Why does |s=1,m=1> equal |++> ?

[Ethan0718]

Sorry...I know it's a very stupid problem .. =.=

But i just don't know why...

ihttp://imageshack.us/photo/my-images/594/imagewyt.jpg/

it's mentioned at p205 of Modern Quantum Mechanics by J.J.Sakurai...＝＝＝＝＝＝＝

sorry... i found i post at wrong place after i received the PM from PF mentor...

please forgive me @@... i won't do this again!..

 

[SpectraCat]

Sorry...I know it's a very stupid problem .. =.=

But i just don't know why...

ihttp://imageshack.us/photo/my-images/594/imagewyt.jpg/

it's mentioned at p205 of Modern Quantum Mechanics by J.J.Sakurai...

Because it is defined that way. You are talking about the total spin angular momentum and its projection for a system consisting of two spin-1/2 particles. Spin-up or "+" denotes a projection of +1/2 and spin-down "-" denotes a projection of -1/2. If you consider the case where both particles are spin-up (i.e. |++>), then the total spin is 1, and the projection m=(1/2)+(1/2)=1. If you consider the case where both are spin-down (i.e. |-->), then the total spin is still 1, but the projection is now m=(-1/2)+(-1/2)=-1.

Does that help?

 

[Ethan0718]

Thank you very much...but... I still have some questions >"<

i know the following information

S² = (S₁+S₂)² : s(s+1) * h'
S_Z = S_1Z + S_2Z : m * h'
S_1Z : m₁ * h'
S_2Z : m₂ * h'

h' means h bar

I can deduce m = 1 from the fact that |++> means two spin up ; 1/2 + 1/2 = 1

but i don't know how to know s = 1... @@

 

[SpectraCat]

Thank you very much...but... I still have some questions >"<

i know the following information

S² = (S₁+S₂)² : s(s+1) * h'
S_Z = S_1Z + S_2Z : m * h'
S_1Z : m₁ * h'
S_2Z : m₂ * h'

h' means h bar

I can deduce m = 1 from the fact that |++> means two spin up ; 1/2 + 1/2 = 1

but i don't know how to know s = 1... @@

Spins are angular momenta, so you have to follow the rules for their addition. Specifically, any two angular momenta j₁ and j₂, can add together to give a resultant total angular momentum in the set, {(j₁+j₂, j₁+j₂ - 1, j₁+j₂ - 2, ... , j₁- j₂+1, j₁ - j₂}.

In QM, angular momenta can often be treated "as if" they are vectors (although they are *not* vectors), particularly with respect to coupling (addition) of multiple angular momenta. Note that the above expression is just the usual triangle inequality for vector addition, with the additional stipulation that angular momentum is quantized such that the smallest allowable step size is hbar.

For this case, we have s₁=s₂=1/2, so possible values for s _{are 0 and 1.}