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Homework Statement
I need to show: [tex]S^2 \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)[/tex] has an eigenvalue of zero.
The attempt at a solution
[tex]S_1^2 \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)=\hbar^2 \frac{3}{4} \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right) [/tex]
[tex]S_2^2 \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)=\hbar^2 \frac{3}{4} \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)[/tex]
[tex]2S_{1z}S_{2z} \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)=-\hbar^2 \frac{1}{2} \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right) [/tex]
[tex]\left( S_{1+}S_{2+}+S_{1-}S_{2-} \right) \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)=0[/tex]
As you can see, these values do not add up to zero. [tex]\hbar^2 \frac{3}{4} + \hbar^2 \frac{3}{4} -\hbar^2 \frac{1}{2} =2 \hbar^2 [/tex]
I am not sure where my calculations went off, if you would like to see more work, please ask.
I need to show: [tex]S^2 \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)[/tex] has an eigenvalue of zero.
The attempt at a solution
[tex]S_1^2 \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)=\hbar^2 \frac{3}{4} \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right) [/tex]
[tex]S_2^2 \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)=\hbar^2 \frac{3}{4} \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)[/tex]
[tex]2S_{1z}S_{2z} \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)=-\hbar^2 \frac{1}{2} \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right) [/tex]
[tex]\left( S_{1+}S_{2+}+S_{1-}S_{2-} \right) \left( \frac{1}{\sqrt{2}} \left[ \alpha(1) \beta(2) - \beta(1) \alpha(2) \right] \right)=0[/tex]
As you can see, these values do not add up to zero. [tex]\hbar^2 \frac{3}{4} + \hbar^2 \frac{3}{4} -\hbar^2 \frac{1}{2} =2 \hbar^2 [/tex]
I am not sure where my calculations went off, if you would like to see more work, please ask.