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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 .
Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem . [ 25 ]
The "time" axis gives the angular frequency (rad⋅s −1) and the "space" axis represents the angular wavenumber (rad⋅m −1). Green and indigo represent left and right polarization. In empty space, the photon moves at c (the speed of light) and its energy and momentum are related by E = pc, where p is the magnitude of the momentum vector p.
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects, such as how different observers perceive where and when events ...
That is, the probability of finding an electron at a given place is also a function of the electron's average momentum at that point, since high electron momentum at a given position tends to "localize" the electron in that position, via the properties of electron wave-packets (see the Heisenberg uncertainty principle for details of the mechanism).
The kinetic momentum, a gauge invariant physical quantity, can be expressed in terms of the canonical momentum, the scalar potential φ and vector potential A: [6] ^ = The expression above is called minimal coupling. For electrically neutral particles, the canonical momentum is equal to the kinetic momentum.
The momentum space for the theory is another copy of the same algebra and there is a certain 'braided addition' of momentum on it expressed as the structure of a braided Hopf algebra or quantum group in a certain braided monoidal category).
Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms. [2]: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science.