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Quantum optics studies light using quantum states of the electromagnetic field and their interaction with matter. Photon statistics asks how detections fluctuate; the quantum theory of optical coherence relates those detections to field correlations. A coherent state, thermal light, and a Fock state can have the same mean photon number yet different pair counts. This distinction makes photon counting useful for testing sources intended to emit one photon at a time.

A Floquet Hamiltonian generates the evolution over one period of a time-periodic Quantum Hamiltonian. For period $T$, choose $H_F$ such that $U(T,0)=e^{-i{H}_{F}T/\hslash }$. Its eigenvalues are quasienergies, defined modulo $\hslash \omega$ with $\omega =2\pi /T$.

Taking a logarithm of the same unitary permits different branches, so the quasienergy convention must be specified. A complete description within the period also needs the periodic micromotion; the stroboscopic generator alone does not determine every intermediate observable. At sufficiently high drive frequency, an inverse-frequency expansion can approximate $H_F$, but resonances or long-time heating can limit its validity. The construction is useful for separating repeated-cycle dynamics from motion within a cycle.

By Barton Zwiebach, explicitly credited as B. Zwiebach on PDF p. 1, dated 4 November 2013. Chapter 6 notes in Quantum Physics II (8.05), Fall 2013, listed for lectures 12–16. Massachusetts Institute of Technology issues the course; MIT OpenCourseWare hosts this authored document.

Online text: ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/79a8091ef4f18d8e3ff76a097e8db33c_MIT8_05F13_Chap_06.pdf

Cited passages: §§2.1–2.2, printed pp. 10–14; §2.3, printed pp. 15–17, equations (2.45), (2.46) and (2.52).

By Richard Fitzpatrick. Page footer dated 22 January 2016; cited equations (7.57), (7.65).

By Richard Fitzpatrick. Page footer dated 22 January 2016; cited §7.6, equations (7.82)–(7.94).

An energy level is an allowed energy eigenvalue of a Quantum Hamiltonian. Its corresponding energy eigenstates satisfy $H|n⟩=E_n|n⟩$. Several independent energy eigenstates may share one energy level; their number is the energy-level degeneracy. A quantum superposition of different energy eigenstates need not have a definite energy, even though an ideal energy measurement yields one of the allowed outcomes.

Quantum confinement often produces discrete energy levels: the quantum harmonic oscillator has $E_n=\hslash \omega (n+1/2)$. An unconfined free particle instead has a continuous spectrum. In spectroscopy, a transition between two energy levels exchanges approximately $|E_f-E_i|=h\nu$ with a photon; recoil and external fields can affect the measured frequency. Selection rules determine which energy differences produce observable spectral lines.

Wave mechanics represents a quantum state by a wavefunction and observables by operators acting on it. For a nonrelativistic spinless particle in a potential energy $V$, the Schrödinger equation readsThe spatial shape and boundary conditions determine possible energy eigenfunctions; the Born rule gives their experimental interpretation as probability amplitudes.

For the infinite square well, vanishing at both walls selects standing sine waves and hence discrete energy levels. This explains quantization through a boundary-value problem rather than by imposing a separate allowed-orbit rule. Erwin Schrödinger's first 1926 paper developed quantization as an eigenvalue problem for the hydrogen atom. Matrix and wave mechanics equivalence explains how the same physical predictions arise from matrix mechanics in a suitable orthonormal basis.^[ref]

Wave-particle duality names the coexistence of localized detection events and wave-like interference in quantum mechanics. A photon or electron is detected as an individual event, yet the distribution of many events depends on probability amplitudes from alternative paths. If the alternatives remain coherent, adding the probability amplitudes before applying the Born rule gives an interference term; simply adding the separate path probabilities does not.

In the single-electron experiment of Akira Tonomura and collaborators, an electrostatic biprism divided and overlapped the electron wave, and a position-sensitive detector recorded individual arrivals. Successive accumulated records developed interference fringes. This makes the apparent tension concrete: localization describes the detector events, while matter-wave interference describes their statistical distribution.^[ref]

BKS theory was a 1924 attempt to describe light interacting with chemical atoms through continuous radiation fields, while allowing energy conservation and momentum conservation to hold statistically rather than in every individual atomic event. Its crucial experimental consequence was the absence of the event-by-event correlations expected from a photon collision picture of Compton scattering.

Walther Bothe and Hans Geiger tested whether a recoil electron and the photoelectron produced by the scattered radiation appeared together. Their coincident detections contradicted the predicted independent event times. In his 1925 response, John C. Slater explicitly discussed this conflict and also the observed correlation of the scattered radiation's direction with that of the recoil electron. BKS theory is therefore a useful example of testing a physical interpretation through individual-event correlations, even when average radiation intensities alone do not decide between pictures.^[ref]

A Landau-Zener transition occurs when a changing parameter sweeps two coupled quantum states through an avoided crossing. In a two-state description,the uncoupled energy levels cross at $t=0$, but the coupled energy eigenvalues have minimum separation $2|\Delta |$. The adiabatic theorem predicts following an instantaneous energy eigenstate during a sufficiently slow sweep; a fast sweep can retain the original uncoupled-state character and thus change instantaneous energy eigenstate.

