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Definition in classical mechanics

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Orbital angular momentum in two dimensions edit Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum p is proportional to mass m and linear speed v , p = m v , {\displaystyle p=mv,} angular momentum L is proportional to moment of inertia I and angular speed ω measured in radians per second. L = I ω . {\displaystyle L=I\omega .} Unlike mass, which depends only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation and the shape of the matter. Unlike linear velocity, which does not d
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Analogy to linear momentum

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Angular momentum can be described as the rotational analog of linear momentum. Like linear momentum it involves elements of mass and displacement. Unlike linear momentum it also involves elements of position and shape. Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product, ( amount of inertia ) × ( amount of displacement ) = amount of (inertia⋅displacement) mass × velocity = momentum m × v = p {\displaystyle {\begin{aligned}({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\t
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Angular momentum in quantum mechanics

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Angular momentum in quantum mechanics differs in many profound respects from angular momentum in classical mechanics. In relativistic quantum mechanics, it differs even more, in which the above relativistic definition becomes a tensorial operator. Spin, orbital, and total angular momentum edit The classical definition of angular momentum as L = r × p {\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} } can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator. L is then an operator, specifically called the orbital angular momentum operator . The components of the angular momentum operator satisfy the commutation relations of the Lie algebra so(3). Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum Hilbert space. (See also the discussion below of the angular momentum operators as the generators of rotations.) However, in quantum physi
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