Editing Angular momentum (section)

=== Quantization ===
{{Further|Angular momentum operator}}
In [[quantum mechanics]], angular momentum is [[angular momentum quantization|quantized]] – that is, it cannot vary continuously, but only in "[[Quantum number|quantum leaps]]" between certain allowed values. For any system, the following restrictions on measurement results apply, where <math>\hbar</math> is the [[reduced Planck constant]] and <math>\hat n</math> is any [[Euclidean vector]] such as x, y, or z:
{| class="wikitable"
|-
| '''If you [[measurement in quantum mechanics|measure]]...'''
| '''The result can be...'''
|-
| <math>L_\hat{n}</math>
| <math>\ldots, -2\hbar, -\hbar, 0, \hbar, 2\hbar, \ldots</math>
|-
| <math>S_\hat{n}</math> or <math>J_\hat{n}</math>
| <math>\ldots, -\frac{3}{2}\hbar, -\hbar, -\frac{1}{2}\hbar, 0, \frac{1}{2}\hbar, \hbar, \frac{3}{2}\hbar, \ldots</math>
|-
| <math>\begin{align}
      &L^2 \\
  ={} &L_x^2 + L_y^2 + L_z^2
\end{align}</math>
| <math>\left[\hbar^2 n(n + 1)\right]</math>, where <math>n = 0, 1, 2, \ldots</math>
|-
| <math>S^2</math> or <math>J^2</math>
| <math>\left[\hbar^2 n(n + 1)\right]</math>, where <math>n = 0, \tfrac{1}{2}, 1, \tfrac{3}{2}, \ldots</math>
|}
[[File:Circular Standing Wave.gif|thumb|right|In this [[standing wave]] on a circular string, the circle is broken into exactly 8 [[wavelength]]s. A standing wave like this can have 0, 1, 2, or any integer number of wavelengths around the circle, but it ''cannot'' have a non-integer number of wavelengths like 8.3. In quantum mechanics, angular momentum is quantized for a similar reason.]]
The [[reduced Planck constant]] <math>\hbar</math> is tiny by everyday standards, about 10<sup>−34</sup> [[Joule-second|J s]], and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure of [[electron shell]]s and subshells in chemistry is significantly affected by the quantization of angular momentum.

Quantization of angular momentum was first postulated by [[Niels Bohr]] in [[Bohr model|his model]] of the atom and was later predicted by [[Erwin Schrödinger]] in his [[Schrödinger equation#Quantization|Schrödinger equation]].