Understanding Isospin in Neutron/Proton System: T & T3

[malawi_glenn]

This is from Krane, p 389:

The neutron and the proton are treated as two different states of a single particle, the nucleon. The nucleon is assigned with a fictious spin vector, called isospin.

Nucleon has isospin number t = ½, a proton has m_{t} = 1/2 and neutron has m_{t} = - 1/2.

The isospin obeys the same rules for angular momentum vecotrs.

The third component of a nucleus isospin is:
T_{3} = \frac{1}{2} (Z-N)

For any value on T_{3}, the total isospin T can take any value at least as great as |T_{3} | [/tex].<br /> <br /> We consider as an example the two-nucleon system, which can have T of 0 or 1. There are thus four possible 3-axis components: T_{3} = 1(two protons); T_{3} = - 1(two neutrons), and two combinations with T_{3} = 0(one neutron and one proton). The first two states must have T = 1, while the latter two can have T = 0 and T =1.<br /> <br /> - - - <br /> <br /> Now this is really confusing me. I am think that the according to the statement: <i>For any value on</i> &quot; T_{3}, <i>the total isospin</i> T <i>can take any value at least as great as </i>|T_{3} | [/tex].&amp;quot; The two proton system can therefore have T = 0 or 1. And the same thing regarding the 2N system.&lt;br /&gt; &lt;br /&gt; And also how can there be two combinations of P-N that gives T_{3} = 0? And why isn&amp;#039;t just T = 0 allowed?&lt;br /&gt; &lt;br /&gt; Should I try to think &amp;quot;backwards&amp;quot;: &lt;i&gt;Given a value on T, what values of T_{3} can I have, and what combinations of N and P do they represent?&lt;/i&gt;&lt;br /&gt; &lt;br /&gt; Cheers&lt;img src=&quot;https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f644.png&quot; class=&quot;smilie smilie--emoji&quot; loading=&quot;lazy&quot; width=&quot;64&quot; height=&quot;64&quot; alt=&quot;:rolleyes:&quot; title=&quot;Roll Eyes :rolleyes:&quot; data-smilie=&quot;11&quot;data-shortname=&quot;:rolleyes:&quot; /&gt;

 

[Meir Achuz]

"The two proton system can therefore have T = 0 or 1. And the same thing regarding the 2N system."

The two p system has T_3=+1, so T cannot equal zero.
The two n system has T_3=-1, so T cannot equal zero.

 

[Meir Achuz]

T_3=0 can come from the two different combinations
(pn+np)/sqrt{2} for T=1, and
(pn-np)/sqrt{2} for T=0.

 

[malawi_glenn]

okay, I think I got it now. Thanx a lot dude! =)