# Quantization, Space

## Quantization, Space

in quantum mechanics, the discreteness of the possible spatial orientations of the angular momentum of an atom (or another particle or system of particles) with respect to any arbitrarily selected axis (the *z*-axis).

Space quantization is manifested in that the projection *M*_{z} of the angular momentum *M* on this axis may only assume discrete values equal to an integer (0, 1, 2, …) or a half-integer (1/2, 3/2, 5/2, …) *m* multiplied by the Planck constant *h*, *M*_{z} = *mh*. The other two projections of angular momentum *M*_{x} and *M*_{y} remain indeterminate, since, according to the main principle of quantum mechanics, only the magnitude of the angular momentum and one of its projections can simultaneously have exact values. For the orbital angular momentum, the quantity *m* (*m _{l}*) can assume the values 0, ±

^{1}, ±

^{2}, …, ±

^{l}, where l = 0, 1, 2, … determines the square of the momentum

*M*

_{l}(that is, its absolute magnitude):

*M*

^{2}

_{1}= l (l + 1)

*h*

^{2}. For the total angular momentum

*M*(orbital plus spin momentum),

*m*(

*m*

_{j}) assumes values separated by unity—

*j*to +

*j*, where

*j*determines the magnitude of the total momentum:

*M*

^{2}=

*j*(

*j*’ + 1)

*h*

^{2}; it may be an integer or half-integer.

If an atom is placed in an external magnetic field H, then a well-defined direction in space—the direction of the field (which is taken as the z-axis)—appears. In this case, space quantization leads to quantization of the projection *μ*_{H} of the magnetic moment of the atom jui on the direction of the field, since the magnetic moment is proportional to the mechanical angular momentum (hence the name of *m* —magnetic quantum number). This leads to splitting of the energy levels of the atom in a magnetic field, since the energy of the atom’s magnetic interaction with the field, equal to *μ*_{H}*H*, is added to its energy (*see*ZEEMAN EFFECT).

V. I. GRIGOR’EV