I have a question regarding Dirac's notation in quantum physics.
As far as I understand: $\langle a|b\rangle=(a1^*,a2^*)*(b1,b2)^T$
But what does $\langle1/2,1/2|J|1/2,-1/2\rangle$ mean?
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If $J=\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ then $\langle x,y\mid J\mid z,t\rangle=axz+bxt+cyz+dyt$.
Did
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That's what I tought, but on my textbook it is written that, when $J=\begin{pmatrix}0 & 1 \ 0 & 0\end{pmatrix}$ , $\langle1/2,1/2|J|1/2,-1/2\rangle=sqrt(3/4+1/2*1/2)=1$ ... why is that? – Federico Scanagatta Mar 10 '13 at 16:53
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Which textbook? Which precise formulation? – Did Mar 10 '13 at 16:58
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Well it's an italian text-book.. page 290 of this pdf: http://www.studentifisica.info/corso/file-terzo/Meccanica-Quantistica/Meccanica-quantistica-Casalbuoni-2010semestrale.pdf , and it does regard spins.. but unfortunately apart from saying it's written in dirac's notation there's not much more. – Federico Scanagatta Mar 10 '13 at 17:01
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Equation (15.92) is the answer. – Did Mar 10 '13 at 17:07
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@Did: Excuse the basic question, but is this the same as $\binom{x}{y}^T J \binom{z}{t}$? I guess I am asking if this is mainly a notational device? – copper.hat Mar 10 '13 at 17:20
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Yes, with the exception that your text seems first to rewrite $\binom{x}{y}$ and $\binom{z}{t}$ as linear combinations of the two basis vectors, then to apply $J$. – Did Mar 10 '13 at 18:59