Spatial quantization

In quantum mechanics, spatial quantization is the quantization of angular momentum in three-dimensional space. It results from the fact that the angular momentum of a rigid rotor 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 operators 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 cone 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 {-2, -1, 0, 1, 2}, 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 mechanics, 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.

References

* Thomas Engel. "Quantum Chemistry and Spectroscopy." ISBN 0-8053-3843-8. Pearson Education, 2006. Pages 124-126.