The given problem requires proving the formula for the nucleon's gyromagnetic ratio [tex]\( g_j \)[/tex]:
[tex]\[
g_j = g_s \frac{j(j+1)-l(l+1)+s(s+1)}{2j(j+1)} + g_l \frac{j(j+1)+l(l+1)-s(s+1)}{2j(j+1)}
\][/tex]
We are provided with the specific case where [tex]\( j = l + s \)[/tex]. By analyzing this, we can see if there is a simplification that occurs under this condition.
Firstly, let's define the quantum numbers involved:
- [tex]\( j \)[/tex] is the total angular momentum,
- [tex]\( l \)[/tex] is the orbital angular momentum, and
- [tex]\( s \)[/tex] is the spin angular momentum.
According to the given values and their relationships, when [tex]\( j = l + s \)[/tex], we use the quantum mechanical relationships for the magnitudes of these angular momenta:
[tex]\[
j^2 = l^2 + s^2 + 2\mathbf{l} \cdot \mathbf{s}
\][/tex]
From which we can deduce:
[tex]\[
j(j+1) = l(l+1) + s(s+1) + 2\mathbf{l} \cdot \mathbf{s}
\][/tex]
Next, consider the requirement for the magnetic moment formula involving the following terms:
[tex]\[
j(j+1), \quad l(l+1), \quad \text{and}, \quad s(s+1)
\][/tex]
Under the condition [tex]\( j = l + s \)[/tex]:
- For term 1:
[tex]\[
j(j+1) - l(l+1) + s(s+1) \rightarrow \text{substitute } j(j+1):
\][/tex]
[tex]\[
[l(l+1) + s(s+1) + 2ls] - l(l+1) + s(s+1) = s(s+1) + s(s+1) + 2ls
\][/tex]
[tex]\[
= 2s(s+1) + 2\mathbf{l} \cdot \mathbf{s}
\][/tex]
- For term 2:
[tex]\[
j(j+1) + l(l+1) - s(s+1) \rightarrow \text{substitute } j(j+1):
\][/tex]
[tex]\[
[l(l+1) + s(s+1) + 2\mathbf{l} \cdot \mathbf{s}] + l(l+1) - s(s+1)
\][/tex]
[tex]\[
= l(l+1) + l(l+1) + 2\mathbf{l} \cdot \mathbf{s}
\][/tex]
[tex]\[
= 2l(l+1) + 2\mathbf{l} \cdot \mathbf{s}
\][/tex]
Finally, consolidate these values back into the formula [tex]\( g_j \)[/tex]:
\begin{align}
g_j &= \left( g_s \frac{2\mathbf{s(s+1)} + 2\mathbf{l} \cdot \mathbf{s}}{2j(j+1)} \right) + \left(g_l \frac{2\mathbf{l(l+1)} + 2\mathbf{l} \dot \mathbf{s} }{2j(j+1)} \right)
\end{align}
Since [tex]\( j(j+1) = l(l+1) + s(s+1) + 2 \mathbf{l} \cdot \mathbf{s} \)[/tex], you can identify part of these values simplifications step-by-step.
We can conclude specific simplifications:
[tex]\[
g_j = \frac{1}{2}(g_l + g_s), \quad \text{if} \quad j = l + s
\][/tex]
Thus the formula for [tex]\( g_j \)[/tex] highly depends on the specific [tex]\(j=l+s\)[/tex]. This provides simplicity as
[tex]\[
g_j = \frac{g_l + g_s}{2}
\][/tex]
This far simplifies the original formula into manageable factor context and adjustments.
