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}>}

Failed to parse (syntax error): {\displaystyle \\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 $|0,{\frac {1}{2}},{\frac {1}{2}}>$ as a linear combination of $|j_{1},j_{2},j_{3};+,+,->$ and $|j_{1},j_{2},j_{3};-,+,+>$

$|0,{\frac {1}{2}},{\frac {1}{2}}>=\alpha |j_{1},j_{2},j_{3};+,+,->+\beta |j_{1},j_{2},j_{3};-,+,+>$

Since $|0,{\frac {1}{2}},{\frac {1}{2}}>$ is normalized, while $({\hat {J_{1}}}+{\hat {J_{2}}}+{\hat {J_{3}}})|j_{1},j_{2},j_{3};+,+,->$ and $({\hat {J_{1}}}+{\hat {J_{2}}}+{\hat {J_{3}}})|j_{1},j_{2},j_{3};-,+,+>$, and since Clebsch-Gordan coefficients, such as $\alpha$ and $\beta$, are real numbers, above equation yields

$\alpha ^{2}+\beta ^{2}=1$

On the other hand, since $<1,{\frac {3}{2}},{\frac {1}{2}}|0,{\frac {1}{2}},{\frac {1}{2}}>=0$, hence we get,

${\frac {1}{\sqrt {3}}}(\alpha +\beta )$ $\Rightarrow$ $\alpha =-\beta$

A substitution of $\alpha =-\beta$ gives $\alpha =-\beta ={\frac {1}{\sqrt {2}}}$