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.