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Avodyne
#2
Dec6-07, 02:19 AM
Sci Advisor
P: 1,204
If we label the eigenstates of [itex]S_z[/itex] as [itex]|{+}\rangle[/itex] and [itex]|{-}\rangle[/itex], so that [itex]S_z|{\pm}\rangle=\pm\frac12 |{\pm}\rangle[/itex], then
[tex]S_+|{-}\rangle=|{+}\rangle[/tex]
[tex]S_-|{+}\rangle=|{-}\rangle[/tex]
Also,
[tex]S_+|{+}\rangle=0[/tex]
[tex]S_-|{-}\rangle=0[/tex]
That is, [itex]S_+[/itex] raises the value of [itex]S_z[/itex], and [itex]S_-[/itex] lowers it. That is how the raising and lowering operators are defined, and [itex]S_x \pm i S_y[/itex] is just what they work out to be.