Chapter 15, Problem 15.12E

Interpretation Introduction

(a)

Interpretation:

The possible observable values of orbital and spin angular momenta and their $z$ components for a single $d$ electron are to be calculated. The values of $j$ and $m_j$ are to be calculated.

Concept introduction:

The magnetic fields generated by the spin angular momentum and the orbital angular momentum of an electron interact with each other to generate an overall angular momentum. The interaction between the orbital angular momentum and the spin angular momentum is termed as spin-orbit coupling.

Answer to Problem 15.12E

The possible observable values of orbital angular momenta for a single $d$ electron is $2$. The possible value of spin angular momenta for a single $d$ electron is $\frac{1}{2}$. The values of $z$ components of orbital angular momentum, $m_l$ for a single $d$ electron, are $-2$, $-1$, $0$, $1$, $2$. The values of $z$ components of spin angular momentum, $m_s$ for a single $d$ electron, are $\frac{1}{2},-\frac{1}{2}$. The values of $j$ are $\frac{5}{2}\text{or}\frac{3}{2}$. The values of $m_j$ are $-\frac{5}{2},-2,-\frac{3}{2},-1,-\frac{1}{2},0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2}$.

Explanation of Solution

The value of orbital angular momentum $\left(l\right)$ for a $d$ electron is $2$.

The total number of $z$ components of orbital angular momentum $\left(m_l\right)$ is given by the formula shown below.

$m_l=2l+1$ …(1)

Substitute the value of $l$ of a $d$ electron in equation (1).

$\begin{array}{rll}{m}_l& =& 2\times 2+1\\ & =& 5\end{array}$

The value of $z$ components of orbital angular momentum is given below.

$m_l=-l\to l$

Therefore, the values of $z$ components of orbital angular momentum, $m_l$ are $-2$, $-1$, $0$, $1$, $2$.

The spin angular momentum $\left(s\right)$ is $\frac{1}{2}$ for electrons in all subshells.

The total number of $z$ components of spin angular momentum $\left(m_s\right)$ is given by the formula below.

$m_s=2s+1$ …(2)

Substitute the value of $s$ in equation (2).

$\begin{array}{rll}{m}_s& =& 2\times \frac{1}{2}+1\\ & =& 2\end{array}$

The value of $z$ components of spin angular momentum is given below.

$m_s=-s\to s$

Therefore, the values of $z$ components of spin angular momentum, $m_s$ are $\frac{1}{2},-\frac{1}{2}$.

The total angular momentum $\left(j\right)$ can have two values as shown below.

$\begin{array}{l}j=l+s\text{or}l-s\\ j=2+\frac{1}{2}\text{or}2-\frac{1}{2}\text{For}d\text{subshell,}l=2;s=\pm \frac{1}{2}\\ j=\frac{5}{2}\text{or}\frac{3}{2}\end{array}$

The value of $z$ components of total orbital angular momentum is given below.

$m_j=-j\to j$

Therefore, the values of $z$ components of total orbital angular momentum, $m_j$ are $-\frac{5}{2},-2,-\frac{3}{2},-1,-\frac{1}{2},0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2}$.

Conclusion

The possible observable values of orbital angular momenta for a single $d$ electron is $2$. The possible value of spin angular momenta for a single $d$ electron is $\frac{1}{2}$. The values of $z$ components of orbital angular momentum, $m_l$ for a single $d$ electron, are $-2$, $-1$, $0$, $1$, $2$. The values of $z$ components of spin angular momentum, $m_s$ for a single $d$ electron, are $\frac{1}{2},-\frac{1}{2}$. The values of $j$ are $\frac{5}{2}\text{or}\frac{3}{2}$. The values of $m_j$ are $-\frac{5}{2},-2,-\frac{3}{2},-1,-\frac{1}{2},0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2}$.

Interpretation Introduction

(b)

Interpretation:

The possible observable values of orbital and spin angular momenta and their $z$ components for a single $f$ electron are to be calculated. The values of $j$ and $m_j$ are to be calculated.

Concept introduction:

The magnetic fields generated by the spin angular momentum and the orbital angular momentum of an electron interact with each other to generate an overall angular momentum. The interaction between the orbital angular momentum and the spin angular momentum is termed as spin-orbit coupling.

Answer to Problem 15.12E

The possible observable values of orbital angular momenta for a single $f$ electron is $3$. The possible value of spin angular momenta for a single $f$ electron is $\frac{1}{2}$. The values of $z$ components of orbital angular momentum, $m_l$ for a single $f$ electron, are $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$. The values of $z$ components of spin angular momentum, $m_s$ for a single $f$ electron, are $\frac{1}{2},-\frac{1}{2}$. The values of $j$ are $\frac{7}{2}\text{or}\frac{5}{2}$. The values of $m_j$ are $-\frac{7}{2},-3,-\frac{5}{2},-2,-\frac{3}{2}-1,0,1,\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2}$.

Explanation of Solution

The value of orbital angular momentum $\left(l\right)$ for an $f$ electron is $3$.

The total number of $z$ components of orbital angular momentum $\left(m_l\right)$ is given by the formula shown below.

$m_l=2l+1$ …(1)

Substitute the value of $l$ of an $f$ electron in equation (1).

$\begin{array}{rll}{m}_l& =& 2\times 3+1\\ & =& 7\end{array}$

The value of $z$ components of orbital angular momentum is given below.

$m_l=-l\to l$

Therefore, the values of $z$ components of orbital angular momentum, $m_l$ are $-3,$ $-2$, $-1$, $0$, $1$, $2$, $3$.

The spin angular momentum $\left(s\right)$ is $\frac{1}{2}$ for electrons in all subshells.

The total number of $z$ components of spin angular momentum $\left(m_s\right)$ is given by the formula below.

