Detailed theory library
This library is designed for deep learning: concepts, equations, physical interpretation, and links to interactive panels that turn passive reading into active reasoning.
A quantum state is a vector in a complex Hilbert space. In position representation it becomes a wavefunction psi(x), with |psi(x)|^2 interpreted as probability density after normalization. Superposition is not ignorance; it is linear structure that produces interference.
Unitary time evolution is generated by the Hamiltonian. The time-dependent equation predicts how amplitudes flow; the time-independent equation finds stationary energy eigenstates. Potentials shape spectra, tunneling, bound states, and scattering.
Measurable quantities correspond to Hermitian operators. Eigenvalues are possible outcomes, while commutators encode whether two observables can be sharply known together. The uncertainty principle follows from this operator structure.
The Born rule maps amplitudes to probabilities. Projection describes ideal measurements, while decoherence explains how environmental entanglement suppresses observable interference between macroscopic alternatives.
Spin-1/2 systems are the natural doorway to qubits. A pure qubit can be visualized on the Bloch sphere, where rotations represent unitary gates and measurement along an axis returns probabilistic outcomes.
Composite systems live in tensor-product spaces. Entanglement appears when the full state cannot be factored into independent parts. Bell inequalities show that quantum correlations cannot be explained by local hidden variables.
Exact solutions are rare. Perturbation theory handles small changes to solvable Hamiltonians, variational methods estimate ground states, and WKB-style semiclassical methods explain tunneling and phase accumulation.
Fermions and bosons obey different exchange symmetries, creating Pauli exclusion, Bose condensation, band theory, superconductivity, topological phases, and entanglement growth in complex systems.
Real devices interact with environments. Density matrices describe mixed states, decoherence channels, relaxation, dephasing, and the practical limits of quantum computation and sensing.
Quantum information uses controllable superposition and entanglement for algorithms, cryptography, metrology, and simulators for molecules, materials, and high-energy models.
QFT treats particles as excitations of fields. It unifies quantum mechanics with special relativity and provides the language for particle physics, condensed matter effective theories, and modern entanglement research.
Continuous improvement
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