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The creation-operator matrix element $⟨n+1|a^{†}|n⟩=\sqrt{n+1}$ makes emission into a radiation mode proportional to $n+1$. The $n$ part is stimulated emission and the remaining part is spontaneous emission. Absorption has the corresponding occupation factor $n$.

For a well $-a<x<a$, choose normalized energy eigenstates ${\varphi }_n=a^{-1/2}\sin [(n+1)\pi (x+a)/(2a)]$, $n=0,1,\dots$. The diagonal position matrix elements vanish by the parity operator. For $k\ne n$,To derive it, set $y=x+a$ and use $2\sin Ay\sin By=\cos [(A-B)y]-\cos [(A+B)y]$. For unequal positive integer indices $r,s$, the constant part of $x=y-a$ integrates to zero by orthogonality, whileTaking the difference for $j=r-s$ and $j=r+s$ yields the formula after setting $r=k+1,s=n+1$. It vanishes for equal parity and gives $x_{01}=-32a/(9{\pi }^2)$ for this phase convention.

Between spin-zero pseudoscalars, Lorentz covariance decomposes a vector current matrix element into $(p+k{)}^{\mu }{f}_{+}(q^2)+(p-k{)}^{\mu }{f}_{-}(q^2)$, with $q=p-k$. Strong interaction parity symmetry in quantum field theory makes the corresponding axial current matrix element vanish.

For the field-free, spin-free linear rigid rotor in one fixed internal quantum state, the electric-dipole interaction is $-\mathbf{d}\cdot \mathbf{E}(t)$. A nonzero permanent electric dipole moment along the molecular axis allows pure rotational photon absorption with $\Delta J=+1$ and photon emission with $\Delta J=-1$. The matrix element between initial and final quantum states must be nonzero: the existence of an energy gap alone is insufficient. Relative to a chosen $z$ axis, polarization of an electromagnetic wave along $z$ selects $\Delta M=0$; transverse polarization of an electromagnetic wave permits $\Delta M=\pm 1$.^[ref]

Hydrogen chloride and carbon monoxide are examples with a permanent electric dipole moment. By symmetry, molecular hydrogen and carbon dioxide have no permanent electric dipole moment in the ideal symmetric internal state. They still rotate and have rotational energy eigenvalues, but the simple permanent-electric dipole spectrum described here is absent. This restriction concerns this interaction and these assumptions, not every possible way of detecting molecular rotation.

The quadratic Stark effect is an energy level shift whose leading term is quadratic in an applied electric field. For a nondegenerate energy eigenstate of definite parity, the diagonal matrix element of an odd electric-dipole perturbation vanishes. Second-order nondegenerate perturbation theory givesEvery term is negative for the ground state. If field reversal is implemented by the parity operator, an analytic nondegenerate energy branch is even in the field. This symmetry explains the absence of odd powers; it does not apply without qualification to a degeneracy split linearly by the field.

For the spin-free linear rigid rotor in thermal equilibrium at temperature $T>0$, assume all integer $J\ge 0$ are allowed and ignore nuclear-spin restrictions. Each $J$ level has $2J+1$ quantum states sharing one energy eigenvalue, an instance of degenerate energy levels, so the Boltzmann distribution givesHere $Q_{\mathrm{rot}}$ is a rotational partition function, $h$ the Planck constant, $k_{\mathrm{B}}$ the Boltzmann constant, and $B$ a frequency in $\mathrm{Hz}$. Thus $hB/(k_{\mathrm{B}}T)$ is dimensionless. The $2J+1$ factor counts magnetic quantum numbers; it does not mean that a whole level is one quantum state. For the stipulated $B=5\mathrm{GHz}$, the most populated levels are $J=2$ at $3\mathrm{K}$ and $J=7$ at $30\mathrm{K}$.

Population alone is insufficient to predict intensity. In weak, unsaturated absorption, the integrated absorption coefficient includes the transition frequency, the squared electric dipole moment matrix element, and the difference of lower and upper substate populations because stimulated emission returns energy to the field. A pulsed FTMW spectroscopy signal additionally depends on excitation, quantum coherence, cavity tuning, and receiver response.^[ref]

For an isotropic sample with polarization of an electromagnetic wave along $z$, write $d_z=d\cos \theta$. Normalized spherical harmonics giveThe last equality follows from ${\sum }_{M=-J}^J{M}^2=J(J+1)(2J+1)/3$. Combining this sum with the per-substate Boltzmann distribution and subtracting stimulated emission gives relative integrated absorption areasThe common factor includes $d^2/3$ and cancels upon normalization. This formula describes integrated weak absorption in a thermal, spin-free linear rigid rotor, not arbitrary peak heights. The example's strongest integrated lines originate at $J=3$ and $J=13$ for $3$ and $30\mathrm{K}$, respectively: they differ from the most populated levels. A missing line can reflect population, coupling, coverage, or sensitivity; it does not prove that the corresponding quantum state is absent.

