Spatial quantization
Encyclopedia
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 Lz (the z-component of angular momentum) simultaneously because these two quantum mechanical operators
commute. However, it is not possible to know Lx and Ly, 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.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, spatial quantization is the quantization
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
of angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
in three-dimensional
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
space. It results from the fact that the angular momentum of a rigid rotor
Rigid rotor
The rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
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 Lz (the z-component of angular momentum) simultaneously because these two quantum mechanical operators
Operator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
commute. However, it is not possible to know Lx and Ly, 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
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...
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
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, 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.