Position space (also real space or coordinate space) is the set of all position vectors r of an object in space (usually 3D). The position vector defines a point in space. If the position vector varies with time it will trace out a path or surface, such as the trajectory of a particle.
Momentum space or k-space is the set of all wavevectors k, associated with particles - free and bound. The terms "momentum" (symbol p, also a vector) and "wavevector" are used interchangeably due to the De Broglie relation p = ħk, meaning they are equivalent up to proportionality, although this is not true in a crystal, see below.
This is an example of Pontryagin duality.
The position vector r has dimensions of length, the k-vector has dimensions of reciprocal length, so k is the frequency analogue of r, just as angular frequency ω is the inverse quantity and frequency analogue of time t. Physical phenomena can be described using either the positions of particles, or their momenta, both formulations equivalently provide the same information about the system in consideration. Usually r is more intuitive and simpler than k, though the converse is also true, such as in solid-state physics.
Read more about Position Space: Position and Momentum Spaces in Quantum Mechanics, Relation Between Space and Reciprocal Space, Unitary Equivalence Between Position and Momentum Operator, Reciprocal Space and Crystals, See Also
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