Main Page/CG coefficients of 3 spin half particles

Problem

(a) Find the total spin of a syatem of three spin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}} particles and derive the corresponding Clebsch-Gordan Coefficients.

(b) Consider a system of three nonidentical spin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}} particles whose Hamiltonian is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}=-\epsilon _{0}(\vec{S}_{1}.\vec{S}_{3}+\vec{S}_{2}.\vec{S}_{3})/\hbar^{2}} . Find the system's energy levels and their degeneracies.

Solution:-

(a) To add Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_{1}=\frac{1}{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_{2}=\frac{1}{2}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_{3}=\frac{1}{2}} , we begin by coupling Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_{1}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_{2}} to form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_{12}=j_{1}+j_{2}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{1}-j_{2}|\leq j_{12}\leq |j_{1}+j_{2}|} , hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_{12}=0,1} . Then we add jFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{12}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_{3}} ; this leads to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{12}-j_{3}|\leq j\leq |j_{12}+j_{3}|}

We are going to denote the joint eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{1}}^{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{2}}^{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{3}}^{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{12}}^{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J}^{2}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_{z}} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{12},j,m> } and the joint eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{1}}^{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{2}}^{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{3}}^{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{1_{z}}}^{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{2_{z}}}^{2}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{3_{z}}}^{2}} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{1},J_{2},j_{3};m_{1},m_{2},m_{3}>} ; since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_{1}=j_{2}=j_{3}=\frac{1}{2}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{1}=\pm \frac{1}{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{2}=\pm \frac{1}{2}} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{3}=\pm \frac{1}{2}} ,we will be using throughout this problem the lighter notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{1},j_{2},j_{3};\pm ,\pm,\pm> } to abbreviate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\frac{1}{2},\frac{1}{2},\frac{1}{2};\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2}> }

In total there are eight states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{12}, j, m>} since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2j_{1}+1)(2j_{2}+1)(2j_{3}+1)=8} . Four of these correspond to the subspace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j=\frac{3}{2}:|1,\frac{3}{2},\frac{3}{2}>,|1,\frac{3}{2},\frac{1}{2}>,|1,\frac{3}{2},-\frac{1}{2}>,|1,\frac{3}{2},-\frac{3}{2}>} . The remaining four belong to the subspace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j=\frac{1}{2}:|0,\frac{1}{2},\frac{1}{2}>,|0,\frac{1}{2},-\frac{1}{2}>,|1,\frac{1}{2},\frac{1}{2}>,|1,\frac{1}{2},-\frac{1}{2}>} . To construct the states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{12},j,m> } in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{1},j_{2},j_{3};\pm,\pm,\pm} , we are going to consider the two subspaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j=\frac{3}{2}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j=\frac{1}{2}} separately.

Subspace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j=\frac{3}{2}}

First, the states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{3}{2},\frac{3}{2}>} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{3}{2},-\frac{3}{2}>} are clearly given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{3}{2},\frac{3}{2}>=|j_{1},j_{2},j_{3};+,+,+> } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{3}{2},-\frac{3}{2}>=|j_{1},j_{2},j_{3};-,-,-> }

To obtain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{3}{2},\frac{1}{2}>} , we need to apply, on the one hand, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{-}}} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{3}{2},\frac{3}{2}>}

${\hat {J_{-}}}|1,{\frac {3}{2}},{\frac {3}{2}}>=\hbar {\sqrt {{\frac {3}{2}}\left({\frac {3}{2}}+1\right)-{\frac {3}{2}}\left({\frac {3}{2}}-1\right)}}|1,{\frac {3}{2}},{\frac {3}{2}}>=\hbar {\sqrt {3}}|1,{\frac {3}{2}},{\frac {1}{2}}>$

and, on the other hand, apply $({\hat {J_{1-}}}+{\hat {J_{2-}}}+{\hat {J_{3-}}})$ on $|j_{1},j_{2},j_{3};+,+,+>$

This yields

$({\hat {J_{1-}}}+{\hat {J_{2-}}}+{\hat {J_{3-}}})|j_{1},j_{2},j_{3};+,+,+>=\hbar \left(|j_{1},j_{2},j_{3};-,+,+>+|j_{1},j_{2},j_{3};+,-,+>+|j_{1},j_{2},j_{3};+,+,->\right)$

