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The magnetization $\mathbf{M}$ is magnetic dipole moment per unit volume. In magnetostatics it producesThe magnetic field intensity $\mathbf{H}=\mathbf{B}/{\mu }_0-\mathbf{M}$ obeysso $\mathbf{H}$ isolates free currents while $\mathbf{M}$ accounts for bound currents. For a linear isotropic material, $\mathbf{B}=\mu \mathbf{H}$.

Ampere's law for a circular loop around the free line current givesin both media. ThereforeAt $r=R$, the net bound surface-current density isHence its magnitude isand for a positive current along $+{\mathbf{e}}_z$ it points along $-{\mathbf{e}}_z$ when ${\mu }_1>{\mu }_2$, reversing when ${\mu }_1<{\mu }_2$.

The electric polarization $\mathbf{P}$ is electric dipole moment per unit volume. It producesThe electric displacement field $\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}$ is useful becauseso it separates free charge from the bound charge represented by $\mathbf{P}$. In a linear isotropic dielectric, $\mathbf{D}=ϵ\mathbf{E}$.

Spherical symmetry and Gauss's law for $\mathbf{D}$ give, on both sides of the interface,ThusWith $\mathbf{n}={\mathbf{e}}_r$ pointing from medium one to medium two, the net bound sheet charge at $r=R$ is the polarization jumpIf one assigns surface charge separately to the inner body and outer medium, their contributions are ${\mathbf{P}}_1\cdot \mathbf{n}$ and $-{\mathbf{P}}_2\cdot \mathbf{n}$; their sum is the displayed physical interface charge.

A dielectric is an electrically insulating material whose bound positive and negative charges can be displaced slightly by an electric field, producing electric polarization without sustained conduction current. Its macroscopic electric response is described by its permittivity.

At coexistence, $G_{\mathrm{I}}(T_0,p,N)=G_{\mathrm{II}}(T_0,p,N)$. Stability saysThus $G_{\mathrm{I}}-G_{\mathrm{II}}$ crosses zero with negative slope as $T$ increases through $T_0$. On the two smooth phase branches,ThereforeThe transition from phase II to phase I consequently absorbs positive latent heat $T_0(S_{\mathrm{I}}-S_{\mathrm{II}})$.

A first-order phase transition occurs where two phases have equal Gibbs free energy but a first derivative of that free energy is discontinuous. Typical jumps occur inA nonzero entropy jump gives a nonzero latent heat $T\Delta S$.

The fundamental thermodynamic relation isBecause $E(S,V,N)$ is extensive, the Euler theorem for homogeneous functions givesSubstitution into the Gibbs free energy $G=E+pV-TS$ therefore yieldsFor a one-component equilibrium phase, the intensive chemical potential depends on the intensive state variables $T$ and $p$.

The Carnot theorem states that no engine between two fixed heat reservoirs can be more efficient than a reversible engine, and that all reversible engines between those reservoirs have the same efficiency. Thus the heat ratio of a reversible engine depends only on the two reservoirs. Its multiplicative consistency for three reservoirs permits a state variable $T$, unique up to an overall scale, such thatFixing one reference value defines the thermodynamic temperature scale, and the reversible efficiency is

The radial factoris sharply peaked, for large $n$, nearRequiring the orbit to lie inside a disk of radius $R$ permitsThus the total degeneracy at fixed $k_z$ is, up to the negligible boundary correction,Since the disk has area $\pi R^2$, the degeneracy of a Landau level per unit area isFor either sign of the charge-field product, replace $eB$ by $|eB|$.

The factor $e^{i{k}_{z}z}$ is an eigenstate of longitudinal momentum with $p_z=\hslash k_z$. The integer $n$ is the azimuthal angular momentum quantum number, since $L_z=-i\hslash {\partial }_{\theta }$ has eigenvalue $n\hslash$.

Fordirect differentiation givesRegularity and cancellation of the $r^{-2}$ term require $\alpha =n$ for $n\ge 0$. Cancellation of the $r^2$ terms requiresassuming $eB>0$. The remaining radial energy is independent of $n$:This is the lowest Landau level. If $eB<0$, the same family uses nonpositive angular-momentum quantum numbers and $|eB|$ replaces $eB$.

With the magnetic minimal coupling convention for charge $e$,Here $\nabla \cdot \mathbf{A}=0$, ${\mathbf{A}}^2=B^2(x^2+y^2)/4$, andTherefore in Cartesian coordinatesIn cylindrical coordinates this isThis choice of $\mathbf{A}$ is the symmetric gauge.

The required dipole matrix element iswhere differentiating the elementary Laplace integral for $\sin (kx)$ evaluates the last integral. Hence

Put $E_f=E_0+\hslash \omega >0$ and $k_f=\sqrt{2m{E}_f}/\hslash$. The continuum transition probability at long times usesSincethe total escape probability becomesFor the attractive delta potential,Substitution gives the requested leading-order result:

To first order in $F$, time-dependent perturbation theory replaces the state on the right-hand side by its unperturbed value $e^{-i{E}_{0}t/\hslash }|0⟩$. The bound state is even, while the position operator $x$ is odd. The parity selection rule therefore givesso ${\dot{b}}_k=0$ and, with no initial continuum component,

Projection onto the odd state givesWriting ${\Omega }_k=(E_k-E_0-\hslash \omega )/\hslash$ and integrating from zero to $t$ yields

Insert the stated expansion into the Time-dependent Schrödinger equationWhen the derivative acts on each free phase, its energy factor is cancelled by the action of $H_0$ on the corresponding eigenstate. The remaining terms obeyDivision by $i\hslash$ gives

There are two bifurcations in $0\le k\le 1/2$, both at $k=0$.

Near $(0,0,0)$, append $\dot{k}=0$. The plane $y=0$ is invariant and tangent to the extended center subspace, so it is an extended center manifold. The reduced equation is exactlyIts leading terms $-kx-3x^2$ have two branches, $x=0$ and $x=-k/3+O(k^2)$, which cross and exchange their center-direction stability. Hence this is a transcritical bifurcation.

For the bifurcation at $(1,2,0)$, setThe extended system becomesSolving the centre-manifold invariance equation for $v=h(X,k)$ givesSubstitution into the $X$ equation yieldsFor $k>0$ this has two nearby equilibria $X=\pm \sqrt{k}+O(k)$, while for $k<0$ it has none. It is therefore a saddle-node bifurcation. These reductions are collected in bifurcations of the 2022 Cambridge quadratic-cubic system.