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I can't seem to reconcile a part of the vector model of spin and some of its operators.

to quote wiki just above the "Bohr model" section:

http://en.wikipedia.org/wiki/Vector_model_of_the_atom#Mathematical_background_of_angular_momenta

"2.The magnitude of the vectors must be constant (for a specified state corresponding to the quantum number),

so the two indeterminate components of each of the vectors must be confined to a circle,

in such a way that the measurable and un-measurable components (at an instant of time)

allow the magnitudes to be constructed correctly, for all possible indeterminate components."

I would assume pythagorean theorem holds true and "the two indeterminate components (S_x & S_y)

of each of the vectors must be confined to a circle" of radius r:

magnitude = sqrt(s(s+1))

r = sqrt(s(s+1) - m^2)

however the operators are defined as:

S+ = sqrt(s(s+1) - m(m+1))

S- = sqrt(s(s+1) - m(m-1))

http://en.wikipedia.org/wiki/Spin_(physics)#Spin_operator

http://en.wikipedia.org/wiki/Anti-symmetric_operator#Spin

I must be missing something obvious. Please help.

A visual of spin 5/2 with some notes:

http://dl.dropbox.com/u/13155084/SPIN/SPIN-5-2-ladder-crop.png [Broken]

to quote wiki just above the "Bohr model" section:

http://en.wikipedia.org/wiki/Vector_model_of_the_atom#Mathematical_background_of_angular_momenta

"2.The magnitude of the vectors must be constant (for a specified state corresponding to the quantum number),

so the two indeterminate components of each of the vectors must be confined to a circle,

in such a way that the measurable and un-measurable components (at an instant of time)

allow the magnitudes to be constructed correctly, for all possible indeterminate components."

I would assume pythagorean theorem holds true and "the two indeterminate components (S_x & S_y)

of each of the vectors must be confined to a circle" of radius r:

magnitude = sqrt(s(s+1))

r = sqrt(s(s+1) - m^2)

however the operators are defined as:

S+ = sqrt(s(s+1) - m(m+1))

S- = sqrt(s(s+1) - m(m-1))

http://en.wikipedia.org/wiki/Spin_(physics)#Spin_operator

http://en.wikipedia.org/wiki/Anti-symmetric_operator#Spin

I must be missing something obvious. Please help.

A visual of spin 5/2 with some notes:

http://dl.dropbox.com/u/13155084/SPIN/SPIN-5-2-ladder-crop.png [Broken]

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