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

$$t_{pn}= _{s-f}\langle p |\sum _i\frac{\tau_i}{2}|n\rangle_{s-f} \tag{3.29}$$

"where the subscript indicates a spin-flavor matrix element only"

$$t_{pn}= \langle p |\tau_i/2|n\rangle \tag{3.30}$$

Can anyone guide me through the calculation between 3.29 and 3.30?

Answer 1

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:

$\tau |n\rangle=-|p\rangle$

Don't forget the $1/2$ factor and the project this result on $|p\rangle$

Once everything is computed one can see that 3.29 and 3.30 wield the same result