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Momentum is the Noether charge of translational invariance. As such, even fields as well as other things can have momentum, not just particles. However, in curved spacetime which is not asymptotically Minkowski, momentum isn't defined at all.
In classical mechanics, momentum (traditionally written as p) is defined as the product of mass and velocity. It is thus a vector quantity.
The change in momentum, called the impulse, is equal to force times the change in time.
The SI unit of momentum is newton-seconds, which can alternatively be expressed with the units kg·m/s.
An impulse changes the momentum of an object. An impulse is calculated as the integral of force with respect to duration.
using the definition of force yields:
See also angular momentum.
It is commonly believed that the physical laws should be invariant under translationEuclidean geometry In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shiftings. Thus, the definition of momentum was changed when EinsteinAlbert Einstein ( March 14 1879 April 18 1955) was a theoretical physicist who is widely regarded as the greatest scientist of the 20th century. He proposed the theory of relativity and also made major contributions to the development of quantum mechanics formulated Special relativitySpecial relativity (SR or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. It replaced Newtonian notions of space and time, and incorporated electromagnetism as represented by Maxwell's equations. The theory is so that its magnitude would remain invariant under relativistic transformations. See physical conservation law. We now define a vector, called the 4-momentum thus:
where E is the total energy of the system, and p is called the "relativistic momentum" defined thus:
where
Setting velocity to zero, one derives the result that objects have a rest mass which is related by the expression
The "length" of the vector that remains constant is defined thus:
Massless objects such as photonFor the Japanese anime video, see Photon (anime). In physics, the photon (from Greek φοτος, meaning light is a quantum of excitation of the quantised electromagnetic field and is one of the elementary particles studied by qus also carry momentum; the formula is p=E/c, where E is the energy the photon carries and c is the speed of light.
In quantum mechanics momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics position and momentum are interchangeble.
For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as
where is the gradient operator. This is a commonly encountered form of the momentum operator, though not the most general one.
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