Main Page/CG coefficients of 3 spin half particles: Difference between revisions
SuvadipDas (talk | contribs) (New page: Problem (a) Find the total spin of a syatem of three spin <math>\frac{1}{2}</math> particles and derive the corresponding Clebsch-Gordan Coefficients. (b) Consider a system of three noni...) |
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We are going to denote the joint eigenstates of <math>\hat{J_{1}}^{2}</math>,<math>\hat{J_{2}}^{2}</math>,<math>\hat{J_{3}}^{2}</math>,<math>\hat{J_{12}}^{2}</math>,<math>\hat{J}^{2}</math> and <math>J_{z}</math> by <math>|j_{12},j,m> </math> and the joint eigenstates of <math>\hat{J_{1}}^{2}</math>,<math>\hat{J_{2}}^{2}</math>,<math>\hat{J_{3}}^{2}</math>,<math>\hat{J_{1_{z}}}^{2}</math>,<math>\hat{J_{2_{z}}}^{2}</math> and <math>\hat{J_{3_{z}}}^{2}</math> by <math>|j_{1},J_{2},j_{3};m_{1},m_{2},m_{3}></math>; since <math>j_{1}=j_{2}=j_{3}=\frac{1}{2}</math> and <math>m_{1}=\pm \frac{1}{2}</math>,<math>m_{2}=\pm \frac{1}{2}</math>,<math>m_{3}=\pm \frac{1}{2}</math>,we will be using throughout this problem the lighter notation <math>|j_{1},j_{2},j_{3};\pm ,\pm,\pm> </math> to abbreviate <math>|\frac{1}{2},\frac{1}{2},\frac{1}{2};\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2}> </math> | We are going to denote the joint eigenstates of <math>\hat{J_{1}}^{2}</math>,<math>\hat{J_{2}}^{2}</math>,<math>\hat{J_{3}}^{2}</math>,<math>\hat{J_{12}}^{2}</math>,<math>\hat{J}^{2}</math> and <math>J_{z}</math> by <math>|j_{12},j,m> </math> and the joint eigenstates of <math>\hat{J_{1}}^{2}</math>,<math>\hat{J_{2}}^{2}</math>,<math>\hat{J_{3}}^{2}</math>,<math>\hat{J_{1_{z}}}^{2}</math>,<math>\hat{J_{2_{z}}}^{2}</math> and <math>\hat{J_{3_{z}}}^{2}</math> by <math>|j_{1},J_{2},j_{3};m_{1},m_{2},m_{3}></math>; since <math>j_{1}=j_{2}=j_{3}=\frac{1}{2}</math> and <math>m_{1}=\pm \frac{1}{2}</math>,<math>m_{2}=\pm \frac{1}{2}</math>,<math>m_{3}=\pm \frac{1}{2}</math>,we will be using throughout this problem the lighter notation <math>|j_{1},j_{2},j_{3};\pm ,\pm,\pm> </math> to abbreviate <math>|\frac{1}{2},\frac{1}{2},\frac{1}{2};\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2}> </math> | ||
In total there are eight states <math>|j_{12},j,m></math> since <math>(2j_{1}+1)(2j_{2}+1)(2j_{3}+1)=8</math>. Four of these correspond to the subspace | In total there are eight states <math>|j_{12}, j, m></math> since <math>(2j_{1}+1)(2j_{2}+1)(2j_{3}+1)=8</math>. Four of these correspond to the subspace <math>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}></math>. The remaining four belong to the subspace <math>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}></math>. To construct the states <math>|j_{12},j,m> </math> in terms of <math>|j_{1},j_{2},j_{3};\pm,\pm,\pm</math>, we are going to consider the two subspaces <math>j=\frac{3}{2}</math> and <math>j=\frac{1}{2}</math> separately. | ||
Subspace <math>j=\frac{3}{2}</math> | |||
First, the states <math>|1,\frac{3}{2},\frac{3}{2}></math> and <math>|1,\frac{3}{2},-\frac{3}{2}></math> are clearly given by | |||
<math>|1,\frac{3}{2},\frac{3}{2}>=|j_{1},j_{2},j_{3};+,+,+> </math>, <math>|1,\frac{3}{2},-\frac{3}{2}>=|j_{1},j_{2},j_{3};-,-,-> </math> | |||
To obtain | |||
Revision as of 19:07, 26 April 2010
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