A coherent superposition of distinct optical coherent amplitudes, for example $N(|\alpha ⟩+e^{i\varphi }|-\alpha ⟩)$. It differs from the corresponding incoherent mixture through cross terms in observables and its phase-space interference. For large $|\alpha |$, branch distinguishability is large while ordinary quadrature squeezing can become exponentially small.

Ideal conditional application of a creation operator, followed by normalization. It reweights the input number distribution as well as raising each occupied number by one. A heralded physical implementation has a success probability; the algebraic creation operator alone is not a trace-preserving deterministic quantum operation.

Ideal conditional application of an annihilation operator, defined only for positive initial mean photon number. Number-dependent weighting means that the conditional mean need not fall. Vacuum input gives the zero vector and therefore no successful conditional state.

A state of light is nonclassical when it is not a classical mixture of coherent states. The Glauber-Sudarshan representation states this criterion in phase space. Examples include nonvacuum number states and quadrature squeezing states. A coherent state is on the classical boundary despite its vacuum fluctuations; thermal light is a classical mixture despite its non-Poisson fluctuations. None of these nonclassicality witnesses is a necessary condition for nonclassicality.

A measurement with small information and disturbance per trial, often implemented by coupling an observable weakly to a meter so that its eigenvalue-dependent pointer distributions overlap. Repeated observations can still accumulate substantial measurement backaction. A weak value additionally requires specified preparation and postselection; a weak measurement without postselection need not involve one.

The coherent interaction that correlates a system observable with detector records before selecting an outcome. An ideal coupling maps $|s_j⟩|d_0⟩$ to $|s_j⟩|d_j⟩$. Linearity then maps a superposition to ${\sum }_j{c}_j|s_j{d}_j⟩$. The full state can remain pure and entangled, whereas a nonselective projective measurement removes coherence between different record sectors. Conditioning on one observed record is a third, selective operation.

The spin-one-half quantum XX chain has equal nearest-neighbour transverse exchange. Its continuous spin rotation symmetry conserves total longitudinal magnetization. The Jordan–Wigner transformation maps it to number-conserving free fermions. This quantum chain is distinct from the classical XY model.

Quantum communication transmits or distributes quantum states and the correlations they carry between separated systems. An optical photon's polarization or an excitation shared between stationary qubits can serve as a carrier. Loss and noise are represented by a quantum channel and can degrade the transmitted state or its entanglement.

Quantum teleportation transfers an unknown state using shared entanglement, a local measurement and a classical message. The classical message is necessary, so the protocol does not send usable information faster than light. The no-cloning theorem prevents repairing an arbitrary unknown state by simply making redundant copies, motivating quantum error correction and carefully designed communication protocols.

Consider $-y^{\prime \prime }-(n-1)y^{\prime }/x+x^{2}y=\lambda y$ for a positive integer $n$, with regular radial data at zero and square-integrable decay at infinity in the measure $x^{n-1}dx$. Its Sturm-Liouville form has $p=w=x^{n-1}$ and $q=x^{n+1}$. For the Gaussian trial $y=e^{a{x}^2}$, $a<0$, the Rayleigh quotient isThe Gamma function recurrence evaluates the ratio, and the minimum is $n$ at $a=-1/2$. It is the exact lowest eigenvalue: after the endpoint term $[x^n{y}^2{]}_0^{\infty }$ vanishes, integration by parts givesEquality gives the displayed positive Gaussian eigenfunction. For $n=1$, the radial regularity condition is the Neumann boundary condition at zero, not a vanishing value there. Changing the boundary condition changes the lowest eigenvalue and whether this trial is admissible.

For the normalized ground state of a reduced quantum harmonic oscillator, ${\varphi }_0(q)={\pi }^{-1/4}{e}^{-q^2/2}$ and $E_0=1/2$. Its classical turning points are $q=\pm 1$. The Born rule assigns probability $P=2{\pi }^{-1/2}{\int }_1^{\infty }{e}^{-q^2}dq=\mathrm{erfc}(1)\simeq 0.157299$ to the classically forbidden region. This is a stationary probability of detection outside the classical interval. It is not a transmitted flux or an escape rate: the real stationary wavefunction has zero probability current.

A jump from positive frequency $\omega$ to $\Omega$ keeps an oscillator mode and its first derivative continuous. A normalized positive-frequency incident solution has outgoing coefficients $\alpha =(\Omega +\omega )/(2\sqrt{\Omega \omega })$ and $\beta =(\Omega -\omega )/(2\sqrt{\Omega \omega })$, satisfying ${\alpha }^2-{\beta }^2=1$. This temporal matching is not a sum-one spatial-scattering probability rule.

With oscillator length $\ell =\sqrt{\hslash /(m\omega )}$, the eigenfunctions are proportional to $H_n(x/\ell )e^{-x^2/(2{\ell }^2)}$. They have $n$ real nodes, parity $(-1{)}^n$, and energy $\hslash \omega (n+1/2)$.

An energy eigenstate has $⟨x⟩=⟨p⟩=0$, $⟨x^2⟩=\hslash (n+1/2)/(m\omega )$ and $⟨p^2⟩=m\hslash \omega (n+1/2)$. Thus $\Delta x\Delta p=\hslash (n+1/2)$; only the ground state in this sequence minimizes the Heisenberg uncertainty principle.

For $H=p^2/(2m)+k{x}^2/2-qEx$, completing the square translates the well by $x_0=qE/k$ and shifts every energy by $-q^2{E}^2/(2k)$. The wavefunction of the ground state is translated without changing its width. Consequently the induced electric dipole moment is exactly linear in $E$, even when a low-order perturbative approximation to the state is inaccurate.

For identical masses and bare frequencies with coupling $kxy$, coordinates $(x\pm y)/\sqrt{2}$ diagonalize the Hamiltonian. A stable normalizable ground state requires $|k|<m{\omega }^2$. Its wave function is the product of Gaussians of frequencies ${\omega }_{\pm }$ in these normal coordinates; a two-mode squeeze of the original bare oscillators alone is generally insufficient.

The ground-state turning points are $\pm \sqrt{\hslash /(m\omega )}$. In dimensionless position $y=x\sqrt{m\omega /\hslash }$, the probability outside them is $(2/\sqrt{\pi }){\int }_1^{\infty }{e}^{-y^2}dy=\mathrm{erfc}(1)\simeq 0.15730$. This tail demonstrates that a bound quantum state need not be confined to its classically allowed region.

A canonical quantum harmonic oscillator has geometric number probabilities and mean occupation $(e^{\beta \hslash \omega }-1{)}^{-1}$. The position operator then gives the displayed variance. It interpolates between zero-point fluctuations $\hslash /(2m\omega )$ and the classical variance $k_{B}T/(m{\omega }^2)$. The kinetic and potential energies are equal at every temperature, although classical equipartition describes their values only at high temperature.

At imaginary duration $\beta \hslash$, $K_E(q,q)=\sqrt{m\omega /[2\pi \hslash \sinh x]}\exp [-m\omega q^2\tanh (x/2)/\hslash ]$, $x=\beta \hslash \omega$. Integrating the common endpoint yields $[2\sinh (x/2){]}^{-1}$.

A classical oscillator of amplitude $A$ has energy $m{\omega }^2{A}^2/2$. Comparing with the oscillator spacing $\hslash \omega$ gives the large-occupation scale displayed. A coherent state with this classical amplitude has exactly this mean excitation occupation and a Poisson number distribution; its total mean energy also includes the zero-point term. An observed macroscopic amplitude does not, by itself, establish a pure coherent state.

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.