| nonrelativistic quantum mechanics | |
|---|---|
| Name | Nonrelativistic quantum mechanics |
| Field | Quantum mechanics |
| Introduced | 1920s |
| Notable figures | Erwin Schrödinger, Werner Heisenberg, Paul Dirac, Max Born, Niels Bohr |
nonrelativistic quantum mechanics
Nonrelativistic quantum mechanics is the quantum theory describing particles and systems at speeds much less than the speed of light, using the Schrödinger equation and operator methods. It provides the foundation for understanding atoms, molecules, and low-energy condensed matter phenomena and forms an essential component of modern Quantum Physics and physical chemistry.
The theory is built on a small set of postulates formalized in the 1920s by figures associated with the Copenhagen interpretation, including Niels Bohr, Werner Heisenberg, and Max Born. State vectors live in a separable complex Hilbert space and evolve unitarily under the Schrödinger picture via the time-dependent Schrödinger equation. Observables correspond to self-adjoint operators; measurement outcomes are eigenvalues with probabilities given by the Born rule. The formalism emphasizes conservation laws derived from symmetries via Noether's theorem and uses the Hamiltonian as the generator of time evolution. Foundational debates involve the measurement problem, contrasting approaches such as de Broglie–Bohm theory and objective-collapse models, and pragmatic treatments in laboratories like Los Alamos National Laboratory and CERN where nonrelativistic approximations often apply in low-energy regimes.
The core mathematical structures include Hilbert space, linear operators, spectral theory, and functional analysis techniques developed in institutions such as Princeton University and Cambridge University. Wavefunctions ψ(x,t) represent states in the position basis; momentum-space representations use the Fourier transform. The eigenvalue problem for the Hamiltonian reduces to time-independent formulations, and operator algebra is handled via commutators and the canonical commutation relations [x,p]=iħ. Techniques from Sturm–Liouville theory and special functions (e.g., Hermite polynomials, Laguerre polynomials, Spherical harmonics) solve many textbook problems. Mathematical rigor was advanced by workers like John von Neumann and Paul Dirac, while numerical methods draw on work at Bell Labs and computational centers.
Classic solvable models include the particle in a box, the harmonic oscillator, and the hydrogen atom, whose exact solutions underpin atomic spectroscopy and are tied to the work of Erwin Schrödinger and Arnold Sommerfeld. The finite potential well, delta potential, and scattering from simple potentials provide pedagogical cases for bound and scattering states; the Lippmann–Schwinger equation and partial wave analysis address scattering theory. Exactly solvable models also touch on supersymmetric quantum mechanics and the algebraic methods introduced by groups such as École Normale Supérieure researchers. These models are essential for interpreting spectra from experiments at institutions like the National Institute of Standards and Technology (NIST).
Practical problems require approximation schemes: time-independent and time-dependent perturbation theory, the variational method, and the WKB approximation. Rayleigh–Schrödinger perturbation theory and degenerate perturbation theory treat small corrections to solvable Hamiltonians; the Fermi Golden Rule describes transition rates induced by perturbations. The Born–Oppenheimer approximation separates electronic and nuclear motion in molecules, foundational to computational chemistry programs developed at IBM and in academic groups. Numerical diagonalization, Hartree–Fock methods, and density functional approaches approximate many-electron systems, while semiclassical methods connect to classical mechanics via the Ehrenfest theorem.
Nonrelativistic theory incorporates spin as an intrinsic degree of freedom introduced by Samuel Goudsmit and George Uhlenbeck. Identical particles obey symmetry constraints: Fermi–Dirac statistics for fermions and Bose–Einstein statistics for bosons, leading to the Pauli exclusion principle and phenomena like Bose–Einstein condensation observed at labs such as MIT and JILA. Second quantization and the occupation number representation provide efficient formalisms for many-body problems, connecting to models such as the Hubbard model and Heisenberg model central to condensed matter physics.
Measurement theory in nonrelativistic quantum mechanics centers on projection postulates and decoherence, with contributions from researchers like Wojciech Zurek. Interpretational frameworks include the Copenhagen interpretation, Many-worlds interpretation, and hidden variable theories; experimental tests of foundations have been pursued using Bell's theorem and Bell test experiments at laboratories including University of Innsbruck and Imperial College London. The correspondence principle and semiclassical approximations explain the classical limit; the role of coherent states and the WKB approximation illuminate how classical trajectories emerge from quantum dynamics.
Nonrelativistic quantum mechanics underpins atomic and molecular structure theory, spectroscopy, chemical bonding, and reaction dynamics, with practical tools such as Molecular orbital theory and computational packages developed in academic and industrial settings. In condensed matter, it explains band structure via Bloch's theorem, effective mass, and transport; key institutions include Stanford University, University of Cambridge, and corporate research labs like Bell Labs. Phenomena such as superconductivity, the quantum Hall effect (in its low-energy descriptions), and magnetism are modeled using many-body nonrelativistic techniques. Technologies from semiconductor devices to laser cooling and quantum dots trace design principles to nonrelativistic quantum mechanics and its extensions into quantum information science.