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For a spin-one Hamiltonian $H=A{S}_z^2+B(S_x^2-S_y^2)$, Cartesian basis energies are ${\hslash }^2(A-B)$, ${\hslash }^2(A+B)$ and zero along the $x,y,z$ basis labels. The $S_x=\hslash$ state combines the latter two basis vectors equally. Their relative phase produces the displayed spin-reversal probability. Complete reversal occurs at odd multiples of $\pi /(|A+B|\hslash )$ if $A+B\ne 0$; when it vanishes those two components remain degenerate. Individual matrix signs depend on basis conventions, while the commutators and physical probability must remain consistent.

The spin-rotation coupling contribution to a rotating-frame Hamiltonian couples angular velocity to angular momentum. When discussing spin alone, identify which part of the total angular momentum enters the effective model.

The perturbation $\lambda \hslash \omega (a^r+(a^{†}{)}^r)$ couples the oscillator vacuum only to $|r⟩$ at first order. Its squared coupling is $(\hslash \omega {)}^{2}r!$ and the energy denominator is $-r\hslash \omega$, giving the displayed formal second-order coefficient. For $r>2$ and nonzero real $\lambda$, coherent-state expectations contain $R^2+2\lambda R^r\cos r\theta$; choosing its phase negative proves the Hamiltonian is unbounded below. The formal vacuum branch must therefore not be called an exact ground state. For $r=2$ boundedness requires $|\lambda |<1/2$, while the displaced $r=1$ oscillator is stable for every real strength.

At ${\mathbf{K}}_{+}$ and positive energy, a normalized eigenvector for $\mathbf{q}=|\mathbf{q}|(\cos \varphi ,\sin \varphi )$ is $(1,e^{i\varphi })/\sqrt{2}$. Its Pauli matrix expectation values give the displayed direction. This pseudospin records the relative amplitudes and phase on the two sublattices. Real spin supplies another label; confusing these two labels would double-count states and misinterpret the two-band Hamiltonian.

In the weak, non-crossing, leading-eccentricity planar model, exterior orders one to three have positive resonant disturbing function coefficients $B_q$. Interior eccentricity-type coefficients of these orders have sign $(-1{)}^q$. A local resonant Hamiltonian $H=-A{J}^2/2-B{e}^q\cos \varphi$, with $A>0$, gives $\ddot{\varphi }=AB{e}^q\sin \varphi$. Linear stability analysis therefore selects $\varphi =\pi$ for $B>0$ and $\varphi =0$ for $B<0$. Exterior symmetric centres are $\pi$; interior centres are $0,\pi ,0$ for orders one, two and three. These are conditional leading-harmonic predictions, not universal centres at arbitrary orbital eccentricity; asymmetric resonant-argument libration changes the picture.

An operator implementing a symmetry of quantum states and observables. The ordinary spatial symmetries considered in molecular spectroscopy are unitary and commute with the Hamiltonian. More generally, an antiunitary symmetry such as time reversal needs its corresponding treatment.

Four equivalent $s$ atomic orbitals at the vertices of a regular tetrahedron carry the tetrahedral symmetry representation $A_1\oplus T_2$. The normalized equal sum is $A_1$; three coordinate-sign combinations span $T_2$. Every normalized coefficient vector has entries $\pm 1/2$ in a suitable orthonormal real basis. These functions simplify a Hamiltonian invariant under the tetrahedral operations, but equivalent symmetry copies on other sites may still mix.

In an orthonormal basis of localized atomic orbitals, a ring of three equivalent sites with real hopping $t$ has Hamiltonian $H=(E_0-t)I+tJ$, where $J$ is the three-by-three all-ones matrix. The normalized symmetric state $(|1⟩+|2⟩+|3⟩)/\sqrt{3}$ has energy $E_0+2t$. Its two-dimensional orthogonal complement has energy $E_0-t$. Thus the symmetric state is the unique spatial ground state for $t<0$, with an energy difference $3|t|$ to the excited eigenstates. Equivalently, the three discrete Bloch states have energies $E_0+2t\cos (2\pi l/3)$, $l=0,\pm 1$. Nonorthogonal atomic orbitals instead require a generalized eigenvalue problem.

