A finite-dimensional density operator $\rho$ represents a quantum state on a Hilbert space. It is Hermitian, positive semidefinite, and has trace one:The density matrix is its matrix in a chosen orthonormal basis. In dimension $N$ it has $N^2-1$ real parameters: a Hermitian matrix has $N^2$, and fixing its trace removes one; positivity restricts the allowed values.

A quantum state ensemble of normalized pure states $|{\psi }_i⟩$ with positive probabilities $p_i$ givesZero-weight terms can be omitted. Conversely, the spectral theorem gives such an ensemble for every density operator, using its orthonormal eigenvectors and nonnegative eigenvalues, with zero eigenvalues omitted. A pure state has $\rho =|\psi ⟩⟨\psi |$; a mixed state needs at least two nonzero eigenvalues. ^[ref]

For a projective measurement with orthogonal projections ${\Pi }_a$, the Born rule isThis is the average of the component pure state probabilities. Positivity makes each outcome probability nonnegative, and ${\sum }_a{\Pi }_a=I$ makes them sum to one. The same density operator therefore predicts every measurement on the system without requiring a unique preparation history. ^[ref]

For a closed system, Schrödinger picture evolution under a unitary operator iswhen ${\rho }_H={\rho }_S(0)$ denotes the time-independent Heisenberg picture state and $U(0)=I$. In the Schrödinger picture the density operator evolves; in the Heisenberg picture the observables evolve instead, with identical measurement predictions. This unitary operator preserves the eigenvalues and hence the purity of a density operator. ^[ref]

A historical source for the probabilistic formulation of quantum mechanics is von Neumann's 1927 paper Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. ^[ref]

Minimal coupling introduces a gauge potential through a covariant derivative, or replaces canonical momentum by $p-qA$. The associated scalar potential supplies the electric coupling. Magnetic minimal coupling is the magnetic-only nonrelativistic case.

The normalized Hadamard transform maps a vector indexed by binary strings through entries $2^{-n/2}(-1{)}^{x\cdot y}$, where the dot product is evaluated modulo two. It is its own inverse and an orthogonal transformation, hence also a unitary operator on complex input vectors. The Hadamard gate is the $n=1$ quantum case, and the Hadamard basis is the image of the computational basis. The Walsh-Hadamard transform gives the same tensor product construction.

For two qubits, applying one Hadamard gate to each prepares the uniform quantum superpositionEach probability amplitude is $1/2$, so each computational-basis measurement outcome has probability $1/4$. Applying the Hadamard transform again reconstructs $|00⟩$: signed contributions cancel at the other three outputs. Its usefulness lies in combining simple local quantum gates into an operation on exponentially many amplitudes. Extracting useful information still requires an appropriate subsequent transformation and measurement.

By Barton Zwiebach.

Online text: ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/d87c768c4e1181fde8d4366112169719_MIT8_04S16_LecNotes6.pdf

Cited passages: Lecture 6 by B. Zwiebach, section 3, p. 5, equations (3.4)-(3.6): probability continuity equation and probability current.

The Hartree equations are self-consistent one-particle equations obtained by approximating a many-body wave function by a tensor product of orbital wave functions. Each orbital sees the potential averaged over the other particles' probability densities; attractive Newtonian mean-field variants include the Gravitational Hartree equation.

Perturbation theory expands a quantum problem around a solvable Hamiltonian. Time-independent perturbation theory expands stationary energies and eigenstates; time-dependent perturbation theory expands dynamical amplitudes and transitions.

A rectangular potential barrier has a constant elevated potential over a finite interval and a lower constant potential outside it. Matching wavefunctions and derivatives at the interval endpoints gives its reflection and transmission amplitudes. It is a special potential barrier.

An energy level is degenerate when its eigenspace has dimension greater than one: several linearly independent energy eigenstates share the same energy eigenvalue. Any quantum superposition within that eigenspace is also an energy eigenstate with the same energy.

For a particle in a ring without applied flux, clockwise and anticlockwise momentum eigenstates have equal kinetic energy, except for the single zero-momentum state. A field or other perturbation may split such a pair if it distinguishes the two states. Degenerate perturbation theory identifies the combinations selected by a weak perturbation. The energy-level degeneracy counts independent states, rather than the number of different basis descriptions one can write for them.

