Question

(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.

Short answer

(a) The Clebsch-Gordan coefficients are, $\mathrm{A}=\sqrt{\frac{{\mathrm{s}}_2+{\displaystyle \frac{1}{2}}\pm \mathrm{m}}{2{\mathrm{s}}_2+1}}$, and $\mathrm{B}=\pm \sqrt{\frac{{\mathrm{s}}_2+{\displaystyle \frac{1}{2}}\pm \mathrm{m}}{2{\mathrm{s}}_2+1}}$

(b) The result is checked for few entries.

Step-by-step solution

Introduction to few formulae

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.

(a) Determination of the Clebsch-Gordan coefficients for the case s1=1/2,s2= anything.

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.