What is electron spin?

Electron spin is an intrinsic property of an electron in which it exists in what is known as two distinct spin states, one known as "spin up" and the other, "spin down". Contrary to what you might expect, it is not actually as if it were literally spinning and precessing in a classical manner.

Basically, from the Pauli Exclusion Principle, an electron cannot exist in an orbital with the same spin as the other electron.

We depict spin-up and spin-down for electrons as:

#m_s = +"1/2":# #ul(uarr color(white)(darr))#

#m_s = -"1/2":# #ul(color(white)(uarr) darr)#

There exists a z-spin operator, #hatS_z#, analogous to the z-angular momentum operator, #hatL_z#, that returns an eigenvalue #m_sℏ#, or #pmℏ"/"2#, similar to how #hatL_z# returns #m_lℏ#, or #{-l, -l + 1, . . . , -1, 0, +1, . . . , l - 1, l}ℏ#.

That is:

#hatS_(iz) vecalpha(i) = ℏ/2vecalpha(i)#
#hatS_(iz) vecbeta(i) = -ℏ/2vecbeta(i)#

where #vecalpha(i)# and #vecbeta(i)# are spin-up and spin-down eigenfunctions, respectively, for electron #i#, while #hatS_(iz)# operates only on electron #i#.

Or, in implied units of #ℏ#, we would report these eigenvalues to mean that:

#m_s = pm"1/2"#

or that the electron spin can be #+"1/2"# or #-"1/2"#. We depict that as:

#m_s = +"1/2":# #ul(uarr color(white)(darr))#

#m_s = -"1/2":# #ul(color(white)(uarr) darr)#