Position operator

The quantum observable for a particle’s spatial location. In the position basis, it acts by multiplying a wavefunction by the coordinate.

Position representation: A representation of a quantum state as a wavefunction over spatial coordinates. In this representation, the position operator acts by coordinate multiplication.
Observable: A measurable physical quantity represented in quantum mechanics by a self-adjoint operator. Position is one such quantity, with measurement outcomes tied to its spectrum.
Momentum operator: The quantum observable for linear momentum, represented by a spatial derivative in the position basis. Unlike position’s multiplication action, momentum differentiates the position-space wavefunction.
Wavepacket: A localized quantum state formed by superposing waves with different momenta or frequencies. Its spatial profile is evaluated through the position operator’s representation.
Position eigenstate: A generalized eigenstate of the position operator with a definite coordinate eigenvalue. Its eigenvalue identifies the location associated with an idealized position measurement.
Hilbert space: A complete inner-product vector space used to represent quantum states. Position operators act on state vectors in a suitable Hilbert space.
Position measurement: A quantum measurement that produces a spatial outcome according to a system’s position probability distribution. The operator formalism connects its possible outcomes to the position spectrum.
Quantum harmonic oscillator: A quantum system with a potential energy proportional to the square of displacement from equilibrium. Its position operator connects energy eigenstates to spatial motion and observables.
Position eigenvalue: A possible measurement value of the position operator, corresponding to a spatial coordinate. The operator’s spectrum supplies the coordinates that position measurements can return.
Eigenvalue: A scalar associated with a linear operator and a nonzero vector that the operator maps to a scalar multiple of itself. Position measurements correspond to eigenvalues of the position operator.

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