Your email address will not be published. \begin{matrix} {{{\hat{L}}}_{0}}\left| \psi \right\rangle =a\left| \psi \right\rangle \\ {{{\hat{L}}}_{m>0}}\left| \psi \right\rangle =0 \\ \end{matrix} \right\} $, $ \begin{align} & 0={{{\hat{L}}}_{1}}\left| \Phi \right\rangle \\ & \text{ }={{{\hat{L}}}_{1}}{{{\hat{L}}}_{-1}}\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=\left[ {{{\hat{L}}}_{-1}},{{{\hat{L}}}_{1}} \right]\left| {{\chi }_{1}} \right\rangle +{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{1}}\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=\left[ {{{\hat{L}}}_{-1}},{{{\hat{L}}}_{1}} \right]\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=2{{{\hat{L}}}_{0}}\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=2{{c}_{0}}\left( a-1 \right)\left| {{\chi }_{1}} \right\rangle \\ \end{align} $, $ \begin{align} & \left[ {{{\hat{L}}}_{1}},{{{\hat{L}}}_{-2}}+k{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right]\left| \psi \right\rangle =\left( 3{{{\hat{L}}}_{-1}}+2k{{{\hat{L}}}_{0}}{{{\hat{L}}}_{-1}}+2k{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{0}} \right)\left| \psi \right\rangle \text{ } \\ & \text{ }=\left( 3-2k \right){{{\hat{L}}}_{-1}}+4k{{{\hat{L}}}_{0}}{{{\hat{L}}}_{-1}}\text{ }\left( 3-2k \right){{{\hat{L}}}_{-1}}+4k{{{\hat{L}}}_{0}}{{{\hat{L}}}_{-1}}\text{ } \\ & 0={{{\hat{L}}}_{1}}\left| \psi \right\rangle ={{{\hat{L}}}_{1}}\left( {{{\hat{L}}}_{-2}}+k{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right)\left| {{\chi }_{1}} \right\rangle =\left( \left( 3-2k \right){{{\hat{L}}}_{-1}}+4k{{{\hat{L}}}_{0}}{{{\hat{L}}}_{-1}} \right)\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=\left( \left( 3-2k \right){{{\hat{L}}}_{-1}}+4k{{{\hat{L}}}_{-1}}\left( {{{\hat{L}}}_{0}}+1 \right) \right)\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=\left( 3-2k \right){{{\hat{L}}}_{-1}}\left| {{\chi }_{1}} \right\rangle \\ & 2k=3 \\ & k=\frac{3}{2} \\ \end{align} $, $ \begin{align} & {{{\hat{L}}}_{2}}\left| \Phi \right\rangle =0 \\ & {{{\hat{L}}}_{2}}\left( {{{\hat{L}}}_{-2}}+\frac{3}{2}{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right)\left| {{\chi }_{2}} \right\rangle =0 \\ & \left[ {{{\hat{L}}}_{2}},{{{\hat{L}}}_{-2}}+\frac{3}{2}{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right]\left| {{\chi }_{2}} \right\rangle +\left( {{{\hat{L}}}_{-2}}+\frac{3}{2}{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right){{{\hat{L}}}_{2}}\left| {{\chi }_{2}} \right\rangle =0 \\ & \left[ {{{\hat{L}}}_{2}},{{{\hat{L}}}_{-2}}+\frac{3}{2}{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right]\left| {{\chi }_{2}} \right\rangle =0 \\ & \left( 13{{{\hat{L}}}_{0}}+9{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}}+\frac{D}{2} \right)\left| {{\chi }_{2}} \right\rangle =0 \\ & \frac{D}{2}=13 \\ & D=26 \\ \end{align} $.
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