(a) Work out the Clebsch-Gordan coefficients for the case ${\mathbf{s}}_{\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{/}\mathbf{2}\mathbf{,}{\mathbf{s}}_{\mathbf{2}}$=anything. Hint: You're looking for the coefficients A and Bin

$\mathbf{|}\mathbf{sm}\mathbf{⟩}\mathbf{=}\mathbf{A}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{⟩}\mathbf{|}{\mathbf{s}}_{\mathbf{2}}\mathbf{}\left(m-\frac{1}{2}\right)\mathbf{⟩}\mathbf{+}\mathbf{B}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\left(-\frac{1}{2}\right)\mathbf{⟩}\mathbf{|}{\mathbf{s}}_{\mathbf{2}}\left(m+\frac{1}{2}\right)\mathbf{⟩}$

such that$\mathbf{|}\mathbf{sm}\mathbf{⟩}$ is an eigenstate of . Use the method of Equations 4.179 through 4.182. If you can't figure out what${\mathit{S}}_{\mathbf{x}}^{\left(2\right)}$ (for instance) does to$\mathbf{|}{\mathbf{s}}_{\mathbf{2}}{\mathbf{m}}_{\mathbf{2}}\mathbf{⟩}$ , refer back to Equation 4.136 and the line before Equation 4.147. Answer:

;role="math" localid="1658209512756" $\mathbf{A}\mathbf{=}\sqrt{\frac{{\mathbf{s}}_{\mathbf{2}}\mathbf{+}{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}\mathbf{\pm }\mathbf{m}}{\mathbf{2}{\mathbf{s}}_{\mathbf{2}}\mathbf{+}\mathbf{1}}}\mathbf{;}\mathbf{}\mathbf{B}\mathbf{=}\mathbf{\pm }\sqrt{\frac{{\mathbf{s}}_{\mathbf{2}}\mathbf{+}{\displaystyle \frac{1}{2}}\mathbf{\pm }\mathbf{m}}{\mathbf{2}{\mathbf{s}}_{\mathbf{2}}\mathbf{+}\mathbf{1}}}$

where, the signs are determined by$\mathbf{s}\mathbf{=}{\mathbf{s}}_{\mathbf{2}}\mathbf{\pm }\mathbf{1}\mathbf{/}\mathbf{2}$ .

(b) Check this general result against three or four entries in Table 4.8.

Before solving the problem, some formulae should be known,

${\mathbf{S}}_{\mathbf{\pm }}\mathbf{|}\mathbf{sm}\mathbf{⟩}\mathbf{=}\mathbf{\hslash }\sqrt{\mathbf{s}\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{-}\mathbf{m}\mathbf{(}\mathbf{m}\mathbf{\pm }\mathbf{1}\mathbf{)}}\mathbf{|}\mathbf{sm}\mathbf{\pm }\mathbf{1}\mathbf{⟩}$

Here, $S_{\pm }$is the raising and lowering operators with an eigenstate$⟨|s⟩$

${\mathbf{S}}^{\mathbf{2}}\mathbf{|}\mathbf{sm}\mathbf{⟩}\mathbf{=}{\mathbf{\hslash }}^{\mathbf{2}}\mathbf{s}\mathbf{(}\mathbf{s}\mathbf{+}\mathbf{1}\mathbf{\left)}\mathbf{\right|}\mathbf{sm}\mathbf{⟩}$

Here,$S^2$is the spin angular momentum operator squared with an eigenvalue${\hslash }^{2}s(s+1)$and an eigenstate$|sm⟩$.

localid="1658209816175" ${\mathbf{S}}_{\mathbf{z}}\mathbf{|}\mathbf{sm}\mathbf{⟩}\mathbf{=}\mathbf{m}\mathbf{\hslash }\mathbf{|}\mathbf{sm}\mathbf{⟩}$

Here,$S_z$is the quantized z -component of spin angular momentum with an eigenvalue$m\hslash$and an eigenstae$|sm⟩$.

s is the spin quantum number which is always a positive integer while m is the magnetic spin quantum number which has always a half-integer value.

A chemical bond is an atom-to-atom attraction. This attraction can be explained by differences in the behaviour of atoms' outermost or valence electrons.

