\Omega =\rho /{\rho }_{\mathrm{crit}}$. In a Friedmann universe, its departure from one measures the spatial-curvature term relative to the density or expansion term.

The Friedmann equation relates this ratio to spatial curvature and the expansion rate; the parameter and the evolution equations are distinct concepts.