Position and momentum spaces

Position and momentum spaces are conjugate representations of a quantum state, expressed as wavefunctions over position or momentum and related by a Fourier transform.

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Fourier transform: A mathematical transformation that decomposes a function into its component frequencies or spatial modes. It converts the position-space wavefunction into the momentum-space wavefunction.
Wavefunction: A complex-valued function that encodes the probability amplitudes of a quantum state in a chosen representation. The position and momentum wavefunctions are two representations of the same state.
Schrödinger equation: The equation governing how a nonrelativistic quantum state changes in time. Its form can be simpler in one representation depending on the system's Hamiltonian.
Energy representation: A quantum-state representation whose basis consists of energy eigenstates. It expands a state by energy rather than by position or momentum.
Momentum operator: The quantum operator whose measurement outcomes are momentum values. In position space, its action involves a spatial derivative; in momentum space, it acts by multiplication.
Quantum state: The mathematical object that specifies the probabilities of possible outcomes for a quantum system. A single state can be represented by either of these wavefunctions.
Free particle: An idealized particle subject to no forces or potential energy. Momentum-space states evolve especially simply when the particle is free.
Phase space: A space whose coordinates specify a system's position and momentum together. Classical phase space treats both variables as simultaneous coordinates, unlike quantum wavefunction representations.
Position operator: The quantum operator whose measurement outcomes are position values. In momentum space, its action involves a momentum derivative, unlike its multiplicative action in position space.
Position eigenstate: An idealized state with a definite position and a position wavefunction proportional to a Dirac delta distribution. It illustrates the limiting case of complete position localization.