Solve the issue by creating the following relationship between the spin angular momentum operator's x -component, $S_x$ , the spin angular momentum operator's y -component, $S_y$ , and the spin raising and lowering operators,$S_{\pm }$ ,

$$\begin{gathered} S_x=\frac{1}{2}\left[S_{+}+S_{-}\right] \\ {S}_x\overline{)sm>=\frac{1}{2}\left[S_{+}\overline{)sm>}+S_{-}\overline{)sm>}\right]} \\ =\frac{\sout{h}}{2}\left[\sqrt{s\left(s+1\right)-m\left(m+1\right)}\overline{)sm+1>}\sqrt{s\left(s+1\right)-m\left(m-1\right)}\overline{)sm-1>}\right] \\ {S}_y=\frac{1}{2i}\left[S_{+}+S_{-}\right] \\ {S}_y\overline{)sm>=\frac{1}{2}\left[S_{+}\overline{)sm>}+S_{-}\overline{)sm>}\right]} \\ =\frac{\sout{h}}{2i}\left[\sqrt{s\left(s+1\right)-m\left(m+1\right)}\overline{)sm+1>}\sqrt{s\left(s+1\right)-m\left(m-1\right)}\overline{)sm-1>}\right] \end{gathered}$$

Add angular momenta.

$$\begin{gathered} S=S^{\left(1\right)}+S^{\left(2\right)} \\ {S}^2=\left(S^{\left(1\right)}+S^{\left(2\right)}\right).\left(S^{\left(1\right)}+S^{\left(2\right)}\right) \\ =\left({\left(S^{\left(1\right)}\right)}^2+{\left(S^{\left(2\right)}\right)}^2+2S^{\left(1\right)}.S^{\left(2\right)}\right) \\ =\left({\left(S^{\left(1\right)}\right)}^2+{\left(S^{\left(2\right)}\right)}^2+2\left(S_x^{\left(1\right)}{S}_x^{\left(2\right)}+S_y^{\left(1\right)}{S}_y^{\left(2\right)}+S_z^{\left(1\right)}{S}_z^{\left(2\right)}\right)\right) \\ =S^2\overline{)sm>=}\left({\left(S^{\left(1\right)}\right)}^2+{\left(S^{\left(2\right)}\right)}^2+2\left(S_x^{\left(1\right)}{S}_x^{\left(2\right)}+S_y^{\left(1\right)}{S}_y^{\left(2\right)}+S_z^{\left(1\right)}{S}_z^{\left(2\right)}\right)\right)\overline{)sm>} \\ =\left({\left(S^{\left(1\right)}\right)}^2+{\left(S^{\left(2\right)}\right)}^2+2\left(S_x^{\left(1\right)}{S}_x^{\left(2\right)}+S_y^{\left(1\right)}{S}_y^{\left(2\right)}+S_z^{\left(1\right)}{S}_z^{\left(2\right)}\right)\right)\left(A\overline{)\frac{1}{2}\frac{1}{2}>\overline{)s_{2}m-\frac{1}{2}>+B}\overline{)\frac{1}{2}\frac{1}{2}>\overline{)s_{2}m+\frac{1}{2}>}}}\right) \end{gathered}$$

Distribute the operators over the states.

