Spin angular momentum

The electron, as shown in the Stern-Gerlach experiment, has an additional angular momentum called a spin angular momentum apart from an orbital angular momentum.

The spin angular momentum of an electron is referred to as an intrinsic angular momentum of the electron because it is in certain ways similar to classical spin (e.g. it is associated with a magnetic dipole moment) but is purely a quantum mechanical effect, i.e. it is erroneous to view the electron spinning about its own axis.

The theory of quantum mechanics postulates that spin angular momentum operators \small \hat{S}^{2},\hat{S}_x,\hat{S}_y,\hat{S}_z are linear and Hermitian, and obey the same commutation relations as described by eq99, eq100 and eq101, i.e.:

\small \left [\hat{S}_x,\hat{S}_y\right ]=i\hbar\hat{S}_z\; \; \; \; \; \; \; \; 165

\small \left [\hat{S}_y,\hat{S}_z\right ]=i\hbar\hat{S}_x\; \; \; \; \; \; \; \; 166

\small \left [\hat{S}_z,\hat{S}_x\right ]=i\hbar\hat{S}_y\; \; \; \; \; \; \; \; 167

Therefore, the corresponding equations of eq132, eq133 and eq148 are:

\small \hat{S}_z\vert s,m_s\rangle=m_s\hbar\vert s,m_s\rangle\; \; \; \; \; \; \; \; 168

\small \hat{S}^{2}\vert s,m_s\rangle=s(s+1)\hbar^{2}\vert s,m_s\rangle\; \; \; \; \; \; \; \; 169

\small \hat{S}_\pm\vert s,m_s\rangle=\hbar\sqrt{s(s+1)-m_s(m_s\pm1)}\vert s,m_s\pm1\rangle\; \; \; \; \; \; \; \; 170

respectively, where \small s=0,\frac{1}{2},1,\frac{3}{2},\cdots is the spin angular momentum quantum number; \small m_s=-s,-s+1,\cdots,s-1,s is the spin magnetic quantum number; \small \hat{S}_\pm=\hat{S}_x\pm i\hat{S}_y.

For an electron, \small s has been determined experimentally not to have a range of values but to be specifically equal to \small \frac{1}{2}. We say that electrons have spin \small \frac{1}{2}. Similarly, both protons and neutrons have spin \small \frac{1}{2}.



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