For an ideal linear sweep extending from $t=-\infty$ to $+\infty$, the probability of remaining in the initial uncoupled state is $\exp [-2\pi |\Delta {|}^2/(\hslash |\alpha |)]$. The exponent compares the square of the coupling with the rate of change of the uncoupled energy difference. This model helps estimate when a molecular passage through an avoided crossing or a controlled qubit sweep ceases to obey the adiabatic theorem.

For an isolated quantum system, pure states are normalized rays in a complex Hilbert space, observables are self-adjoint operators, and their probabilities obey the Born rule. A spectral projector $P_a$ has outcome probability $⟨\psi |P_a|\psi ⟩$; an ideal repeatable projective measurement has conditional state $P_a\psi /\sqrt{⟨\psi |P_a|\psi ⟩}$. The Schrödinger equation specifies continuous unitary evolution between such ideal measurements, given a Hamiltonian operator. General measurements need not have the projective state-update rule.

Composite systems use the tensor product of their state spaces. Identical particles additionally obey the bosonic or fermionic exchange symmetry appropriate to their species. The photoelectric effect, Davisson–Germer experiment and double-slit experiment motivate energy quanta, matter-wave phases and probability interference, respectively; these experiments constrain the framework but do not separately prove every abstract postulate or specify every system's Hamiltonian.

Second quantization describes identical-particle quantum states in Fock space using creation operators and annihilation operators. Relative to an orthonormal basis of one-particle states, a one-body operator becomes ${\sum }_{\alpha \beta }{h}_{\alpha \beta }{a}_{\alpha }^{†}{a}_{\beta }$. The canonical commutation relation implements bosonic statistics, while the canonical anticommutation relations implement fermionic statistics. Expanding the quantized field in a complete basis gives a delta-function equal-time bracket; restricting the expansion to finitely many modes replaces that delta function by the corresponding projection operator kernel.

Quantum confinement restricts a particle's wavefunction over a length comparable with its characteristic quantum length scales, altering the allowed energy levels relative to a bulk material. For a parabolic band confined over width $L$, a characteristic spacing is of order ${\hslash }^2/(m^{\ast }{L}^2)$, with boundary-dependent numerical factors. A quantum dot confines all three directions; confinement in only one direction can leave continuous motion in the other two. Finite barrier height and interactions modify the simplest infinite-wall estimates.

A virtual particle is an internal contribution to a perturbative scattering amplitude, commonly represented by an internal line of a Feynman diagram. The line contributes a propagator, and its internal four-momentum is summed or integrated as required by the diagram. It need not satisfy the on-shell relativistic energy-momentum relation of an external detected particle.

For example, a virtual photon connects charged currents in electron scattering. This is a calculational contribution to the full interaction amplitude, rather than a directly observed photon flying between the electrons. Energy conservation and momentum conservation still hold at the vertices; off-shell propagation does not mean energy has been borrowed in violation of conservation. Individual virtual contributions can depend on the gauge or perturbative organization, whereas an observable prediction requires the appropriate complete amplitude.

A quantum dot of InAs embedded in gallium arsenide confines injected electrons and electron holes. Strain-driven island formation supplies dots with lateral dimensions of order tens of nanometres and heights of a few nanometres. In an electrically driven implementation, a dot lies in a junction's intrinsic region. Selecting one dot spatially and one radiative line spectrally turns discrete exciton recombination into a single-photon source. Pulsed injection must avoid re-excitation during the same collection window.

The pioneering electrically driven device used an areal dot density around $5\times 10^8\mathrm{c}{\mathrm{m}}^{-2}$, a GaAs p-i-n structure and an aperture selecting individual-dot light.^[ref]

Rounding a voltage to the nearest level in a uniform grid of spacing $q$ gives a quantization error of magnitude at most $q/2$, provided the input is within the converter's range. This follows because each grid point represents the interval bounded by the midpoints to its two neighbors. Clipping invalidates the bound.

The bound concerns recorded samples, whereas the Nyquist–Shannon sampling theorem concerns reconstruction from exact samples of a bandlimited function. If a reconstruction is $\tilde{v}(t)={\sum }_n{Q}_q(v_n)w_n(t)$, its error is ${\sum }_n{ϵ}_n{w}_n(t)$, so a general bound involves $(q/2){\sum }_n|w_n(t)|$, not simply $q/2$. Thus sampling bandwidth and per-sample rounding precision are distinct specifications. Under an additional uniform independent-error model, the quantization-noise standard deviation is $q/\sqrt{12}$, an rms statement rather than a worst-case reconstruction guarantee.

A stop-transfer sequence interrupts ongoing passage of a polypeptide chain across a membrane and retains a membrane-spanning segment. Protein targeting, insertion and subsequent sorting involve distinguishable sequence instructions.^[ref]

In the figure's cleaved-leader example, assume the N-terminus of the mature chain has entered the ER lumen, a stop-transfer sequence remains in the membrane, and the remaining C-terminus is made on the cytosol side. The outcome differs from releasing the entire mature chain into the ER lumen. The latter is soluble cargo, potentially retained there or transported onward; entry alone does not establish secretion.