Chapter 15, Problem 15.9E

List the possible values of $L$, $M_L$, $S$, $M_S$, $J$, and $M_J$ for the following: (a) two coupled $p$ electrons in different shells, (b) two coupled $f$ electrons in different shells, (c) two coupled electrons, one a $p$ electron and one a $d$ electron. Remember that the $z$-component quantum numbers depend on the values of the total angular momentum quantum numbers.

Interpretation Introduction

(a)

Interpretation:

The possible values of $L$, $M_L$, $S$, $M_S$, $J$, and $M_J$ for two coupled $p$ electrons in different shells are to be stated.

Concept introduction:

The symbol $L$ represents the vector sum of orbital angular momentum. The value of $M_L$ goes from $-L$ to $+L$. The symbol $S$ represents the vector sum of spin angular momentum. The value of $M_S$ goes from $-S$ to $+S$. The symbol $J$ represents the total angular momentum. The value of $M_J$ goes from $-J$ to $+J$.

Answer to Problem 15.9E

The values of $L$ are $2$, $1$ and $0$. The values of $M_L$ are $-2$, $-1$, $0$$+1$, and $+2$. The values of $S$ are $1$ and $0$. The values of $M_S$ are $-1$, $0$ and $1$. The values of $J$ are $3$, $2$, $1$ and $0$. The possible values of $M_J$ are $-3$, $-2$, $-1$, $0$, $1$, $2$ and $3$.

Explanation of Solution

When two coupled $p$ electrons are present in different shells, then the value of principal quantum number for both electrons will be different.

The orbital angular momentum $\left(l\right)$ for $p$ subshell is $1$. For $p^1$ electron configuration, the values of $m_l$ are shown below.

$\begin{array}{l}{m}_l+10-1\\ \boxed{}\boxed{}\boxed{}\end{array}$

An electron can occupy any of the orbitals. Therefore, an electron in the $p$ orbital can have $+1$, $0$, and $-1$ values of $m_l$. When two $p$ electrons present in different shells are coupled with each other, then the combination of $m_l$ value are $\left(+1,+1\right)$, $\left(+1,-1\right)$, $\left(+1,0\right)$, $\left(-1,+1\right)$, $\left(-1,-1\right)$, $\left(-1,0\right)$, $\left(0,+1\right)$, $\left(0,-1\right)$, and $\left(0,0\right)$.

The magnitude of the vector sums for $\left(+1,+1\right)$ and $\left(-1,-1\right)$ is $2$.

The magnitude of the vector sums for $\left(+1,-1\right)$, $\left(-1,+1\right)$ and $\left(0,0\right)$ is $0$.

The magnitude of the vector sums for $\left(-1,0\right)$, $\left(0,+1\right)$, $\left(0,-1\right)$ and $\left(+1,0\right)$ is $1$.

The symbol $L$ represents the vector sum of orbital angular momentum. Therefore, the values of $L$ are $2$, $1$ and $0$.

The value of $M_L$ goes from $-L$ to $+L$. Therefore, the values of $M_L$ are $-2$, $-1$, $0$$+1$, and $+2$.

An electron can have two value of spin angular momentum $\left(s\right)$ that are $\frac{1}{2}$ and $-\frac{1}{2}$.

The possible combinations of spin angular momentum for two electrons are $\left(\frac{1}{2},\frac{1}{2}\right)$, $\left(\frac{1}{2},-\frac{1}{2}\right)$, $\left(\frac{1}{2},-\frac{1}{2}\right)$ and $\left(-\frac{1}{2},-\frac{1}{2}\right)$.

Therefore, the possible values for the vector sum of spin angular momentum $S$ are $1$ and $0$. The value of $M_S$ goes from $-S$ to $+S$. Therefore, the values of $M_S$ are $-1$, $0$ and $1$.

The value of $J$ depends on $L$ and $S$.

The possible combination of $\left(L,S\right)$ are $\left(2,1\right)$, $\left(2,0\right)$, $\left(0,1\right)$, $\left(0,0\right)$, $\left(1,0\right)$ and $\left(1,1\right)$.

The value of $J$ can be calculated by the formula shown below.

$J=L+S\to \left|L-S\right|$

The addition of $\left(2,1\right)$ results in $J=3$.

