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A periodic canonical coordinate paired with an action variable. In an integrable background Hamiltonian $H_0(\mathbf{J})$, the angles advance uniformly at ${\dot{\theta }}_i=\partial H_0/\partial J_i$, while the action variables remain constant. The periodicity permits an integer-indexed Fourier series on each regular orbital torus.

For a positive diagonal quadratic potential $V=m{\sum }_i{\omega }_i^2{x}_i^2/2$, coordinate Hamiltonians commute and wavefunctions separate into one-dimensional oscillator factors. Equality of two squared frequencies gives rotation symmetry in their plane; equality of all three gives full rotation symmetry. Equalities create energy degeneracies and extra conserved angular momentum. Merely finding an accidental equality between particular level energies does not establish a global rotational symmetry.

A dressed state of an atom and its driving field. Within the rotating-wave approximation, the paired configurations $|e,n⟩$ and $|g,n+1⟩$ mix. A classical-drive rotating-frame Hamiltonian $H=\hslash (-\Delta {\sigma }_z+\Omega {\sigma }_x)/2$ has eigenvalues $\pm \hslash \sqrt{{\Delta }^2+{\Omega }^2}/2$. At resonance its eigenvectors are equal-weight combinations of the two bare configurations. Radiative transitions between neighboring manifolds explain the Mollow triplet.

A level of an atomic term with specified total angular momentum $J$. Without an external field, its $2J+1$ magnetic states remain degenerate in a rotationally invariant Hamiltonian.

Band hybridization mixes two orbital-derived bands through an off-diagonal Hamiltonian matrix element. For a real coupling $\alpha$,An allowed bare crossing becomes an avoided crossing with direct separation $2|\alpha |$. Orbital character transfers continuously between branches through the crossing. A direct separation alone does not guarantee a global band gap; the extrema at different wavevectors must also be compared.

The Barnett effect is the change of a body's magnetization caused by mechanical rotation. Turning the whole specimen can bias the internal angular momentum carrying its magnetic moments, even with zero applied magnetic field. It is the reciprocal physical question to the Einstein-de Haas effect: impose rotation and measure magnetization, rather than change magnetization and observe mechanical motion. The relation between the two becomes especially transparent when the same signed gyromagnetic ratio connects magnetic moment and internal angular momentum.

Samuel Jackson Barnett published Magnetization by Rotation on 1 October 1915 in Physical Review. The Barnett rotation magnetometry experiment below provides an inspected modern example. The important distinction is between an equivalent term in a Hamiltonian and the actual external magnetic field measured by a magnetometer: the first biases the material; its resulting magnetization produces the second.

Let the specimen rotate uniformly with angular velocity $\mathbf{\Omega }=\Omega \hat{\mathbf{z}}$. In its rotating reference frame, the internal Hamiltonian containsHere $J_z$ is the internal angular momentum operator about the rotation axis, and ${\mathcal{H}}_0$ contains the internal interactions and any imposed magnetic field expressed in that frame. It does not include the separately treated rigid body's rotational kinetic energy. For a sector with magnetic moment $m_z=\gamma J_z$, the rotational term has exactly the form of a magnetic dipole potential energy:This Barnett equivalent magnetic field has tesla units, with the angular convention for the signed gyromagnetic ratio: $\Omega$ is in radians per second, not revolutions per second. Its sign matters. For an electronic contribution with $\gamma <0$, positive $\Omega$ favors positive $J_z$ and negative $m_z$. A sector with $\gamma >0$ has the opposite magnetic sign. Equal $\Omega$ need not correspond to equal Barnett fields for sectors having different gyromagnetic ratios.

A short derivation fixes the sign independently of any material model. Put $U(t)=\exp (-\mathrm{i}\Omega t{J}_z/\hslash )$ and transform a laboratory quantum state by $|{\psi }_{\mathrm{rot}}⟩=U^{†}|{\psi }_{\mathrm{lab}}⟩$. The Schrödinger equation givesThe transformed laboratory term is denoted by ${\mathcal{H}}_0$; a laboratory perturbation not stationary in the rotating frame can make it time dependent. Establishing a steady thermal response also requires exchange of internal angular momentum with the rotating body, for example through spin-lattice relaxation. A mere relabeling of coordinates cannot polarize isolated, unrelaxed spins.

