math concept
13 topics use this
Math concept
Quantum Mechanics
Core equation
$$i\hbar\frac{\partial}{\partial t}|\psi\rangle = H|\psi\rangle$$
Quantum mechanics is the fundamental theory of physics at the atomic and subatomic scale. Its mathematical framework — Hilbert spaces, operators, and unitary evolution — underpins quantum computing, computational chemistry, and condensed matter physics.
The postulates
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State: a quantum system is described by a state vector $ \psi\rangle$ in a complex Hilbert space $\mathcal{H}$. -
Evolution: $i\hbar\partial_t \psi\rangle = H \psi\rangle$ (Schrödinger equation), where $H$ is the Hamiltonian. -
Measurement: observable $A$ has outcomes equal to its eigenvalues ${a_n}$; probability of $a_n$ is $ \langle a_n \psi\rangle ^2$. -
Born rule: after measuring $a_n$, the state collapses to $ a_n\rangle$.
Wave-particle duality and the uncertainty principle
Position and momentum are Fourier conjugate variables. The Heisenberg uncertainty principle:
\[\sigma_x \sigma_p \geq \frac{\hbar}{2}\]is a mathematical consequence of the non-commutativity $[x, p] = i\hbar$.
Spin and qubits
| A spin-$\frac{1}{2}$ particle is the simplest quantum system: a two-level Hilbert space $\mathbb{C}^2$. A qubit state is $ | \psi\rangle = \alpha | 0\rangle + \beta | 1\rangle$ with $ | \alpha | ^2 + | \beta | ^2 = 1$ — the foundation of quantum computing. |
Fields that use this concept
Physical sciences
Astrophysics
Black Holes
Regions of spacetime where gravity is so extreme that nothing, not even light, can escape beyond the event horizon.
Nucleosynthesis
The nuclear processes that forge every element from hydrogen to uranium inside stars, supernovae, and the Big Bang.
Physical sciences
Computational chemistry
Born-Oppenheimer Approximation
The foundational separation of nuclear and electronic motion that underlies nearly all of computational chemistry.
Coupled Cluster Theory
The gold-standard wavefunction method based on an exponential cluster operator that systematically captures electron correlation.
Density Functional Theory
A quantum mechanical method that replaces the many-body wavefunction with the electron density as the fundamental variable.
Exchange-Correlation Functionals
The approximations to the unknown exchange-correlation energy in DFT, ranging from LDA to hybrid and dispersion-corrected functionals.
Hartree-Fock Theory
The foundational mean-field method for solving the electronic Schrödinger equation using antisymmetrised orbital products.
Path Integral Molecular Dynamics
Feynman's imaginary-time path integral formulation extended to finite-temperature MD, capturing nuclear quantum effects such as tunnelling and zero-point energy.
Perturbation Theory in Quantum Chemistry
Systematic expansion of energies and wavefunctions in powers of a small perturbation, including the Møller-Plesset treatment of electron correlation.
Physical sciences
Quantum computing
Quantum Annealing
A metaheuristic for combinatorial optimisation that uses quantum tunnelling to escape local minima in energy landscapes.
Quantum Key Distribution
Protocols for distributing cryptographic keys with information-theoretic security guaranteed by the laws of quantum mechanics.
Quantum Phase Estimation
A core quantum subroutine that extracts eigenphases of unitary operators, underpinning Shor's algorithm and quantum chemistry simulations.
Shor's Algorithm
A quantum algorithm for integer factorisation running in polynomial time, breaking RSA encryption.