Momentum space | Wikipedia audio article
Channel: wikipedia tts
Date: 29/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Position_and_momentum_space


00:02:11 1 Position and momentum spaces in classical mechanics
00:02:33 1.1 Lagrangian mechanics
00:02:45 1.2 Hamiltonian mechanics
00:02:58 2 Position and momentum spaces in quantum mechanics
00:13:20 3 Relation between space and reciprocal space
00:14:20 3.1 Functions and operators in position space
00:15:57 3.2 Functions and operators in momentum space
00:16:42 4 Unitary equivalence between position and momentum operator
00:21:11 5 Reciprocal space and crystals
00:25:10 6 See also
00:25:44 7 Footnotes
00:27:06 8 References



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SUMMARY
=======
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general could be any finite number of dimensions.
Position space (also real space or coordinate space) is the set of all position vectors r in space, and has dimensions of length. A position vector defines a point in space. If the position vector of a point particle varies with time it will trace out a path, the trajectory of a particle. Momentum space is the set of all momentum vectors p a physical system can have. The momentum vector of a particle corresponds to its motion, with units of [mass][length][time]−1.
Mathematically, the duality between position and momentum is an example of Pontryagin duality. In particular, if a function is given in position space, f(r), then its Fourier transform obtains the function in momentum space, φ(p). Conversely, the inverse transform of a momentum space function is a position space function.
These quantities and ideas transcend all of classical and quantum physics, and a physical system can be described using either the positions of the constituent particles, or their momenta, both formulations equivalently provide the same information about the system in consideration. Another quantity is useful to define in the context of waves. The wave vector k (or simply "k-vector") has dimensions of reciprocal length, making it an analogue of angular frequency ω which has dimensions of reciprocal time. The set of all wave vectors is k-space. Usually r is more intuitive and simpler than k, though the converse can also be true, such as in solid-state physics.
Quantum mechanics provides two fundamental examples of the duality between position and momentum, the Heisenberg uncertainty principle ΔxΔp ≥ ħ/2 stating that position and momentum cannot be simultaneously known to arbitrary precision, and the De Broglie relation p = ħk which states the momentum and wavevector of a free particle are proportional to each other. In this context, when it is unambiguous, the terms "momentum" and "wavevector" are used interchangeably. However, the De Broglie relation is not true in a crystal.