The Barnett field is a useful description of spin-rotation coupling, not a claim that empty laboratory space acquires a uniform magnetic flux density $\Omega /\gamma$. In particular, inserting it into the Lorentz force on an unrelated test electric charge is unjustified. A nearby magnetometer detects the specimen's actual stray magnetic field and backgrounds, rather than directly detecting this internal Hamiltonian parameter.

A Bogoliubov transformation is a canonical transformation of mode operators mixing creation operators and annihilation operators. It diagonalizes quadratic Hamiltonians or relates different vacuum descriptions. Bosonic and fermionic canonical bracket constraints differ. A nonzero creation-mixing coefficient makes its vacuum contain particles relative to the original mode decomposition.

For real even coefficients with $A_k>|B_k|$, the quadratic Hamiltonianis diagonalized by the Bogoliubov transformation $a_k=\cosh {\theta }_k{\alpha }_k-\sinh {\theta }_k{\alpha }_{-k}^{†}$, with $\tanh 2{\theta }_k=B_k/A_k$. The canonical commutation relations follow from ${\cosh }^2{\theta }_k-{\sinh }^2{\theta }_k=1$. The diagonal form is ${\sum }_k[{\omega }_k{\alpha }_k^{†}{\alpha }_k+({\omega }_k-A_k)/2]$, with ${\omega }_k=\sqrt{A_k^2-B_k^2}$. The original-mode vacuum depletion is ${\sinh }^2{\theta }_k=(A_k/{\omega }_k-1)/2$. At $A_k=|B_k|$, the exact zero mode requires separate infrared treatment.

In a bosonic coherent-state path integral, coherent-state resolutions between imaginary-time evolution factors turn a normal-ordered grand Hamiltonian $K=H-\mu N$ into its complex symbol. Coherent-state overlaps supply the first-order temporal term. A thermal trace makes the fields periodic in imaginary time. The chemical potential couples to number, and interactions retain their normal-ordering convention.

In a bosonic Nambu spinor expression, the second diagonal entry commonly contains $a_{-k}{a}_{-k}^{†}$. The canonical commutation relations add one constant per mode when this is normal ordered. For an antiferromagnetic chain coefficient $JS$, rewriting the quadratic Hamiltonian as a full Nambu sum therefore changes the exterior constant from $-NJ{S}^2$ to $-NJS(S+1)$. Omitting the shift corrupts the ground-state energy even when the dispersion is unchanged.

For a normalized bound eigenstate of a factorized Coulomb radial Hamiltonian, the displayed squared norm must be nonnegative. At fixed negative $E$, increasing $l$ indefinitely would make it negative. The intertwining chain must therefore terminate with zero norm, giving $E=-{\hslash }^2/[2m{a}^2(L+1{)}^2]$. The positivity contradiction does not apply to the positive-energy continuum; domain preservation and normalizability are necessary assumptions.

For electronic $J=1/2$, nuclear spin $I$ and conserved angular momentum component $m$, define ${\Delta }_{\mathrm{hf}}=A(I+1/2)$ and the dimensionless field $x$ shown in the title. The hyperfine-plus-magnetic energies areThe factors $g_I,g_J$ use the same Hamiltonian convention and Bohr-magneton units. The two stretched states require the branch consistent with their unique spin configuration. The formula follows by diagonalizing each two-dimensional fixed-$m$ block; it connects weak-field coupled-spin levels to decoupled high-field states. The name refers to Gregory Breit and Isidor Isaac Rabi.^[ref]

For the linear-in-potential charged-scalar Lagrangian, $\pi ={\dot{\varphi }}^{\ast }-ie{A}_0{\varphi }^{\ast }$ and ${\pi }^{\ast }=\dot{\varphi }+ie{A}_0\varphi$. With $A_{\mu }=(A_0,-\mathbf{A})$, the Hamiltonian density isThe first product includes $+e^2{A}_0^2|\varphi {|}^2$ after the Legendre transform; that term can arise in canonical variables even though the original Lagrangian has no interaction quadratic in $A$.

Although the energy in velocities is $m({\dot{r}}^2+r^2{\dot{\varphi }}^2)/2+2{\pi }^{2}k(r-r_0{)}^2$, the Hamiltonian in canonical variables isThe field therefore enters Hamilton's equations through the relation between canonical and mechanical momentum, even though a magnetic field does no work.