The addition of $\left(2,0\right)$ and $\left(1,1\right)$ results in $J=2$.

The addition of $\left(0,1\right)$ and $\left(1,0\right)$ results in $J=1$.

The addition or subtraction of $\left(0,0\right)$ results in $J=0$.

The subtraction of $\left(2,0\right)$ results in $J=2$.

The subtraction of $\left(1,1\right)$ results in $J=0$.

The subtraction of $\left(2,1\right)$, $\left(0,1\right)$ and $\left(1,0\right)$ results in $J=1$.

Therefore, the values of $J$ are $3$, $2$, $1$ and $0$.

The value of $M_J$ goes from $-J$ to $+J$.

Therefore, the possible value of $-3$, $-2$, $-1$, $0$, $1$, $2$ and $3$.

Conclusion

The values of $L$ are $2$, $1$ and $0$. The values of $M_L$ are $-2$, $-1$, $0$$+1$, and $+2$. The values of $S$ are $1$ and $0$. The values of $M_S$ are $-1$, $0$ and $1$. The values of $J$ are $3$, $2$, $1$ and $0$. The possible values of $M_J$ are $-3$, $-2$, $-1$, $0$, $1$, $2$ and $3$.

Interpretation Introduction

(b)

The possible values of $L$, $M_L$, $S$, $M_S$, $J$, and $M_J$ for two coupled $f$ electrons in different shells are to be stated.

Concept introduction:

The symbol $L$ represents the vector sum of orbital angular momentum. The value of $M_L$ goes from $-L$ to $+L$.The symbol $S$ represents the vector sum of spin angular momentum. The value of $M_S$ goes from $-S$ to $+S$. The symbol $J$ represents the total angular momentum. The value of $M_J$ goes from $-J$ to $+J$.

Answer to Problem 15.9E

The values of $L$ are $6$, $5$, $4$, $3$, $2$, $1$ and $0$. The values of $M_L$ are $-6$, $-5$, $-4$, $-3$, $-2$, $-1$, $0$$+1$, $+2$, $+3$, $+4$, $+5$, and $+6$.The values of $S$ are $1$ and $0$. The values of $M_S$ are $-1$, $0$ and $1$. The values of $J$ are $7$, $6$, $5$, $4$$3$, $2$, $1$ and $0$. The possible value of $M_J$ are $-7$, $-6$, $-5$, $-4$, $-3$, $-2$, $-1$, $0$$+1$, $+2$, $+3$, $+4$, $+5$, $+6$ and $+7$.

Explanation of Solution

When two coupled $f$ electrons are present in different shells, then the value of principal quantum number for both electrons will be different.

The orbital angular momentum $\left(l\right)$ for $f$ subshell is $3$. For $f^1$ electron configuration, the values of $m_l$ is shown below.

$\begin{array}{cccccccc}{m}_l& +3& +2& +1& 0& -1& -2& -3\\ & \boxed{}& \boxed{}& \boxed{}& \boxed{}& \boxed{}& \boxed{}& \boxed{}\end{array}$

An electron can occupy any of the orbital. Therefore, an electron in the orbital $f$ can have $+3$, $+2$, $+1$, $0$, $-1$, $-2$, and $-3$ values of $m_l$. When two $f$ electrons present in different shells are coupled with each other, then the combination of $m_l$ value are $\left(+3,+3\right)$, $\left(+3,+2\right)$, $\left(+3,+1\right)$, $\left(+3,0\right)$, $\left(+3,-3\right)$, $\left(+3,-2\right)$, $\left(+3,-1\right)$, $\left(+2,+3\right)$, $\left(+2,+2\right)$, $\left(+2,+1\right)$, $\left(+2,0\right)$, $\left(+2,-3\right)$, $\left(+2,-2\right)$, $\left(+2,-1\right)$, $\left(+1,+3\right)$, $\left(+1,+2\right)$, $\left(+1,+1\right)$, $\left(+1,0\right)$, $\left(+1,-3\right)$, $\left(+1,-2\right)$, $\left(0,-1\right)$, $\left(0,+3\right)$, $\left(0,+2\right)$, $\left(0,+1\right)$, $\left(0,0\right)$, $\left(0,-3\right)$, $\left(0,-2\right)$, $\left(0,-1\right)$, $\left(-1,-1\right)$, $\left(-1,+3\right)$, $\left(-1,+2\right)$, $\left(-1,+1\right)$, $\left(-1,0\right)$, $\left(-1,-3\right)$, $\left(-1,-2\right)$, $\left(-1,-1\right)$, $\left(-2,-1\right)$, $\left(-2,+3\right)$, $\left(-2,+2\right)$, $\left(-2,+1\right)$, $\left(-2,0\right)$, $\left(-2,-3\right)$, $\left(-2,-2\right)$, $\left(-2,-1\right)$, $\left(-3,-1\right)$, $\left(-3,+3\right)$, $\left(-3,+2\right)$, $\left(-3,+1\right)$, $\left(-3,0\right)$, $\left(-3,-3\right)$, $\left(-3,-2\right)$, and $\left(-3,-1\right)$.

