Neutron to proton transition matrix element

Question

I am reading the book Advances on Nuclear Physics vol 13 by J. w. Negele and Erich Vogt.
On page 33, one is going to calculate a matrix element corresponding to a transition of a neutron to a proton.
I am having a hard time going from equation 3.29 to equation 3.30
$$t_{pn}= _{s-f}langle p |sum _ifrac{tau_i}{2}|nrangle_{s-f} tag{3.29}$$
"where the subscript indicates a spin-flavor matrix element only"
$$t_{pn}= langle p |tau_i/2|nrangle tag{3.30}$$
Can anyone guide me through the calculation between 3.29 and 3.30?

cmmigl · Accepted Answer

Well I have managed to understand  this.
Basically one must act with $tau$ in every quark of the neutron wave function (https://wikimedia.org/api/rest_v1/media/math/render/svg/d01d80212401fe5d60053ac7dd8ffd0816c7e748 you can check it here)
Since we are comparing matrix elements, one can only use the $tau+$.
the isospin operator $tau+$ will either transform a d quark to a u quark or give  0 when acting on a u quark.
Once one acts in the neutron wave function, we project on the proton wave function (https://wikimedia.org/api/rest_v1/media/math/render/svg/d89ecb243a57bedbf3043da030615a1f733847c3)
Equation 3.30 we just act with the $tau+$ operator on a neutron state, getting:
$tau |nrangle=-|prangle$
Don't forget the $1/2$ factor and the project this result on $|prangle$
Once everything is computed one can see that 3.29 and 3.30 wield the same result

Neutron to proton transition matrix element

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