91Ó°ÊÓ

Angular momentum with ${\mathrm{J}}_1$and ${\mathrm{J}}_2$ is $\frac{3}{2}$.

The quantum number for the total angular momentum${\mathbf{j}}_{\mathbf{T}}$given as:

${\mathbf{j}}_{\mathbf{T}}\mathbf{=}\left|j_1-j_2\right|\mathbf{,}\left|j_1-j_2\right|\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\left|j_1+j_2\right|\mathbf{+}\mathbf{1}\mathbf{,}\left|j_1+j_2\right|$

Here data-custom-editor="chemistry" ${\mathbf{j}}_{\mathbf{1}}$and data-custom-editor="chemistry" ${\mathbf{j}}_{\mathbf{2}}$are the quantum numbers for angular momentum vectors data-custom-editor="chemistry" ${\mathbf{J}}_{\mathbf{1}}$and data-custom-editor="chemistry" ${\mathbf{J}}_{\mathbf{2}}$respectively.

The quantum number for the projection of the total angular momentum onto the z-axis data-custom-editor="chemistry" ${{\mathbf{m}}_{\mathbf{j}}}_{\mathbf{T}}$given as, data-custom-editor="chemistry" ${{\mathbf{m}}_{\mathbf{j}}}_{\mathbf{T}}\mathbf{=}\mathbf{-}\mathbf{jT}\mathbf{,}\mathbf{-}\mathbf{jT}\mathbf{+}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{-}\mathbf{jT}\mathbf{+}\mathbf{1}\mathbf{,}\mathbf{-}\mathbf{jT}$.

Here data-custom-editor="chemistry" ${\mathbf{j}}_{\mathbf{T}}$is the quantum number for the total angular momentum.

The total number of them can just be given as, data-custom-editor="chemistry" $\mathbf{2}{\mathbf{j}}_{\mathbf{T}}\mathbf{+}\mathbf{1}$.

The magnitude data-custom-editor="chemistry" ${\mathbf{J}}_{\mathbf{T}}$of the total angular momentum is, data-custom-editor="chemistry" ${\mathbf{J}}_{\mathbf{T}}\mathbf{=}\sqrt{{\mathbf{j}}_{\mathbf{T}}\left(j_T+1\right)}\sout{\mathbf{h}}$.

Here data-custom-editor="chemistry" ${\mathbf{j}}_{\mathbf{T}}$is the quantum number for the total angular momentum anddata-custom-editor="chemistry" $\sout{\mathbf{h}}$is Planck's reduced constant.

(a)

If ${\mathrm{j}}_1$is 2 and ${\mathrm{j}}_2$is 3/2, the minimum and maximum that ${\mathrm{j}}_{\mathrm{T}}$can be:

$$\begin{gathered} {\mathrm{j}}_{\mathrm{T}\min }=\left|{\mathrm{j}}_1-{\mathrm{j}}_2\right|,{\mathrm{j}}_{\mathrm{T}\max }={\mathrm{j}}_1+{\mathrm{j}}_2 \\ {\mathrm{j}}_{\mathrm{T}\min }=\left|\left(2\right)-\left(\frac{3}{2}\right)\right|,{\mathrm{j}}_{\mathrm{T}\max }=\left(2\right)+\left(\frac{3}{2}\right) \\ {\mathrm{j}}_{\mathrm{T}\min }=\frac{1}{2},{\mathrm{j}}_{\mathrm{T}\max }=\frac{7}{2} \end{gathered}$$

The other values that ${\mathrm{j}}_{\mathrm{T}}$can have are found by adding 1 to the minimum ${\mathrm{j}}_{\mathrm{T}}$in increments until you reach the maximum ${\mathrm{j}}_{\mathrm{T}}$.

$$\begin{gathered} {{\mathrm{j}}_{\mathrm{T}}}_2={\mathrm{j}}_{\mathrm{T}\min }+1 \\ {{\mathrm{j}}_{\mathrm{T}}}_2=\frac{1}{2}+1 \\ {{\mathrm{j}}_{\mathrm{T}}}_2=\frac{3}{2} \end{gathered}$$