The magnitude of the vector sums for $\left(+3,+3\right)$ and $\left(-3,-3\right)$ is $6$.

The magnitude of the vector sums for $\left(+3,+2\right)$ and $\left(-3,-2\right)$ is $5$.

The magnitude of the vector sums for $\left(+3,+1\right)$, $\left(+2,+2\right)$, $\left(+1,+3\right)$, $\left(-2,-2\right)$ and $\left(-3,-1\right)$ is $4$.

The magnitude of the vector sums for $\left(+3,0\right)$, $\left(+2,+1\right)$, $\left(0,+3\right)$, $\left(-1,-2\right)$, $\left(-2,-1\right)$, $\left(0,-3\right)$, and $\left(-3,0\right)$ is $3$.

The magnitude of the vector sums for $\left(+3,-1\right)$, $\left(+2,0\right)$, $\left(+1,+1\right)$, $\left(+1,-3\right)$, $\left(0,+2\right)$, $\left(0,-2\right)$, $\left(-1,-1\right)$, $\left(-1,+3\right)$, $\left(-2,0\right)$, and $\left(-3,+1\right)$ is $2$.

The magnitude of the vector sums for $\left(+3,-2\right)$, $\left(+2,-3\right)$, $\left(+2,-1\right)$, $\left(+1,0\right)$, $\left(+1,-2\right)$, $\left(0,-1\right)$, $\left(0,+1\right)$, $\left(0,-1\right)$, $\left(-1,+2\right)$, $\left(-1,0\right)$, $\left(-2,+3\right)$, $\left(-2,+1\right)$, and $\left(-3,+2\right)$ is $1$.

The rest combinations result in the value of the magnitude of the vector sums equal to zero.

The symbol $L$ represents the vector sum of orbital angular momentum. Therefore, the values of $L$ are $6$, $5$, $4$, $3$, $2$, $1$ and $0$.

The value of $M_L$ goes from $-L$ to $+L$. Therefore, the values of $M_L$ are $-6$, $-5$, $-4$, $-3$, $-2$, $-1$, $0$$+1$, $+2$, $+3$, $+4$, $+5$, and $+6$.

An electron can have two value of spin angular momentum $\left(s\right)$ that are $\frac{1}{2}$ and $-\frac{1}{2}$.

The possible combinations of spin angular momentum for two electrons are $\left(\frac{1}{2},\frac{1}{2}\right)$, $\left(\frac{1}{2},-\frac{1}{2}\right)$, $\left(\frac{1}{2},-\frac{1}{2}\right)$ and $\left(-\frac{1}{2},-\frac{1}{2}\right)$.

Therefore, the possible values for the vector sum of spin angular momentum $S$ are $1$ and $0$. The value of $M_S$ goes from $-S$ to $+S$. Therefore, the values of $M_S$ are $-1$, $0$ and $1$.

The value of $J$ depends on $L$ and $S$.

The possible combination of $\left(L,S\right)$ are $\left(6,1\right)$, $\left(6,0\right)$, $\left(5,1\right)$, $\left(5,0\right)$, $\left(4,1\right)$, $\left(4,0\right)$, $\left(3,0\right)$, $\left(3,1\right)$, $\left(2,1\right)$, $\left(2,0\right)$, $\left(0,1\right)$, $\left(0,0\right)$, $\left(1,0\right)$ and $\left(1,1\right)$.

The value of $J$ can be calculated by the formula shown below.

$J=L+S\to \left|L-S\right|$

The addition of $\left(6,1\right)$ results in $J=7$.