$$\begin{gathered} S^2\overline{)sm}>=\left[\begin{array}{c}A{\left(S^{\left(1\right)}\right)}^2\overline{)\frac{1}{2}\frac{1}{2}>{\overline{)s_{2}m-\frac{1}{2}>+B\left(S^{\left(1\right)}\right)}}^2\overline{)\frac{1}{2}-\frac{1}{2}>}}\overline{)s_{2}m+\frac{1}{2}>}\\ +A\overline{)\frac{1}{2}\frac{1}{2}>{\left(S^{\left(2\right)}\right)}^2\overline{)s_{2}m-\frac{1}{2}>+B}\overline{)\frac{1}{2}-\frac{1}{2}>{\left(S^{\left(2\right)}\right)}^2}}\overline{)s_{2}m+\frac{1}{2}>}\\ +2\left\{\begin{array}{l}A{S}_x\overline{)\frac{1}{2}\frac{1}{2}>S_x}\overline{)s_{2}m-\frac{1}{2}>+B{S}_x}\overline{)\frac{1}{2}-\frac{1}{2}>S_x\overline{)s_{2}m+\frac{1}{2}>}}\\ \end{array}\right.\\ +A{S}_y\overline{)\frac{1}{2}\frac{1}{2}>S_y}\overline{)s_{2}m-\frac{1}{2}>+B{S}_y}\overline{)\frac{1}{2}-\frac{1}{2}>S_y\overline{)s_{2}m+\frac{1}{2}>}}\\ +A{S}_z\overline{)\frac{1}{2}\frac{1}{2}>S_z}\overline{)s_{2}m-\frac{1}{2}>+B{S}_z}\overline{)\frac{1}{2}-\frac{1}{2}>S_z\overline{)s_{2}m+\frac{1}{2}>}}\end{array}\right] \\ =\left[\begin{array}{c}A{\left(S^{\left(1\right)}\right)}^2\overline{)\frac{1}{2}\frac{1}{2}>{\overline{)s_{2}m-\frac{1}{2}>+}}^{}\overline{)\frac{1}{2}-\frac{1}{2}>}}{\left(S^{\left(2\right)}\right)}^2\overline{)s_{2}m+\frac{1}{2}>}\\ +2\left.\begin{array}{r}\left[S_x\overline{)\frac{1}{2}\frac{1}{2}>S_x}\overline{)s_{2}m-\frac{1}{2}>+S_y}\overline{)\frac{1}{2}\frac{1}{2}>S_y}\overline{)s_{2}m-\frac{1}{2}>+S_z}\overline{)\frac{1}{2}\frac{1}{2}>S_z}\overline{)s_{2}m-\frac{1}{2}>}\right]\\ \end{array}\right\}\\ +B\left\{\begin{array}{l}A{\left(S^{\left(1\right)}\right)}^2\overline{)\frac{1}{2}\frac{1}{2}>{\overline{)s_{2}m-\frac{1}{2}>+}}^{}\overline{)\frac{1}{2}-\frac{1}{2}>}}{\left(S^{\left(2\right)}\right)}^2\overline{)s_{2}m+\frac{1}{2}>}\\ \end{array}\right.\\ +2\left.\begin{array}{r}\left[S_x\overline{)\frac{1}{2}\frac{1}{2}>S_x}\overline{)s_{2}m-\frac{1}{2}>+S_y}\overline{)\frac{1}{2}\frac{1}{2}>S_y}\overline{)s_{2}m-\frac{1}{2}>+S_z}\overline{)\frac{1}{2}\frac{1}{2}>S_z}\overline{)s_{2}m-\frac{1}{2}>}\right]\\ \end{array}\right\}\end{array}\right] \end{gathered}$$uncaught exception: Invalid chunk

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('9f895d4bd17c841...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('Invalid chunk

