In the two-spin exchange spectrum with unequal signed magnetic moments, the axial Hamiltonian conserves $M=S_{1z}+S_{2z}$. Parallel-spin states have $M=\pm \hslash$, while the two mixed states lie in the $M=0$ block. Levels in different $M$ sectors may cross as the axial magnetic field changes because the conserved projection forces their coupling matrix element to vanish.

This is a symmetry qualification of the non-crossing rule. A nonzero isotropic exchange interaction does not by itself turn every field-dependent crossing into an avoided crossing. A transverse perturbation that breaks conservation of $M$ can mix the sectors and open a gap, provided its relevant matrix element is nonzero. Plotting the energy eigenvalues by their conserved projection therefore distinguishes protected crossings from the avoided splitting already present within the $M=0$ block.

absorption of two photons whose summed energy matches an atomic transition. In the weak-field regime its rate scales with the product of the two intensities. Two electric-dipole interactions preserve the initial parity.

Closely spaced disks undergo torsional oscillations in He-II. Viscous entrainment of the normal component contributes a temperature-dependent moment of inertia, while vortex-free superfluid need not rotate with the disks. The interpretation requires the normal component to follow the drive under the apparatus's spacing and frequency conditions. The experiment is associated with Elephter Andronikashvili.

A quadratic-dispersion Fermi gas in two dimensions has constant density of states $g$. At zero temperature, $N=g{E}_F$ and $U=g{E}_F^2/2$. Hence the mean energy is half the Fermi energy, independent of a common spin-degeneracy factor.

Each allowed wavevector occupies area $(2\pi {)}^2/A$ under periodic boundary conditions, for each independent spin state. For parabolic dispersion, the number below energy $ϵ$ is $g_{s}Amϵ/(2\pi {\hslash }^2)$. Thus the often-quoted coefficient $Am/(2\pi {\hslash }^2)$ is per spin. Unpolarized electrons have $g_s=2$; the same convention must be used in particle counting and the Fermi energy. A linear Dirac dispersion has a different energy dependence.

For a constant total density of states $D$ above a zero-energy band bottom, integrating the Fermi-Dirac occupation gives $N=D{k}_{B}T\ln (1+e^{\mu /(k_{B}T)})$. With ${ϵ}_F=N/D$, inversion gives the displayed chemical potential. It approaches ${ϵ}_F$ from below with an exponentially small correction at low temperature, rather than the usual three-dimensional quadratic correction. This formula assumes an ideal parabolic band extending to arbitrarily high energy.

The first two optically active hydrogenic levels have separation $32R^{\ast }/9$ in the planar model, versus $3R^{\ast }/4$ in the bulk model. The separation cancels an uncertain common continuum threshold, making it useful when rounded band parameters do not reproduce every absolute peak energy.

A short electrostatically defined constriction connecting two electron reservoirs. Transverse confinement produces transport modes, and sufficiently narrow contacts can collimate carriers for electron-focusing experiments. Its quantum conductance depends on mode transmission; an opening described only as narrower than a micrometre is not automatically in a single-mode limit.

A magnetic field curves ballistic carrier trajectories between an injector and a collector. For openings separated along a specular edge by $L$, peaks occur near $L=2j{r}_c$ with $r_c=\hslash k_F/(e|B|)$, giving $|B_j|=2j\hslash k_F/(eL)$. Field polarity and carrier sign select the connection direction. Electron phase coherence can produce fine structure but is not required for the basic semiclassical focusing geometry.

A gate separated by dielectric thickness $d$ has geometric capacitance per area $C/A=ϵ/d$ in the parallel-plate approximation. An electron-density change is $\Delta n_s=(ϵ/ed)\Delta V_g$ with the stated polarity/reference convention. Fringing, trapped charge, depletion and finite quantum capacitance can modify the relation. An absolute gate voltage additionally requires the zero-density offset or another calibration.

For planar points $A,B,P$, the sign of $(B-A)\times (P-A)$, using the two-dimensional cross product, determines which side of the directed line from $A$ to $B$ contains $P$. Zero means collinearity. A point is strictly inside a nondegenerate triangle exactly when these signs for its three cyclically ordered sides are all strictly positive or all strictly negative.

For dichotomic observables with $A_i^2=B_j^2=I$, let $\mathcal{B}=A_0\otimes (B_0+B_1)+A_1\otimes (B_0-B_1)$. Expanding gives ${\mathcal{B}}^2=4I-[A_0,A_1]\otimes [B_0,B_1]$. Each commutator has operator norm at most two, so $\Vert \mathcal{B}{\Vert }^2\le 8$ and $|⟨\mathcal{B}⟩|\le 2\sqrt{2}$. Appropriate spin measurements of a Bell state attain equality. The local hidden-variable bound of the corresponding CHSH inequality is two.

The three plate velocities plotted in a common reference frame form a triangle whose directed edges are the pairwise relative velocities. Adding the same reference velocity translates the triangle without changing its edges.

For a boundary with in-plane normal $n$, an admissible junction velocity $U$ obeys $(U-v_{\mathrm{boundary}})\cdot n=0$. A ridge uses the mean of adjacent plate velocities in the symmetric model, a trench uses the overriding plate velocity, and a transform uses either adjoining plate velocity. Compatibility requires a common intersection of the three velocity-space lines.

For pure normal symmetric spreading, the two ridge constraints are perpendicular bisectors of two edges of the plate velocity triangle. The transform constraint lies along the third edge. For a nondegenerate triangle their common intersection exists only when the angle opposite that third edge is a right angle; the junction velocity is then the midpoint of the transform edge.

A junction whose persistence depends on the two overriding-plate trench constraints and the transform velocity line meeting at one point. Specifying only the boundary-type letters does not specify the compatibility geometry or trench polarities.

The period-two extension of $1-|x|$ on $[-1,1]$ is even. Its Fourier cosine series has $a_0=1$ and $a_n=2{\int }_0^1(1-x)\cos (n\pi x)dx=2[1-(-1{)}^n]/(n^2{\pi }^2)$ by integration by parts. This proves the displayed series, with only odd nonconstant harmonics. Continuity and piecewise smoothness give pointwise equality through the Dirichlet-Jordan convergence theorem; the summable coefficient magnitudes also imply uniform convergence by the Weierstrass M-test. At $x=0$, it gives ${\sum }_{m\ge 0}(2m+1{)}^{-2}={\pi }^2/8$.

For an equilibrium overdamped harmonic coordinate with covariance $⟨x(t)x(0)⟩=V{e}^{-|t|/\tau }$, a camera exposure $\Delta$ records the average over that interval. Integrating the covariance twice gives $\mathrm{Var}(\bar{x})=2V{\Delta }^{-2}{\int }_0^{\Delta }(\Delta -u)e^{-u/\tau }du$. The displayed reduction follows with $s=\Delta /\tau$. In the tether calibration $F=k_{B}Th/V$, ignoring exposure averaging overestimates the inferred force. Independent localization-noise variance instead adds to the measured variance and biases it in the opposite direction. Other filters require their actual response functions.

An imaging approach resolving fine specimen structure through transmitted-electron interference. Contrast depends on the specimen and imaging transfer, so visible fringes require interpretation rather than direct identification with a new atomic spacing.

Imaging newly emerging peptide chains at individual RNA molecules. Release and run-off controls help distinguish translation-dependent signals from aggregates or completed proteins.