The addition of $\left(6,0\right)$ and $\left(5,1\right)$ results in $J=6$.

The addition of $\left(5,0\right)$ and $\left(4,1\right)$ results in $J=5$.

The addition of $\left(4,0\right)$ and $\left(3,1\right)$ results in $J=4$.

The addition of $\left(3,1\right)$ and $\left(2,1\right)$ results in $J=3$.

The addition of $\left(2,0\right)$ and $\left(1,1\right)$ results in $J=2$.

The addition of $\left(0,1\right)$ and $\left(1,0\right)$ results in $J=1$.

The addition or subtraction of $\left(0,0\right)$ results in $J=0$.

The subtraction of $\left(6,1\right)$ and $\left(5,0\right)$ results in $J=5$.

The subtraction of $\left(6,0\right)$ results in $J=6$.

The subtraction of $\left(5,1\right)$ and $\left(4,0\right)$ results in $J=4$.

The subtraction of $\left(3,0\right)$ and $\left(4,1\right)$ results in $J=3$.

The subtraction of $\left(3,1\right)$ and $\left(2,0\right)$ results in $J=2$.

The subtraction of $\left(1,1\right)$ results in $J=0$.

The subtraction of $\left(0,1\right)$ and $\left(1,0\right)$ results in $J=1$.

Therefore, the values of $J$ are $7$, $6$, $5$, $4$$3$, $2$, $1$ and $0$.

The value of $M_J$ goes from $-J$ to $+J$.

Therefore, the possible value of are $-7$, $-6$, $-5$, $-4$, $-3$, $-2$, $-1$, $0$$+1$, $+2$, $+3$, $+4$, $+5$, $+6$ and $+7$.

Conclusion

The values of $L$ are $6$, $5$, $4$, $3$, $2$, $1$ and $0$.The values of $M_L$ are $-6$, $-5$, $-4$, $-3$, $-2$, $-1$, $0$$+1$, $+2$, $+3$, $+4$, $+5$, and $+6$.The values of $S$ are $1$ and $0$. The values of $M_S$ are $-1$, $0$ and $1$. The values of $J$ are $7$, $6$, $5$, $4$$3$, $2$, $1$ and $0$. The possible value of $M_J$ are $-7$, $-6$, $-5$, $-4$, $-3$, $-2$, $-1$, $0$$+1$, $+2$, $+3$, $+4$, $+5$, $+6$ and $+7$.

Interpretation Introduction

(c)

The possible values of $L$, $M_L$, $S$, $M_S$, $J$, and $M_J$ for two coupled electrons, one in $p$ orbital and one in $d$ orbital are to be stated.

Concept introduction:

The symbol $L$ represents the vector sum of orbital angular momentum. The value of $M_L$ goes from $-L$ to $+L$.The symbol $S$ represents the vector sum of spin angular momentum. The value of $M_S$ goes from $-S$ to $+S$. The symbol $J$ represents the total angular momentum. The value of $M_J$ goes from $-J$ to $+J$.

Answer to Problem 15.9E

The values of $L$ are $3$, $2$, $1$ and $0$. The values of $M_L$ are $-3$$-2$, $-1$, $0$$+1$, $+2$, and $+3$. The values of $M_S$ are $-1$, $0$ and $1$. The values of $J$ are $4$, $3$, $2$, $1$ and $0$. The possible values of $M_J$ are $-4$, $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, and $4$.

Explanation of Solution

When two coupled $p$ electrons are present in different shells, then the value of principal quantum number for both electrons will be different.

The orbital angular momentum $\left(l\right)$ for $p$ subshell is $1$. For $p^1$ electron configuration, the values of $m_l$ is shown below.

$\begin{array}{l}{m}_l+10-1\\ \boxed{}\boxed{}\boxed{}\end{array}$

An electron can occupy any of the orbital. Therefore, an electron in the orbital $p$ can have $+1$, $0$, and $-1$ values of $m_l$.

The orbital angular momentum $\left(l\right)$ for $d$ subshell is $2$. For $d^1$ electron configuration, the values of $m_l$ is shown below.

$\begin{array}{cccccc}{m}_l& +2& +1& 0& -1& -2\\ & \boxed{}& \boxed{}& \boxed{}& \boxed{}& \boxed{}\end{array}$

An electron can occupy any of the orbital. Therefore, an electron in the orbital $d$ can have $+2$, $+1$, $0$, $-1$, and $-2$ values of $m_l$.