A linear change of creation operators and annihilation operators is canonical when it preserves their defining brackets. For finitely many bosonic modes, the canonical commutation relations require $U{U}^{†}-V{V}^{†}=I$ and $U{V}^T-V{U}^T=0$. For fermions, the canonical anticommutation relations instead require $U{U}^{†}+V{V}^{†}=I$ and $U{V}^T+V{U}^T=0$. A Bogoliubov transformation uses such mixing to diagonalize a quadratic Hamiltonian or relate different vacuum descriptions. Infinite-mode unitary implementation has further conditions, as in bosonic mode mixing implementability.

The free transverse photon field is a Hermitian sum of two transverse-polarization plane-wave modes with creation and annihilation operators. Their standard bosonic commutator gives the transverse equal-time commutator. The normal-ordered Hamiltonian is a sum of positive oscillator energies.

For $\mathbf{d}\ne 0$, put $\hat{\mathbf{d}}=\mathbf{d}/|\mathbf{d}|$. The lower-band orthogonal projector is $P_{-}=(I-\hat{\mathbf{d}}\cdot \mathbf{\sigma })/2$. Applying the Pauli matrix multiplication law to $i\mathrm{Tr}(P_{-}[{\partial }_{k_x}{P}_{-},{\partial }_{k_y}{P}_{-}])$ givesThus the Chern number is the signed number of times the Brillouin zone covers a two-sphere under $\hat{\mathbf{d}}$. The lower energy eigenstate's actual Bloch vector is $-\hat{\mathbf{d}}$. The Berry curvature has the displayed positive sign for this lower band and the stated Berry connection; switching to the upper band changes its sign.

A useful local calculation exposes what each valley contributes. Write $p=\hslash v|\mathbf{q}|$. The linear Dirac Hamiltonian givesEach unbounded continuum cone gives a half-integer in this calculation. It is not a separately realizable half-integer Chern number of an isolated lattice energy band. The full honeycomb lattice supplies both valleys and the large-momentum completion. Starting with a trivial large-$|M|$ state, each single-valley mass crossing changes the lattice integer by the corresponding unit. There are no other band touchings. This establishes the exact lattice resultfor $t_2>0$, excluding the boundaries. This sign belongs to our explicit bond vectors, orientation and occupied-band convention.

For an original worked example, choose $t_1=1$, $t_2=0.08$, $\varphi =\pi /2$ and $M=0$. Then $\mu \simeq 0.415692$, $(m_{+},m_{-})=(-\mu ,+\mu )$, both valley gaps are $2\mu \simeq 0.831384$, and $C=-1$. Choosing instead $t_2=0$, $M=\mu$ gives exactly the same valley-gap magnitudes but $C=0$. The first state's two valley contributions reinforce; the second state's cancel.

An independent eigenvector-overlap calculation on periodic $31\times 31$, $61\times 61$ and $121\times 121$ momentum grids gives $C=-1$ for the first example and $C=0$ for the second. Reversing $\varphi$ in the first example gives $C=+1$. Direct integration of the analytic lattice Berry curvature agrees. For the overlap calculation one can remove the embedding phase from $f$, replacing it by $1+e^{i\mathbf{k}\cdot {\mathbf{a}}_1}+e^{i\mathbf{k}\cdot {\mathbf{a}}_2}$; this makes the cell-basis Hamiltonian periodic and prevents a false boundary-seam contribution. The local Berry curvature can depend on this basis convention; the integrated Chern number agrees. All mass-zero parameters are excluded before evaluating any integer.

For a structureless particle with Hamiltonian $p^2/(2m)+ar$, $a>0$, the classical phase space integral gives the displayed one-particle partition function. Its spatial integral is $4\pi {\int }_0^{\infty }{r}^2{e}^{-ar/(k_{B}T)}dr=8\pi (k_{B}T/a{)}^3$, while the momentum integral gives ${\lambda }_T^{-3}$. For $N$ indistinguishable dilute particles, $Z_N=Z_1^N/N!$, $F\simeq -N{k}_{B}T[\ln (Z_1/N)+1]$, $U=(9/2)N{k}_{B}T$ and $S\simeq N{k}_B[\ln (Z_1/N)+11/2]$. These thermodynamic formulas use leading Stirling accuracy; the classical phase-space integral additionally requires temperatures above the discrete trap level scale.

For a finite-dimensional Hamiltonian system, integrability is normally expressed by enough independent first integrals in mutual Poisson bracket involution. In classical field theories, an infinite hierarchy of compatible conserved quantities is the corresponding structure. For Sine-Gordon theory, Bäcklund transformations generate local conservation laws and explicit soliton solutions. Conservation of arbitrary quantities alone should not be confused with proof of their Hamiltonian involution.