An additional time-dependent operator in a quantum Hamiltonian. It may couple stationary states or shift their phases. It is the physical operator being treated, whereas time-dependent perturbation theory is a method for approximating the response.

For a time-independent Hamiltonian $H$, the explicit time partial derivative of a smooth time-dependent conserved quantity is again a conserved quantity. Here conservation means the identity $D_{t}f=f_t+\{f,H\}=0$ on phase space, with the standard Poisson bracket. Differentiating this identity at fixed canonical coordinates givesBecause $H_t=0$, this is exactly $D_t(f_t)=0$, proving the assertion. Iteration gives the same result for all higher explicit time partial derivatives for which the differentiations are defined. By Hamiltonian energy balance, requiring $H$ itself to be a conserved quantity on all trajectories is equivalent to $H_t=0$.

A top-hat wavefunction is constant on an interval and zero elsewhere. It has a uniform position distribution and a sinc function momentum amplitude. Its discontinuous edges place it outside the finite-kinetic-energy domain of the free Hamiltonian. Smooth-edged localized packets can approximate its position profile while having different momentum tails.

Topological quantum matter has robust low-energy properties controlled by global topology rather than local order parameters.

The central question is which quantum states can be joined by a smooth change of a local Hamiltonian while keeping a nonzero bulk spectral gap. A Chern insulator makes this question concrete: two band insulators can have different integer Chern numbers, forcing a gap closure between them. The Haldane phase supplies a one-dimensional example in which a protecting symmetry matters. These are distinct from intrinsic topological order, with its anyons and topology-dependent ground-state degeneracy. The Berezinskii–Kosterlitz–Thouless transition concerns a separate mechanism involving vortices; topology organizes more than one kind of physical phenomenon.

For the rotating-wave coupling in the Jaynes-Cummings model, one atomic excitation is exchanged for one photon, so the displayed operator commutes with the Hamiltonian. Its eigenvalue zero labels $|g,0⟩$ alone, and each eigenvalue $n+1\ge 1$ labels the paired sector $|e,n⟩,|g,n+1⟩$.

The total spin is the vector sum of subsystem spin operators and generates their simultaneous rotation. For a rotationally invariant Heisenberg model, both its squared magnitude and any chosen component commute with the Hamiltonian. An anisotropic exchange or field can preserve only a component.

In a gauge with potentials independent of y, the canonical momentum $p_y$ commutes with the Hamiltonian. A simultaneous eigenbasis can be chosen as $e^{iky}{\Phi }_k(x)$, with $p_y=\hslash k$. Periodic length $L_y$ gives $k=2\pi n/L_y$. The kinetic y-momentum is instead $\hslash k-q{A}_y(x)$ and is not spatially constant. Confusing these two momenta loses the magnetic contribution to the velocity.

An admissible wavefunction chosen from a tractable family for the variational method. Its Hamiltonian expectation gives an upper bound on the ground-state energy; parameters can be adjusted to lower that bound.

The type-two generating function for a canonical transformation $F_2(q,P)=\cos q\sin P$ defines $p=-\sin q\sin P$ and $Q=\cos q\cos P$. On any branch with $0<q<\pi$ and $\cos P\ne 0$, its inverse in mixed coordinates isThe square root equals $\sin q$ on this branch. Differentiating $F_2$ proves the generating equations, and $F_{2,qP}=-\sin q\cos P\ne 0$ proves local invertibility by the inverse function theorem. Also $d(F_2-PQ)=pdq-PdQ$, so the symplectic form obeys $dq\wedge dp=dQ\wedge dP$. This proves the transformation is canonical locally. It is not a single global chart across the inverse-trigonometric branch boundaries.

Since $F_2$ is independent of time, the transformed Hamiltonian is obtained by substitution, with zero generating-function time correction. An added function of time alone is an irrelevant Hamiltonian freedom, since it leaves Hamilton's equations unchanged.