When one $p$ electron coupled one $d$ electron, then the combination of $m_l$ value are, $\left(+2,+1\right)$, $\left(+2,0\right)$, $\left(+2,-1\right)$, $\left(+1,+1\right)$, $\left(+1,0\right)$, $\left(0,-1\right)$, $\left(0,+1\right)$, $\left(0,0\right)$, $\left(0,-2\right)$, $\left(0,-1\right)$, $\left(-1,-1\right)$, $\left(-1,+1\right)$, $\left(-1,0\right)$, $\left(-1,-1\right)$, $\left(-2,-1\right)$, $\left(-2,+1\right)$, $\left(-2,0\right)$, and $\left(-2,-1\right)$.

The magnitude of the vector sums for $\left(+2,+1\right)$ and $\left(-2,-1\right)$ is $3$.

The magnitude of the vector sums for $\left(+2,0\right)$, $\left(+1,+1\right)$, $\left(-1,-1\right)$, and $\left(-2,0\right)$ is $2$.

The magnitude of the vector sums for $\left(+2,-1\right)$, $\left(+1,0\right)$, $\left(0,-1\right)$, $\left(0,+1\right)$, $\left(0,-1\right)$, $\left(-1,0\right)$, and $\left(-2,+1\right)$ is $1$.

The symbol $L$ represents the vector sum of orbital angular momentum. Therefore, the values of $L$ are $3$, $2$, $1$ and $0$.

The value of $M_L$ goes from $-L$ to $+L$. Therefore, the values of $M_L$ are $-3$$-2$, $-1$, $0$$+1$, $+2$, and $+3$.

An electron can have two value of spin angular momentum $\left(s\right)$ that are $\frac{1}{2}$ and $-\frac{1}{2}$.

The possible combinations of spin angular momentum for two electrons are $\left(\frac{1}{2},\frac{1}{2}\right)$, $\left(\frac{1}{2},-\frac{1}{2}\right)$, $\left(\frac{1}{2},-\frac{1}{2}\right)$ and $\left(-\frac{1}{2},-\frac{1}{2}\right)$.

Therefore, the possible values for the vector sum of spin angular momentum $S$ are $1$ and $0$. The value of $M_S$ goes from $-S$ to $+S$. Therefore, the values of $M_S$ are $-1$, $0$ and $1$.

The value of $J$ depends on the $L$ and $S$.

The possible combination of $\left(L,S\right)$ are $\left(3,1\right)$, $\left(3,0\right)$, $\left(2,1\right)$, $\left(2,0\right)$$\left(2,1\right)$, $\left(2,0\right)$, $\left(0,1\right)$, $\left(0,0\right)$, $\left(1,0\right)$ and $\left(1,1\right)$.

The value of $J$ can be calculated by the formula shown below.

$J=L+S\to \left|L-S\right|$

The addition of $\left(3,1\right)$ results in $J=4$.

The addition of $\left(3,0\right)$ and $\left(2,1\right)$ results in $J=3$.

The addition of $\left(2,0\right)$ and $\left(1,1\right)$ results in $J=2$.

The addition of $\left(0,1\right)$ and $\left(1,0\right)$ results in $J=1$.

The addition or subtraction of $\left(0,0\right)$ results in $J=0$.

The subtraction of $\left(3,1\right)$ and $\left(2,0\right)$ results in $J=2$.

The subtraction of $\left(1,1\right)$ results in $J=0$.

The subtraction of $\left(2,1\right)$, $\left(0,1\right)$ and $\left(1,0\right)$ results in $J=1$.

Therefore, the values of $J$ are $4$, $3$, $2$, $1$ and $0$.

The value of $M_J$ goes from $-J$ to $+J$.

Therefore, the possible value of $-4$, $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, and $4$.

Conclusion

The values of $L$ are $3$, $2$, $1$ and $0$. The values of $M_L$ are $-3$$-2$, $-1$, $0$$+1$, $+2$, and $+3$. The values of $M_S$ are $-1$, $0$ and $1$. The values of $J$ are $4$, $3$, $2$, $1$ and $0$.The possible values of $M_J$ are $-4$, $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, and $4$.