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Magnetic quantum number
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== As a component of angular momentum == [[File:Vector model of orbital angular momentum.svg|250px|right|thumb|Illustration of quantum mechanical orbital angular momentum. The cones and plane represent possible orientations of the angular momentum vector for <math>\ell = 2</math> and <math>m_l = -2, -1, 0, 1, 2</math>. Even for the extreme values of <math>m_l</math>, the <math>z</math>-component of this vector is less than its total magnitude.]] The axis used for the polar coordinates in this analysis is chosen arbitrarily. The quantum number <math>m_l</math> refers to the projection of the angular momentum in this arbitrarily-chosen direction, conventionally called the <math>z</math>-direction or [[quantization axis]]. <math>L_z</math>, the magnitude of the angular momentum in the <math>z</math>-direction, is given by the formula:<ref name=h50/> :<math>L_z = m_l \hbar</math>. This is a component of the atomic electron's total orbital angular momentum <math>\mathbf{L}</math>, whose magnitude is related to the azimuthal quantum number of its subshell <math>\ell</math> by the equation: :<math>L = \hbar \sqrt{\ell (\ell + 1)}</math>, where <math>\hbar</math> is the [[reduced Planck constant]]. Note that this <math>L = 0</math> for <math>\ell = 0</math> and approximates <math>L = \left( \ell + \tfrac{1}{2} \right) \hbar</math> for high <math>\ell</math>. It is not possible to measure the angular momentum of the electron along all three axes simultaneously. These properties were first demonstrated in the [[Stern–Gerlach experiment]], by [[Otto Stern]] and [[Walther Gerlach]].<ref>{{cite web|url=http://www.britannica.com/science/spectroscopy/Types-of-electromagnetic-radiation-sources#ref620216|title=Spectroscopy: angular momentum quantum number|publisher=Encyclopædia Britannica}}</ref>
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