Use the independent spin-one-half two-level system with fixed $N$ distinguishable sites and Hamiltonian levels $E_{\pm }=\pm \mu B$, where $\mu >0$ is the magnitude of the field-axis magnetic moment and $B>0$ is a uniform externally controlled magnetic field. Assume positive temperature, no interactions between sites, and no appreciable lattice or load entropy. The two levels differ by $2\mu B$. With inverse temperature $\beta$, the canonical ensemble gives the one-site and total partition functionsHere the minus subscript denotes the lower energy $-\mu B$. Factorization of the partition function requires additive energy and independent sites; there is no particle-exchange factor $N!$ for these fixed distinguishable sites.

Differentiate the partition function, holding $B$ fixed when differentiating with respect to $\beta$. The internal energy, Helmholtz free energy, total field-axis magnetic moment $m$, and magnetic entropy areThe extensive moment $m$ is distinct from magnetization, the moment per unit volume. Direct substitution of $p_{\pm }$ into the Gibbs entropy gives the same $s(x)$ independently of the free-energy derivative.

At fixed $B$, the magnetic heat capacity isStrict decrease makes $s$ invertible on $x>0$. Constant magnetic entropy therefore fixes $x$, not $B$ or $T$ separately:The partition function may be used to label these equilibrium states even though the isolated cooling leg has no external heat bath. Actual thermalization of the lattice and load additionally requires a sufficiently fast spin-lattice relaxation; perfectly uncoupled spins by themselves cannot refrigerate an attached object.

The limiting cases are instructive. For $x\ll 1$, $s(x)=\ln 2-x^2/2+O(x^4)$, $m\simeq N{\mu }^{2}B/(k_{B}T)$, and $C_{B,\mathrm{mag}}\simeq N{k}_B{x}^2$. The moment response reproduces Curie's law. For $x\gg 1$, $s(x)\simeq (1+2x)e^{-2x}$ and the upper-state occupation is exponentially small. Taking $B\to 0$ at fixed $T>0$ gives $s\to \ln 2$; taking $T\to 0$ at fixed $B>0$ gives $s\to 0$. The origin has a path-dependent limit. Extrapolating a line of constant $x$ to $B=T=0$ does not construct a usable zero-temperature state or prove that a real material retains independent degenerate spins there.

Choose two Ising spins with Hamiltonian $H=-J{s}_1{s}_2$, where $J$ independently takes $J_0$ or $2J_0$ with equal prior probabilities and $J_0>0$. This original finite example has no geometrical frustration and no phase transition. Its four spin assignments give $Z_J=4\cosh (\beta J)$. ThereforeThe concave function $\log$ puts $F_{\mathrm{a}}\le F_{\mathrm{q}}$. For this two-value case the claim is just $(Z_1+Z_2)/2\ge \sqrt{Z_1{Z}_2}$, obtained by squaring $\sqrt{Z_1}-\sqrt{Z_2}$. Thus no formal continuation of replicas is needed to see the distinction.

In the annealed disorder ensemble, the stronger bond has probability $Z_{2J_0}/(Z_{J_0}+Z_{2J_0})$, approaching one at low temperature. A quenched disorder collection instead retains one half at each coupling. As $T\to 0$, $F_{\mathrm{q}}\to -3J_0/2$ and $F_{\mathrm{a}}\to -2J_0$: averaging samples and allowing the material variables to equilibrate describe different physics.

On a regular bipartite graph with coordination number $q$, replace each bond product by ${\sigma }_B{\sigma }_W\simeq M_W{\sigma }_B+M_B{\sigma }_W-M_B{M}_W$. For $N$ sites there are $Nq/2$ bonds, so the constant in the effective Hamiltonian is $-NqJ{M}_B{M}_W/2$. The mean-field self-consistency equations are $M_B=\tanh \beta (h+g-qJ{M}_W)$ and $M_W=\tanh \beta (h-g-qJ{M}_B)$. At zero fields the antisymmetric branch has a continuous mean-field Ising critical bifurcation at $k_B{T}_c=qJ$.