$m_s=2s+1$ …(2)

Substitute the value of $s$ in equation (2).

$\begin{array}{rll}{m}_s& =& 2\times \frac{1}{2}+1\\ & =& 2\end{array}$

The value of $z$ components of spin angular momentum is given below.

$m_s=-s\to s$

Therefore, the values of $z$ components of spin angular momentum, $m_s$ are $\frac{1}{2},-\frac{1}{2}$.

The total angular momentum $\left(j\right)$ can have two values as shown below.

$\begin{array}{l}j=l+s\text{or}l-s\\ j=3+\frac{1}{2}\text{or}3-\frac{1}{2}\text{For}f\text{subshell,}l=3;s=\pm \frac{1}{2}\\ j=\frac{7}{2}\text{or}\frac{5}{2}\end{array}$

The value of $z$ components of total orbital angular momentum is given below.

$m_j=-j\to j$

Therefore, the values of $z$ components of total orbital angular momentum, $m_j$ are $-\frac{7}{2},-3,-\frac{5}{2},-2,-\frac{3}{2}-1,0,1,\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2}$.

Conclusion

The possible observable values of orbital angular momenta for a single $f$ electron is $3$. The possible value of spin angular momenta for a single $f$ electron is $\frac{1}{2}$. The values of $z$ components of orbital angular momentum, $m_l$ for a single $f$ electron, are $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$. The values of $z$ components of spin angular momentum, $m_s$ for a single $f$ electron, are $\frac{1}{2},-\frac{1}{2}$. The values of $j$ are $\frac{7}{2}\text{or}\frac{5}{2}$. The values of $m_j$ are $-\frac{7}{2},-3,-\frac{5}{2},-2,-\frac{3}{2}-1,0,1,\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2}$.

Interpretation Introduction

(c)

Interpretation:

The possible observable values of orbital and spin angular momenta and their $z$ components for a single $g$ electron are to be calculated. The values of $j$ and $m_j$ are to be calculated.

Concept introduction:

The magnetic fields generated by the spin angular momentum and the orbital angular momentum of an electron interact with each other to generate an overall angular momentum. The interaction between the orbital angular momentum and the spin angular momentum is termed as spin-orbit coupling.

Answer to Problem 15.12E

The possible observable values of orbital angular momenta for a single $g$ electron is $4$. The possible value of spin angular momenta for a single $g$ electron is $\frac{1}{2}$. The values of $z$ components of orbital angular momentum, $m_l$ for a single $g$ electron, are $-4$, $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, $4$ . The values of $z$ components of spin angular momentum, $m_s$ for a single $g$ electron, are $\frac{1}{2},-\frac{1}{2}$. The values of $j$ are $\frac{9}{2}\text{or}\frac{7}{2}$. The values of $m_j$ are $-\frac{9}{2},4,-\frac{7}{2},-3,-\frac{5}{2},-2,-\frac{3}{2}-1,0,1,\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2},4,\frac{9}{2}$.

Explanation of Solution

The value of orbital angular momentum $\left(l\right)$ for a $g$ electron is $4$.

The total number of $z$ components of orbital angular momentum $\left(m_l\right)$ is given by the formula shown below.

$m_l=2l+1$ …(1)

Substitute the value of $l$ of a $g$ electron in equation (1).

$\begin{array}{rll}{m}_l& =& 2\times 4+1\\ & =& 9\end{array}$

The value of $z$ components of orbital angular momentum is given below.

$m_l=-l\to l$

Therefore, the values of $z$ components of orbital angular momentum, $m_l$ are $-4$, $-3,$ $-2$, $-1$, $0$, $1$, $2$, $3$, $4$.

The spin angular momentum $\left(s\right)$ is $\frac{1}{2}$ for electrons in all subshells.

The total number of $z$ components of spin angular momentum $\left(m_s\right)$ is given by the formula below.

$m_s=2s+1$ …(2)

Substitute the value of $s$ in equation (2).

$\begin{array}{rll}{m}_s& =& 2\times \frac{1}{2}+1\\ & =& 2\end{array}$

The value of $z$ components of spin angular momentum is given below.

$m_s=-s\to s$

Therefore, the values of $z$ components of spin angular momentum, $m_s$ are $\frac{1}{2},-\frac{1}{2}$.

The total angular momentum $\left(j\right)$ can have two values as shown below.

$\begin{array}{l}j=l+s\text{or}l-s\\ j=4+\frac{1}{2}\text{or}4-\frac{1}{2}\text{For}g\text{subshell,}l=4;s=\pm \frac{1}{2}\\ j=\frac{9}{2}\text{or}\frac{7}{2}\end{array}$

The value of $z$ components of total orbital angular momentum is given below.

$m_j=-j\to j$

Therefore, the values of $z$ components of total orbital angular momentum, $m_j$ are $-\frac{9}{2},4,-\frac{7}{2},-3,-\frac{5}{2},-2,-\frac{3}{2}-1,0,1,\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2},4,\frac{9}{2}$.

Conclusion

The possible observable values of orbital angular momenta for a single $g$ electron is $4$. The possible value of spin angular momenta for a single $g$ electron is $\frac{1}{2}$. The values of $z$ components of orbital angular momentum, $m_l$ for a single $g$ electron, are $-4$, $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, $4$. The values of $z$ components of spin angular momentum, $m_s$ for a single $g$ electron, are $\frac{1}{2},-\frac{1}{2}$. The values of $j$ are $\frac{9}{2}\text{or}\frac{7}{2}$. The values of $m_j$ are $-\frac{9}{2},4,-\frac{7}{2},-3,-\frac{5}{2},-2,-\frac{3}{2}-1,0,1,\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2},4,\frac{9}{2}$.