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('9f895d4bd17c841...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('$$\begin{gathered} S^2\overline{)sm}>=\left[\begin{array}{c}A{\left(S^{\left(1\right)}\right)}^2\overline{)\frac{1}{2}\frac{1}{2}>{\overline{)s_{2}m-\frac{1}{2}>+B\left(S^{\left(1\right)}\right)}}^2\overline{)\frac{1}{2}-\frac{1}{2}>}}\overline{)s_{2}m+\frac{1}{2}>}\\ +A\overline{)\frac{1}{2}\frac{1}{2}>{\left(S^{\left(2\right)}\right)}^2\overline{)s_{2}m-\frac{1}{2}>+B}\overline{)\frac{1}{2}-\frac{1}{2}>{\left(S^{\left(2\right)}\right)}^2}}\overline{)s_{2}m+\frac{1}{2}>}\\ +2\left\{\begin{array}{l}A{S}_x\overline{)\frac{1}{2}\frac{1}{2}>S_x}\overline{)s_{2}m-\frac{1}{2}>+B{S}_x}\overline{)\frac{1}{2}-\frac{1}{2}>S_x\overline{)s_{2}m+\frac{1}{2}>}}\\ \end{array}\right.\\ +A{S}_y\overline{)\frac{1}{2}\frac{1}{2}>S_y}\overline{)s_{2}m-\frac{1}{2}>+B{S}_y}\overline{)\frac{1}{2}-\frac{1}{2}>S_y\overline{)s_{2}m+\frac{1}{2}>}}\\ +A{S}_z\overline{)\frac{1}{2}\frac{1}{2}>S_z}\overline{)s_{2}m-\frac{1}{2}>+B{S}_z}\overline{)\frac{1}{2}-\frac{1}{2}>S_z\overline{)s_{2}m+\frac{1}{2}>}}\end{array}\right] \\ =\left[\begin{array}{c}A{\left(S^{\left(1\right)}\right)}^2\overline{)\frac{1}{2}\frac{1}{2}>{\overline{)s_{2}m-\frac{1}{2}>+}}^{}\overline{)\frac{1}{2}-\frac{1}{2}>}}{\left(S^{\left(2\right)}\right)}^2\overline{)s_{2}m+\frac{1}{2}>}\\ +2\left.\begin{array}{r}\left[S_x\overline{)\frac{1}{2}\frac{1}{2}>S_x}\overline{)s_{2}m-\frac{1}{2}>+S_y}\overline{)\frac{1}{2}\frac{1}{2}>S_y}\overline{)s_{2}m-\frac{1}{2}>+S_z}\overline{)\frac{1}{2}\frac{1}{2}>S_z}\overline{)s_{2}m-\frac{1}{2}>}\right]\\ \end{array}\right\}\\ +B\left\{\begin{array}{l}A{\left(S^{\left(1\right)}\right)}^2\overline{)\frac{1}{2}\frac{1}{2}>{\overline{)s_{2}m-\frac{1}{2}>+}}^{}\overline{)\frac{1}{2}-\frac{1}{2}>}}{\left(S^{\left(2\right)}\right)}^2\overline{)s_{2}m+\frac{1}{2}>}\\ \end{array}\right.\\ +2\left.\begin{array}{r}\left[S_x\overline{)\frac{1}{2}\frac{1}{2}>S_x}\overline{)s_{2}m-\frac{1}{2}>+S_y}\overline{)\frac{1}{2}\frac{1}{2}>S_y}\overline{)s_{2}m-\frac{1}{2}>+S_z}\overline{)\frac{1}{2}\frac{1}{2}>S_z}\overline{)s_{2}m-\frac{1}{2}>}\right]\\ \end{array}\right\}\end{array}\right] \end{gathered}$$uncaught exception: Invalid chunk

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('9f895d4bd17c841...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('

$=\left[\begin{array}{c}A{\left(S^{\left(1\right)}\right)}^2\overline{)\frac{1}{2}\frac{1}{2}>{\overline{)s_{2}m-\frac{1}{2}>+}}^{}\overline{)\frac{1}{2}-\frac{1}{2}>}}{\left(S^{\left(2\right)}\right)}^2\overline{)s_{2}m+\frac{1}{2}>}\\ +2\left.\begin{array}{r}\left[S_x\overline{)\frac{1}{2}\frac{1}{2}>S_x}\overline{)s_{2}m-\frac{1}{2}>+S_y}\overline{)\frac{1}{2}\frac{1}{2}>S_y}\overline{)s_{2}m-\frac{1}{2}>+S_z}\overline{)\frac{1}{2}\frac{1}{2}>S_z}\overline{)s_{2}m-\frac{1}{2}>}\right]\\ \end{array}\right\}\\ +B\left\{\begin{array}{l}A{\left(S^{\left(1\right)}\right)}^2\overline{)\frac{1}{2}\frac{1}{2}>{\overline{)s_{2}m-\frac{1}{2}>+}}^{}\overline{)\frac{1}{2}-\frac{1}{2}>}}{\left(S^{\left(2\right)}\right)}^2\overline{)s_{2}m+\frac{1}{2}>}\\ \end{array}\right.\\ +2\left.\begin{array}{r}\left[S_x\overline{)\frac{1}{2}\frac{1}{2}>S_x}\overline{)s_{2}m-\frac{1}{2}>+S_y}\overline{)\frac{1}{2}\frac{1}{2}>S_y}\overline{)s_{2}m-\frac{1}{2}>+S_z}\overline{)\frac{1}{2}\frac{1}{2}>S_z}\overline{)s_{2}m-\frac{1}{2}>}\right]\\ \end{array}\right\}\end{array}\right]$

Act with the operators on the states in the above expression.