Similarly for other ${\mathrm{j}}_{\mathrm{T}}$:

$$\begin{gathered} {{\mathrm{j}}_{\mathrm{T}}}_3={{\mathrm{j}}_{\mathrm{T}}}_2+1 \\ {{\mathrm{j}}_{\mathrm{T}}}_3=\frac{3}{2}+1 \\ {{\mathrm{j}}_{\mathrm{T}}}_3=\frac{5}{2} \end{gathered}$$

Therefore, the full ranges of values that ${\mathrm{j}}_{\mathrm{T}}$can have are ${\mathrm{j}}_{\mathrm{T}}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2}$.

Those values can then be used to find the magnitude of the total angular momentum of ${\mathrm{J}}_{\mathrm{T}}$.

$$\begin{gathered} {\mathrm{J}}_{\mathrm{T}}=\sqrt{{\mathrm{j}}_{\mathrm{T}}\left({\mathrm{j}}_{\mathrm{T}}+1\right)}\sout{\mathrm{h}}{\mathrm{J}}_{\mathrm{T}}=\sqrt{{\mathrm{j}}_{\mathrm{T}}\left({\mathrm{j}}_{\mathrm{T}}+1\right)}\sout{\mathrm{h}} \\ {\mathrm{J}}_{\mathrm{T}}=\sqrt{\frac{1}{2}\left[\frac{1}{2}+1\right]}\sout{\mathrm{h}}{\mathrm{J}}_{\mathrm{T}}=\sqrt{\frac{3}{2}\left[\frac{3}{2}+1\right]}\sout{\mathrm{h}} \\ {\mathrm{J}}_{\mathrm{T}}=\frac{\sqrt{3}}{2}\sout{\mathrm{h}}{\mathrm{J}}_{\mathrm{T}}=\frac{\sqrt{15}}{2}\sout{\mathrm{h}} \end{gathered}$$

Similarly for other${\mathrm{j}}_{\mathrm{T}}$:

$$\begin{gathered} {\mathrm{J}}_{\mathrm{T}}=\sqrt{{\mathrm{j}}_{\mathrm{T}}\left({\mathrm{j}}_{\mathrm{T}}+1\right)}\sout{\mathrm{h}}{\mathrm{J}}_{\mathrm{T}}=\sqrt{{\mathrm{j}}_{\mathrm{T}}\left({\mathrm{j}}_{\mathrm{T}}+1\right)}\sout{\mathrm{h}} \\ {\mathrm{J}}_{\mathrm{T}}=\sqrt{\frac{5}{2}\left[\frac{5}{2}+1\right]}\sout{\mathrm{h}}{\mathrm{J}}_{\mathrm{T}}=\sqrt{\frac{7}{2}\left[\frac{7}{2}+1\right]}\sout{\mathrm{h}} \\ {\mathrm{J}}_{\mathrm{T}}=\frac{\sqrt{35}}{2}\sout{\mathrm{h}}{\mathrm{J}}_{\mathrm{T}}=\frac{\sqrt{63}}{2}\sout{\mathrm{h}} \end{gathered}$$

The full ranges of values that ${\mathrm{j}}_{\mathrm{T}}$can have are ${\mathrm{j}}_{\mathrm{T}}=\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2}$.

The values that ${\mathrm{J}}_{\mathrm{T}}$can take are ${\mathrm{J}}_{\mathrm{T}}=\frac{\sqrt{3}}{2}\sout{\mathrm{h}},\frac{\sqrt{15}}{2}\sout{\mathrm{h}},\frac{\sqrt{35}}{2}\sout{\mathrm{h}},\frac{\sqrt{36}}{2}\sout{\mathrm{h}}$.

