The Pseudomonas aeruginosa quinolone-signaling network involving HHQ/PQS, PqsR and the pqs biosynthetic locus.

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.

For a generic one-parameter Hermitian Hamiltonian, two coupled states of the same conserved symmetry avoid an exact crossing. Opposite symmetry or a vanishing coupling can permit crossings. The rule does not supply numerical energies or prove that every possible reaction path shares the stipulated symmetry.

An antisymmetric single Slater determinant whose occupied orbitals make its energy stationary under occupied-virtual orbital rotations. A stationary solution need not be a stable or global minimum.

Ground-state density-potential uniqueness and a density variational principle for electrons with fixed interaction. The elementary uniqueness proof assumes nondegenerate ground states and fixes the potential only up to a constant.

A density-functional approximation using the energy per particle of a uniform electron gas at the local density. Its exchange-energy-per-electron tail decays exponentially with an exponentially decaying density.

The radius $r_s$ of a sphere containing one electron at a specified density: $4\pi r_s^3\rho /3=1$.

A dimensionless measure of the intensity of a specified optical transition. It combines the transition matrix element and transition energy; selection rules, Polarization and the distribution of electronic states determine which transitions carry appreciable absorption. Peak position alone does not give the absorbed fraction.

Quantum transport studies the flow of particles, electric charge or energy when quantum states, their occupation and their transmission probabilities determine the flow. In a short quantum wire, transverse quantum confinement permits only discrete transverse electronic modes, even though motion along the wire remains possible.

The Landauer formula treats two contacts as reservoirs supplying incident electrons. Their different occupations drive an electric current; a potential barrier changes the transmission probability of each incident mode. At low temperature, opening one perfectly transmitted spin-resolved mode increases the electrical conductance by the spin-resolved conductance quantum $e^2/h$. Quantum tunnelling, interference and Coulomb blockade explain departures from a simple classical resistor picture.

For a stationary bound state with suitable operator domains, $2⟨K⟩=⟨\mathbf{x}\cdot \nabla V⟩$. In a quadratic harmonic potential this gives $⟨K⟩=⟨V⟩=E/2$.

A quantum system with two specified subsystems has a tensor product Hilbert space. A joint density operator determines subsystem $A$'s measurement statistics through its partial trace ${\rho }_A={\mathrm{Tr}}_B{\rho }_{AB}$. A joint pure state need not give a pure reduced state: the Bell state $(|00⟩+|11⟩)/\sqrt{2}$ gives ${\rho }_A=I/2$.

A description of quantum states and dynamics compatible with special relativity. Relativistic one-particle equations have a useful domain, while interactions that create or destroy particles generally require quantum field theory.

A quantum particle is a quantum system whose possible measurement outcomes are described by a quantum state rather than a definite classical trajectory. For a single spinless particle, a wavefunction gives a probability amplitude for each position; the Born rule turns its squared modulus into a probability density. A localized detection does not imply that the particle followed a known path before detection.

An electron sent through an interferometer is a useful example: individual arrivals are localized, while many arrivals reveal matter-wave interference. Changing the relative quantum phase of the available paths changes the detection distribution. For several identical quantum particles, bosonic or fermionic exchange symmetry further restricts the joint quantum state.

A Mott transition is a change between a conducting state and a Mott insulator, driven by the competition between electron motion and electron repulsion. A partially filled energy band need not conduct if repulsion makes moving an electron onto an already occupied site too costly.

The half-filled Hubbard model illustrates the mechanism: hopping lowers the kinetic energy, whereas on-site repulsion $U$ penalizes double occupation. Increasing $U$ relative to the energy-band width can suppress charge motion and open a gap for charge excitations. Changing pressure, composition or carrier concentration can alter that competition, although actual materials may also change their crystal structure or magnetic order. This interaction mechanism differs from a band insulator with filled bands and from disorder-driven Anderson localization.

A spinless charged particle in $B_z=\alpha x$ admits vector potential $A_y=\alpha x^2/2$. Separating a plane wave along y produces a one-dimensional quartic potential energy. For nonzero $q\alpha$, $k/(q\alpha )>0$ gives two minima at $\pm \sqrt{2\hslash k/(q\alpha )}$; the opposite sign gives one minimum at zero. The field gradient creates confinement even though there is no scalar potential. The quartic potential is even, so its nondegenerate eigenstates have definite spatial parity.

An eavesdropper measures quantum carriers and forwards states consistent with the outcomes. The resulting disturbance is tested statistically by the legitimate users. For a singlet measured by Eve on both sides along ${\mathbf{n}}_a,{\mathbf{n}}_b$, the forwarded correlation is $-(\mathbf{a}\cdot {\mathbf{n}}_a)({\mathbf{n}}_a\cdot {\mathbf{n}}_b)(\mathbf{b}\cdot {\mathbf{n}}_b)$. It follows by averaging the product of the conditional local outcome means over Eve's correlated outcomes. This model does not cover every general quantum attack.

Entanglement-based key distribution that tests correlations with a Bell inequality. Equal measurement axes generate correlated key data; other settings test the quantum channel. Authentication, statistical parameter estimation, information reconciliation and privacy amplification are needed to turn observed correlations into a secure shared key under specified assumptions.^[ref]

For a stable relativistic quantum field theory in Minkowski space, a vacuum is a lowest-energy state for the chosen time evolution; a physical vacuum is normally also invariant under spacetime translations. It need not have zero fluctuations or vanish under every field operator. A free field's Fock vacuum is annihilated by its annihilation operators, while an interacting vacuum need not equal the free-theory Fock state. Multiple degenerate vacua permit spontaneous symmetry breaking. In curved spacetime the choice of time evolution and positive-frequency modes introduces additional vacuum ambiguity in a nonstationary spacetime.

Filling a degenerate orbital shell produces a larger addition energy for the next electron. In a circular two-dimensional harmonic potential, shell $q=n_x+n_y$ has $q+1$ orbitals and capacity $2(q+1)$ including spin. Cumulative closed-shell occupancies are $(q+1)(q+2)=2,6,12,20,\dots$. Charging and orbital contributions must be separated, and anisotropy or exchange can split the ideal shells.

The difference between successive electron-addition chemical potentials. In a constant interaction model it is $E_{\mathrm{add}}=e^2/C_{\Sigma }+{ϵ}_{N+1}-{ϵ}_N$, separating electrostatic charging from orbital level spacing. Spin, exchange and correlation can change the orbital contribution.