The approximate fifth-power scaling of an allowed beta decay rate with available energy when electron mass and Coulomb distortion are negligible and the nuclear matrix element is held fixed. It follows from ${\int }_0^Q{p}^2(Q-p{)}^{2}dp=Q^5/30$. Different nuclear matrix elements or nonrelativistic electron kinematics spoil an exact universal law.

A scalar particle has zero intrinsic spin. A scalar-field excitation provides its relativistic description. Spin zero alone does not force its decay matrix element to be independent of internal invariant masses; resonance dynamics can still produce a nonuniform three-body distribution.

Take a star graph $G$ with center $c$ and four leaves $1,2,3,4$. Let $L_G$ be its Laplacian matrix and $I$ the identity matrix. In the order $(c,1,2,3,4)$, the original example isFor a nonzero vector $u$, its quadratic form isThus $A$ is a symmetric positive-definite matrix and either ordering below admits a Cholesky decomposition. Changing the ordering changes the work, rather than the mathematical solvability.

Center first. Eliminating $c$ with pivot element $5$ leaves the Schur complementEvery pair of leaves now has coupling $-1/5$: six new edges, hence six undirected fill-in positions. The four remaining vertices form a clique. In the Cholesky decomposition, the first column has $C_{cc}=\sqrt{5}$ and $C_{ic}=-1/\sqrt{5}$. The rest of the lower triangular matrix is dense. To check that later cancellation does not remove its entries, after $r$ leaves have also been eliminated, the active leaf block isIndeed a leaf step sends $\alpha$ to $\alpha +{\alpha }^2/(2-\alpha )$. All active off-diagonal entries remain negative. The successive pivot elements in ordinary symmetric Gaussian elimination, equivalently the squared diagonals in the Cholesky decomposition, areThe Cholesky decomposition therefore stores $5+4+3+2+1=15$ nonzero entries, including its diagonal.

Leaves first. Permute to $(1,2,3,4,c)$. Each leaf has only one remaining neighbor, so its elimination changes only the center's diagonal, reducing it by $(-1{)}^2/2=1/2$. The successive pivot elements are $2,2,2,2,3$, and the exact Cholesky decomposition isThere is no fill-in, and only $5+4=9$ nonzero entries in $C$. This is a concrete instance of leaf elimination of a tree-pattern positive-definite matrix; its general proof explains why the same idea works on every tree pattern. The minimum degree algorithm also chooses a leaf at every nontrivial stage of this particular example, but that success gives no general optimality guarantee.

The input has 13 nonzero matrix elements when both symmetric halves are counted, or nine in its lower triangle. The six added undirected edges correspond to six added positions in one triangular factor, or 12 positions if both symmetric halves of the filled pattern are displayed. Comparing all 13 input entries directly with the 15 entries of $C$ would mix counting conventions.

A sparse matrix has few nonzero matrix elements in each row or column compared with its dimension. Algorithms normally require an efficient oracle that lists the positions and values of those entries.

A rank-$k$ spherical tensor operator has components $T_q^{(k)}$, $-k\le q\le k$, that transform under rotation like an angular momentum $k$. Equivalently, their commutators with the angular momentum operator satisfyFor a vector operator, the rank is one and a conventional choice is $T_0=V_z$, $T_{\pm 1}=\mp (V_x\pm i{V}_y)/\sqrt{2}$. The Wigner-Eckart theorem separates the matrix elements of a spherical tensor operator into Clebsch-Gordan coefficients and a reduced matrix element.

For $b>0$, the Fourier transform of $f$ is ${\int }_{-\infty }^{\infty }f(t)e^{ixt}dt=2b/(b^2+x^2)$. A weak interaction $D_{0}f(t)\cos \omega t$ between energy levels of gap $\hslash {\omega }_0$ gives final probability amplitude $-idb\{[b^2+({\omega }_0-\omega {)}^2{]}^{-1}+[b^2+({\omega }_0+\omega {)}^2{]}^{-1}\}/\hslash$, where $d$ is the transition matrix element. Only when $b,|{\omega }_0-\omega |\ll {\omega }_0$ does the single resonant lobe give an approximately squared Lorentzian line shape excitation profile. Its full width at half maximum is $2b\sqrt{\sqrt{2}-1}$ in angular frequency.