Since ${\sqrt {{\frac {1}{2}}\left({\frac {1}{2}}+1\right)-{\frac {1}{2}}\left({\frac {1}{2}}-1\right)}}=1$. Equating above equations, we get

$|1,{\frac {3}{2}},{\frac {1}{2}}>={\frac {1}{\sqrt {3}}}\left(|j_{1},j_{2},j_{3};-,+,+>+|j_{1},j_{2},j_{3};+,-,+>+|j_{1},j_{2},j_{3};+,+,->\right)$

Following the same method-applying ${\hat {J_{-}}}$ on $({\hat {J_{1-}}}+{\hat {J_{2-}}}+{\hat {J_{3-}}})$ on the right hand side of above equation and then equating the two results-we find

$|1,{\frac {3}{2}},-{\frac {1}{2}}>={\frac {1}{\sqrt {3}}}\left(|j_{1},j_{2},j_{3};+,-,->+|j_{1},j_{2},j_{3};-,+,->+|j_{1},j_{2},j_{3};-,-,+>\right)$

Subspace--j=1/2

We can write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0,\frac{1}{2},\frac{1}{2}>} as a linear combination of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{1},j_{2},j_{3};+,+,-> } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{1},j_{2},j_{3};-,+,+> }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0,\frac{1}{2},\frac{1}{2}>=\alpha |j_{1},j_{2},j_{3};+,+,-> +\beta |j_{1},j_{2},j_{3};-,+,+> }

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0,\frac{1}{2},\frac{1}{2}>} is normalized, while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\hat{J_{1}}+\hat{J_{2}}+\hat{J_{3}})|j_{1},j_{2},j_{3};+,+,->} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\hat{J_{1}}+\hat{J_{2}}+\hat{J_{3}})|j_{1},j_{2},j_{3};-,+,+>} , and since Clebsch-Gordan coefficients, such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} , are real numbers, above equation yields

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha ^{2}+\beta ^{2}=1}

On the other hand, since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <1,\frac{3}{2},\frac{1}{2}|0,\frac{1}{2},\frac{1}{2}>=0} , hence we get,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\sqrt{3}}(\alpha +\beta )} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha =-\beta }

A substitution of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha =-\beta } gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha =-\beta =\frac{1}{\sqrt{2}}} , and substituting this we get,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0,\frac{1}{2},\frac{1}{2}>= \frac{1}{\sqrt{2}}\left ( |j_{1},j_{2},j_{3};+,+,-> - |j_{1},j_{2},j_{3};-,+,+> \right )}

Following the same procedure i.e. applying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{-}}} on the left of above equation and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\hat{J_{1-}}+\hat{J_{2-}}+\hat{J_{3-}})} on the right hand side and then equating the two results-we find

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0,\frac{1}{2},-\frac{1}{2}>=\frac{1}{\sqrt{2}}\left (- |j_{1},j_{2},j_{3};+,-,->+|j_{1},j_{2},j_{3};-,-,+> \right )}

Now to find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{1}{2},\frac{1}{2}>} , we may write it as a linear combination of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{1},j_{2},j_{3};+,+,->} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{1},j_{2},j_{3};+,-,+>} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_{1},j_{2},j_{3};-,+,+>}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{1}{2},\frac{1}{2}>=\alpha |j_{1},j_{2},j_{3};+,+,-> +\beta |j_{1},j_{2},j_{3};+,-,+> } +\gamma |j_{1},j_{2},j_{3};-,+,+>

This state is orthogonal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0,\frac{1}{2},-\frac{1}{2}>} , and hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha =\gamma }  ; similarly, since this state is also orthogonal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{3}{2},\frac{1}{2}>} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha +\beta +\gamma =0} , and hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\alpha +\beta =0} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta =-2\alpha =-2\gamma } . Now, since all the states of above equation are orthonormal, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha ^{2}+\beta ^{2}+\gamma ^{2}=1} , which when combined with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta =-2\alpha =-2\gamma } leads to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha =\gamma =-\frac{1}{\sqrt{6}} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta =\frac{2}{\sqrt{6}}} . We may thus write as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{1}{2},\frac{1}{2}>=\frac{1}{\sqrt{6}}\left ( -|j_{1},j_{2},j_{3};+,+,-> +2|j_{1},j_{2},j_{3};+,-,+> -|j_{1},j_{2},j_{3};-,+,+> \right )}