A wavefunction is the position representation of a pure state. In the one-dimensional model used here, $\psi (x,t)$ takes values in the complex numbers, and the Born rule givesThe probability density $\rho$ has units of inverse length; the wavefunction has units of inverse square root of length. A value of $\rho$ at a point is not itself a probability: probabilities come from an integral over a region. For the infinite square well ground state, $\rho (L/2)=2/L$, whereas the probability of the exact single point $L/2$ is zero. The interesting question is how much probability a finite interval contains.

Two wavefunctions differing only by a global phase represent the same pure state. Relative quantum phases between components of a quantum superposition affect matter-wave interference, so retaining the full complex number amplitude is essential. This scalar description covers a particle without active spin degrees of freedom; a general electron wavefunction can have two spin components, with the position probability density obtained by adding the squared modulus of each component.

After expressing a driven Quantum Hamiltonian in a suitable interaction or rotating frame, discard terms oscillating rapidly compared with the retained dynamics. Near a selected transition, this leaves resonant terms that accumulate coherently while counter-rotating terms average out to leading order. The drive strength and detuning must be small relative to the discarded oscillation frequencies over the intended regime; strong driving can require the neglected frequency shifts and corrections. The approximation supports resonant quantum control, but it is not exact merely because the carrier is tuned to resonance.

Let $H$ be a self-adjoint Hamiltonian bounded below, $A$ a bounded positive operator and $p_A(t)=⟨e^{-iHt}\psi ,A{e}^{-iHt}\psi ⟩$. The Hegerfeldt theorem states that either $p_A(t)$ vanishes identically or it is nonzero for almost every real time. In particular, vanishing throughout an open interval forces identically vanishing. The proof uses analytic continuation of $A^{1/2}{e}^{-iHt}\psi$ into the lower half-plane. Applied to an assumed sharp localization effect, this obstructs strict finite-speed localization for positive-energy one-particle dynamics. It does not identify a controllable superluminal signal: the localization observables and admissible local preparation procedures need separate analysis.

A particle confined by a Dirichlet boundary condition to a disc of radius $a$ has angular functions $e^{im\varphi }$, $m\in \mathbb{Z}$, and regular radial Bessel functions of the first kind $J_{|m|}(kr)$. Their boundary zeros give the displayed energies. The ground state belongs to $m=0$: nonzero angular modes add a nonnegative centrifugal term to the kinetic-energy form. At the origin exclude the singular radial solution, rather than accepting every square-integrable logarithmic function as a physical eigenstate.

If basis information and test positions are announced before the receiver confirms receipt of quantum carriers, an eavesdropper can store the carriers until learning that information. In a scheme sending one half of each Bell pair after a Hadamard gate encoding, the eavesdropper then forwards test qubits untouched and measures all data qubits in their now-known correct bases. Every test passes, while data are changed to classically correlated product states whose bit labels the eavesdropper knows. This invalidates the sampling argument needed to justify entanglement purification and quantum key distribution. A sound additional purification certification could detect this attack and abort; it cannot generate private Bell pairs from the separable attacked data. Announcing bases only after receipt prevents this particular delayed-carrier attack.

The in-in formalism computes expectation values at a finite time by evolving a state forward and backward along a closed time contour. At first order in the interaction Hamiltonian,An $iϵ$ tilt of the early-time contour projects onto the interacting vacuum.

The joint energy-momentum spectrum lies in the closed forward light cone. In axiomatic quantum field theory, this condition permits analytic continuation of Wightman functions in suitable complex spacetime domains. Together with Lorentz invariance, microcausality and the positivity of the physical state space, it is one of the hypotheses behind the CPT theorem.

Algebraic quantum field theory assigns an observable algebra $\mathcal{A}(O)$ to each suitable spacetime region $O$. Inclusion of regions implies inclusion of algebras, and spacelike-separated algebras commute under microcausality. A state is a positive normalized functional on the observable algebra, so its restriction to a region specifies expectation values and correlations for that algebra. Spectrum conditions restrict translation representations, while primitive causality and time-slice axioms constrain the dynamics; these are different assumptions.

A quantum field is an operator-valued distribution on spacetime: smearing $\Phi (x)$ against a smooth test function gives an operator, whereas pointwise products can require renormalization. For example, a free complex scalar field contains particle annihilation operators and antiparticle creation operators. Its microcausality condition makes field commutators vanish at spacelike separation. For fermions, the corresponding locality condition uses anticommutators. This combines quantum operators with the spacetime locality of a quantum field theory.

Online text: docs.python.org/3.14/reference/simple_stmts.html

Online text: docs.python.org/3.14/reference/expressions.html

Cited passages: §6.11 Boolean operations: not, left operand evaluation, conditional right operand evaluation, returned operands.