Consider an original ideal quantum system with positive scales $ϵ,d$ andInterpret $D$ as one component of an electric dipole moment, so its coupling to the corresponding electric field is the electric-dipole interaction $-\mathcal{E}(t)D$. The gaps are $2ϵ,3ϵ,5ϵ$, but $D_{02}=0$: this interaction has no direct first-order $2\leftrightarrow 0$ coupling. The nonzero squared matrix elements are $|D_{01}{|}^2=d^2$ and $|D_{12}{|}^2=4d^2$. These are coupling weights; actual spectral line intensities also depend on populations and radiation factors. In particular, there is no promise that an experiment populates the upper state of every allowed transition.

A measurement of $D$ itself has possible outcomes $0,\sqrt{5}d,-\sqrt{5}d$, sinceThese are eigenvalues of $D$, not the energy eigenvalues $0,2ϵ,5ϵ$. Moreover, $(D^2{)}_{02}=D_{01}{D}_{12}=2d^2$ although $D_{02}=0$: two factors connect through $|1⟩$. This algebraic statement does not turn $D_{02}$ into a nonzero direct coupling. Higher-order probability amplitudes for transitions require the appropriate dynamical calculation.

For a quantum field with four-momentum operator $P^{\mu }$, spacetime dependence is implemented by the displayed conjugation. Taking a matrix element between momentum eigenstates gives $⟨q|\Phi (x)|p⟩=e^{i(q-p)\cdot x}⟨q|\Phi (0)|p⟩$. Differentiating gives $i{\partial }_t\Phi =[\Phi ,H]$ and $-i\nabla \Phi =[\Phi ,\mathbf{P}]$, connecting the Heisenberg equation of motion to spatial translation covariance.

For an axially split $p$ multiplet with $p_z$ energy $E_0$ and equal $p_x,p_y$ energies $E_t$, a transverse $x$-directed magnetic field leaves $p_x$ unchanged and couples $p_y,p_z$ through the orbital angular momentum operator $L_x$. The matrix element has modulus ${\mu }_{B}B$, giving the two energy eigenvalues in the title. For ${\mu }_{B}B\ll |E_t-E_0|$, the orbital shifts are quadratic despite a linear Zeeman energy operator. A full quadratic-field calculation also includes the diamagnetic term; electron spin gives its own linear contribution.

For two isotropic parabolic subbands, the joint density of states per area and per independent channel is $\mu /(2\pi {\hslash }^2)\Theta (\hslash \omega -E_{\mathrm{edge}})$. The constant is a two-dimensional phase-space effect. Degeneracy and optical matrix elements determine the actual step weights.

In the ideal electric-dipole interaction, a vibrational transition needs a nonzero matrix element of the electric dipole moment. For a diatomic bond stretch, expand the body-axis dipole component as $d(q)=d_e+d_e^{\prime }q+\cdots$. A changing electric dipole moment, rather than a permanent value $d_e$ alone, permits the harmonic fundamental. Keeping only the linear term and harmonic wavefunctions gives $\Delta v=\pm 1$.^[ref]

The derivation is sharper than a picture of a shaking molecule. Orthogonality removes the constant term for $v^{\prime }\ne v$, and creation and annihilation operators giveHere $\delta$ is the Kronecker delta. Thus absorption raises $v$ by one when the dipole derivative and angular factor are nonzero, and its photon satisfiesA nonzero vibrational matrix element is only one factor: rotational electric-dipole selection rules, initial populations, and spectral resolution also constrain visibility.

For an isolated diatomic molecule with identical atomic nuclei, such as molecular hydrogen in its symmetric electronic ground state, stretching preserves zero electric dipole moment. By symmetry, this homonuclear stretch is infrared inactive in the ideal dipole model, although it still vibrates. Other interactions or environments can permit detection.^[ref]

A vibrational overtone excites more than one vibrational quantum: $v=0\to 2$ is the first overtone, and $0\to 3$ the second. Such transitions vanish in the combined harmonic-potential and linear-dipole approximation. Anharmonicity changes wavefunctions and level spacing; nonlinear terms in the electric dipole moment can also supply otherwise missing transition matrix elements.^[ref]

For an original illustrative level model, setThen $E_1-E_0=\hslash \omega (1-2ϵ)$ and $E_2-E_0=2\hslash \omega (1-3ϵ)$. With chosen $ϵ=0.02$, the fundamental gap is $0.96\hslash \omega$ and the first-overtone gap $1.88\hslash \omega$, below twice the fundamental, $1.92\hslash \omega$. This low-level illustration does not determine transition intensity or justify extrapolation to arbitrarily large $v$. It shows why even a well-resolved fundamental band origin need not equal the harmonic curvature scale ${\tilde{\nu }}_{\mathrm{harm}}$.