Spatial quantization: Facts, Discussion Forum, and Encyclopedia Article

Spatial quantization

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	Spatial quantization
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	Topics Spatial quantization Topic Home In quantum mechanicsQuantum mechanics Quantum mechanics is a first quantized quantum theory that supersedes classical mechanics at the atomic and subatomic levels... , spatial quantization is the quantizationQuantization Quantization is the procedure of constraining something to a discrete set of values, such as an integer, rather than a conti... of angular momentumAngular momentum In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the... in three-dimensional3D 3D may refer to:... space. It results from the fact that the angular momentum of a rigid rotorRigid rotor The rigid rotor is a mechanical model that is used to explain rotating systems.... is expressed in three dimensions, and is quantized. For a rigid rotor, it is possible to know L² (the square of the magnitude of angular momentum) and L_z (the z-component of angular momentum) simultaneously because these two quantum mechanical operatorOperator In mathematics, an operator is a function, usually of a special kind depending on the topic.... s commute. However, it is not possible to know L_x and L_y, which are the other two components of angular momentum, simultaneously and exactly. With the magnitude and z-component of angular momentum exactly known, the angular momentum vector points from a single point at a certain angle, but it can end anywhere on a circle. The result is a coneCone (geometry) Overview In common usage and elementary geometry, a cone is a solid object obtained by rotating a right triangle around one of its tw... whose vertex is the origin of the vector, and whose height is the z-component. Since the x and y components are not known, the angular momentum can be represented by any of the vectors that comprise the cone. Spatial quantization results from the fact that only a small number of values for L are allowed in quantum mechanical systems. For example, if the rules of the system require that L be an integer in the set , then there are only five surfaces on which the angular momentum can be found: a flat circle corresponding to L = 0, and two cones above this circle for L = 1 and L = 2, and two cones below this circle for L = -1 and L = -2. In classical mechanicsClassical mechanics Classical mechanics is used to describe the motion of macroscopic objects, from projectiles to parts of machinery, as well a... , spatial quantization does not occur because a large number of values are allowed for L. As the number of allowed values for L approaches infinity, the number of imaginary cones approaches infinity, and the circles form an essentially continuous sphere, so that the momentum vector can be anywhere on the sphere. In quantum mechanics, the angular momentum can only lie on a small number of circles on the imaginary sphere.