A little problem with charge operator

[idontkonw]

I have a problem where it's said that the operator Q is likely to be:

$Q=\sum^3_{i=1}[\frac{1}{2}B_i + I_{3,i}]$

I have to apply this to the proton wave function which is the same as you can see in equation (3.20) here: https://www.google.es/url?sa=t&rct=...mYGwBQ&usg=AFQjCNFeF1HkxpOSTjbWuly9_EJcuTEWeQ

I have only the formula for the first equal. If I apply it because B number is the same for all of them this contribution is 0 and applying the isospin I get also 0 so I have that <p|Q|p>=0 which I assume it's wrong because Q=1. What do I do wrong?

Also, in this formula (3.20) how do they get the second equal? I mean the sum of this terms. I suppose all is about transpose operator but I'm getting quite confused with this.

 

[mfb]

You should get 1 both for "u up u down d up" and for "u up u up d down". They have to be the same (and the same as Q for the whole proton), otherwise the proton would not have a well-defined charge.

 

[idontkonw]

mmm let me explain how I'm doing this. Let's take the last way of 3.20:


$Q(|u\uparrow u\downarrow d\uparrow> -2|u\uparrow u\uparrow d\downarrow>)= \left[ \left(\frac{1}{2}\frac{1}{3} + \frac{1}{2}\right)+\left(\frac{1}{2}\frac{1}{3} + \frac{1}{2}\right)+\left(\frac{1}{2}\frac{1}{3} - \frac{1}{2}\right)\right] -2 \left[ \left(\frac{1}{2}\frac{1}{3} + \frac{1}{2}\right)+\left(\frac{1}{2}\frac{1}{3} + \frac{1}{2}\right)+\left(\frac{1}{2}\frac{1}{3} - \frac{1}{2}\right)\right](|u\uparrow u\downarrow d\uparrow> -2|u\uparrow u\uparrow d\downarrow>)=1-2=-1$

So yes each part is 1 but because of -2 I get -1. Shouldn't be the result 1?

 

[mfb]

You cannot subtract/multiply with prefactors like that.

Your "real" calculation is $\langle \psi|Q|\psi \rangle$. If you expand this, each summand is evaluated independent of the others, and their prefactors are squared. As result, you get $\left(\frac{1}{\sqrt{5}}\cdot 1\right)^2 \cdot 1 + \left(\frac{1}{\sqrt{5}}\cdot 2\right)^2 \cdot 1 = 1$ where I corrected $\sqrt{3} \to \sqrt{5}$.

If you know that the proton has a well-defined charge, both components have to have the same charge, so you can ignore all prefactors and evaluate a single component only with the formula in post 1.

 

[idontkonw]

Oh man, I'm feeling really really genius... I think it's time to stop studying. Enought for today. What a big mistake. All for no writing long expressions...

Thanks a lot!