(b)

Find the total number of angular momentum states possible, you need to check how many states are possible for each for ${\mathrm{j}}_{\mathrm{T}}$and then add them together.

The number of states for each ${\mathrm{j}}_{\mathrm{T}}$is given by $2{\mathrm{j}}_{\mathrm{T}}+1$.

$$\begin{gathered} 2j_T+12j_T+12j_T+12j_T+1 \\ 2\left(\frac{1}{2}\right)+12\left(\frac{3}{2}\right)+12\left(\frac{5}{2}\right)+12\left(\frac{7}{2}\right)+1 \\ 1+13+15+17+1 \\ 2468 \end{gathered}$$

So the total number of states between all of the ${\mathrm{j}}_{\mathrm{T}}$then is $2+4+6+8=20$.

There are 20 total angular momentum states possible.

(c)

Calculate the value for ${\mathrm{m}}_{\mathrm{j}}$for the four values of ${\mathrm{j}}_{\mathrm{T}}$by formula.

$$\begin{gathered} {\mathrm{j}}_{\mathrm{T}}=\frac{1}{2} \\ {\mathrm{m}}_{\mathrm{j}}=-\frac{1}{2}+\frac{1}{2}\sqrt{{\mathrm{b}}^2-4\mathrm{ac}} \end{gathered}$$

The value forfor the four values ofby above equation.

$\begin{array}{rcl}\mathrm{jT}& =& \frac{1}{2}\\ {\mathrm{m}}_{\mathrm{j}}& =& -\frac{1}{2},+\frac{1}{2}\sqrt{{\mathrm{b}}^2-4\mathrm{ac}}\\ & =& \frac{3}{2}\\ {\mathrm{m}}_{\mathrm{j}}& =& -\frac{3}{2},-\frac{1}{2},+\frac{1}{2},+\frac{3}{2}\end{array}$

Calculate further as shown below.

$\begin{array}{rcl}& =& \frac{5}{2}\\ {\mathrm{m}}_{\mathrm{j}}& =& -\frac{5}{2},-\frac{3}{2},-\frac{1}{2},+\frac{1}{2},+\frac{3}{2},+\frac{5}{2}\\ & =& \frac{7}{2}\\ {\mathrm{m}}_{\mathrm{j}}& =& -\frac{7}{2},-\frac{5}{2},-\frac{3}{2},-\frac{1}{2},+\frac{1}{2},+\frac{3}{2},+\frac{5}{2},+\frac{7}{2}\end{array}$

The total angular momentum states $\left({\mathrm{j}}_{\mathrm{T}},{{\mathrm{m}}_{\mathrm{j}}}_{\mathrm{T}}\right)$are:

$$\begin{gathered} \left(\frac{1}{2},+\frac{1}{2}\right),\left(\frac{1}{2},-\frac{1}{2}\right) \\ \left(\frac{3}{2},-\frac{3}{2}\right),\left(\frac{3}{2},-\frac{1}{2}\right)\left(\frac{3}{2},+\frac{1}{2}\right),\left(\frac{3}{2},+\frac{3}{2}\right) \\ \left(\frac{5}{2},-\frac{5}{2}\right),\left(\frac{5}{2},-\frac{3}{2}\right)\left(\frac{5}{2},-\frac{1}{2}\right),\left(\frac{5}{2},+\frac{1}{2}\right)\left(\frac{5}{2},+\frac{3}{2}\right),\left(\frac{5}{2},+\frac{5}{2}\right) \\ \left(\frac{7}{2},-\frac{7}{2}\right),\left(\frac{7}{2},-\frac{5}{2}\right)\left(\frac{7}{2},-\frac{3}{2}\right),\left(\frac{7}{2},-\frac{1}{2}\right)\left(\frac{7}{2},+\frac{1}{2}\right),\left(\frac{7}{2},+\frac{3}{2}\right)\left(\frac{7}{2},+\frac{5}{2}\right),\left(\frac{7}{2},+\frac{7}{2}\right) \end{gathered}$$