Finally, applying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{J_{-}}} on the left side of the above equation and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\hat{J_{1-}}+\hat{J_{2-}}+\hat{J_{3-}})} on the right hand side and equating the two results, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,\frac{1}{2},-\frac{1}{2}>=\frac{1}{\sqrt{6}}\left ( |j_{1},j_{2},j_{3};+,-,-> -2|j_{1},j_{2},j_{3};-,+,-> -|j_{1},j_{2},j_{3};-,-,+> \right )}

(b) Since we have three different (nonidentical) particles, their spin angular momentum mutually commute. We may thus write Hamiltonian as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}=-\frac{\epsilon _{0}}{\hbar^{2}}\left ( \hat{S_{1}} + \hat{S_{2}} \right ).\hat{S_{3}}} . Due to this suggestive form of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle hat{H}} , it is appropriate, as shown in (a), to start by coupling ${\hat {S_{1}}}$ with ${\hat {S_{2}}}$ to obtain ${\hat {S_{12}}}={\hat {S_{1}}}+{\hat {S_{2}}}$, and then add ${\hat {S_{12}}}$ to ${\hat {S_{3}}}$ to generate the total spin; ${\hat {S}}={\hat {S_{12}}}+{\hat {S_{3}}}$. We may thus write ${\hat {H}}$ as

${\hat {H}}=-{\frac {\epsilon _{0}}{\hbar ^{2}}}\left({\hat {S_{1}}}+{\hat {S_{2}}}\right).{\hat {S_{3}}}=-{\frac {\epsilon _{0}}{\hbar ^{2}}}{\hat {S_{12}}}.{\hat {S_{3}}}=-{\frac {\epsilon _{0}}{2\hbar ^{2}}}\left({\hat {S}}^{2}-{\hat {S_{12}}}^{2}-{\hat {S_{3}}}^{2}\right)$

since ${\hat {S_{12}}}.{\hat {S_{3}}}={\frac {1}{2}}[({\hat {S_{12}}}+{\hat {S_{3}}})^{2}-{\hat {S_{12}}}^{2}-{\hat {S_{3}}}^{2}]$. Since the operator ${\hat {H}},{\hat {S}}^{2},{\hat {S_{12}}}^{2}$, and ${\hat {S_{3}}}^{2}$ mutually commute , we may select as their joint eigenstates the kets $|{s_{12}},s,m>$; we have seen in (a) how to construct these states. The eigenvalues of ${\hat {H}}$ are thus given by

${\hat {H}}|s_{12},s,m>=-{\frac {\epsilon _{0}}{\hbar ^{2}}}({\hat {S}}^{2}-{\hat {S_{12}}}^{2}-{\hat {S_{3}}}^{2})|s_{12},s,m>=-{\frac {\epsilon _{0}}{\hbar ^{2}}}[s(s+1)-s_{12}(s_{12}+1-{\frac {3}{4}})]|s_{12},s,m>$

since $s_{3}={\frac {1}{2}}$ and ${\hat {S_{3}}}^{2}|s_{12},s,m>=\hbar ^{2}s_{3}(s_{3}+1)|s_{12},s,m>=({\frac {3\hbar ^{2}}{4}})|s_{12},s,m>$

As shown in above, the energy levels of this system are degenerate with respect to m, since they depend on the quantum numbers $s$ and $s_{12}$ but not m:

$E_{12,s}=-{\frac {\epsilon _{0}}{\hbar ^{2}}}[s(s+1)-s_{12}(s_{12}+1-{\frac {3}{4}})]$

For instance, the energy $E_{12,s}=E_{1,3/2}=-{\frac {\epsilon _{0}}{2}}$ is fourfold degenerate, since it corresponds to four different states: $|s_{12},s,m>=|1,{\frac {3}{2}},\pm {\frac {3}{2}}>$ and $|1,{\frac {3}{2}},\pm {\frac {1}{2}}>$. Similarly, the energy $E_{0,1/2}=0$ is two fold degenerate; the corresponding states are $|0,{\frac {1}{2}},\pm {\frac {1}{2}}>$. Finally, the energy $E_{1,1/2}=\epsilon _{0}$ is also two fold degenerate since it corresponds to $|1,{\frac {1}{2}},\pm {\frac {1}{2}}>$.

(Reference taken from Quantum mechanics Concepts and Applications by Nouredine Zetilli.)