Chapter 15, Problem 15.8E

Interpretation Introduction

(a)

Interpretation:

Whether the transition $\left(1,0,0\right)\to \left(2,0,0\right)$ is allowed or not is to be stated.

Concept introduction:

Electronic spectra are formed by the excitation and relaxation process of electrons. The excitation process is due to the transition of an electron from a lower energy level to a higher energy level. The relaxation process is due to the transition of an electron from a higher energy level to a lower energy level.

Answer to Problem 15.8E

The transition $\left(1,0,0\right)\to \left(2,0,0\right)$ is not allowed.

Explanation of Solution

The selection rules are shown below.

$\begin{array}{l}\Delta l=\pm 1\\ \Delta m_l=0,\pm 1\end{array}$

Where,

• $l$ represents the azimuthal quantum number.

• $m_l$ represents the magnetic quantum number.

The change in principal quantum number $\left(\Delta n\right)$ can have any value.

The electronic transition is $\left(1,0,0\right)\to \left(2,0,0\right)$.

The azimuthal quantum number for the initial and final energy level is $0$. Therefore, the value of $\Delta l$ cannot be $\pm 1$.

The magnetic quantum number for the initial and final energy level is $0$.

The selection will not allow this transition. Therefore, the transition $\left(1,0,0\right)\to \left(2,0,0\right)$ is not allowed.

Conclusion

The transition $\left(1,0,0\right)\to \left(2,0,0\right)$ is not allowed.

Interpretation Introduction

(b)

Interpretation:

Whether the transition $\left(1,0,0\right)\to \left(2,1,0\right)$ is allowed or not is to be stated.

Concept introduction:

Electronic spectra are formed by the excitation and relaxation process of electrons. The excitation process is due to the transition of an electron from a lower energy level to a higher energy level. The relaxation process is due to the transition of an electron from a higher energy level to lower energy level.

Answer to Problem 15.8E

The transition $\left(1,0,0\right)\to \left(2,1,0\right)$ is allowed.

Explanation of Solution

The selection rules are shown below.

$\begin{array}{l}\Delta l=\pm 1\\ \Delta m_l=0,\pm 1\end{array}$

Where,

• $l$ represents the azimuthal quantum number.

• $m_l$ represents the magnetic quantum number.

The change in principal quantum number $\left(\Delta n\right)$ can have any value.

The electronic transition is $\left(1,0,0\right)\to \left(2,1,0\right)$.

The azimuthal quantum number for the initial energy level is $0$.

The azimuthal quantum number for the final energy level is $1$.

The value of $\Delta l$ is calculated as shown below.

$\Delta l=l_{\text{f}}-l_{\text{i}}$ …(1)

Where,

• $l_{\text{i}}$ represents the azimuthal quantum number for the initial energy level.

• $l_{\text{f}}$ represents the azimuthal quantum number for the final energy level.

Substitute the values of $l_{\text{i}}$ and $l_{\text{f}}$ in the equation (1).

$\begin{array}{rll}\Delta l& =& 1-0\\ & =& +1\end{array}$

The value of $\Delta l$ is $+1$.

The magnetic quantum number for the initial and final energy level is $0$.

The value of $\Delta m_l$ is $0$.

The selection will allow this transition. Therefore, the transition $\left(1,0,0\right)\to \left(2,1,0\right)$ is allowed.

Conclusion

The transition $\left(1,0,0\right)\to \left(2,0,0\right)$ is allowed.

Interpretation Introduction

(c)

Interpretation:

Whether the transition $\left(1,0,0\right)\to \left(2,1,-1\right)$ is allowed or not is to be stated.

Concept introduction:

Electronic spectra are formed by the excitation and relaxation process of electrons. The excitation process is due to the transition of an electron from a lower energy level to a higher energy level. The relaxation process is due to the transition of an electron from a higher energy level to a lower energy level.

Answer to Problem 15.8E

The transition $\left(1,0,0\right)\to \left(2,1,-1\right)$ is allowed.

Explanation of Solution

The selection rules are shown below.

$\begin{array}{l}\Delta l=\pm 1\\ \Delta m_l=0,\pm 1\end{array}$

Where,

• $l$ represents the azimuthal quantum number.

• $m_l$ represents the magnetic quantum number.

The change in principal quantum number $\left(\Delta n\right)$ can have any value.

The electronic transition is $\left(1,0,0\right)\to \left(2,1,-1\right)$.

The azimuthal quantum number for the initial energy level is $0$.

The azimuthal quantum number for the final energy level is $1$.

The value of $\Delta l$ is calculated as shown below.

$\Delta l=l_{\text{f}}-l_{\text{i}}$ …(1)

Where,

• $l_{\text{i}}$ represents the azimuthal quantum number for the initial energy level.

• $l_{\text{f}}$ represents the azimuthal quantum number for the final energy level.

Substitute the values of $l_{\text{i}}$ and $l_{\text{f}}$ in the equation (1).

$\begin{array}{rll}\Delta l& =& 1-0\\ & =& +1\end{array}$

The value of $\Delta l$ is $+1$.

The magnetic quantum number for the initial energy level is $0$.

The magnetic quantum number for the final energy level is $-1$.

The value of $\Delta l$ is calculated as shown below.

$\Delta m_l=m_{l,\text{f}}-m_{l,\text{i}}$ …(2)

Where,

• $m_{l,\text{f}}$ represents the magnetic quantum number for the initial energy level.

• $m_{l,\text{i}}$ represents the magnetic quantum number for the final energy level.

Substitute the values of $m_{l,\text{f}}$ and $m_{l,\text{i}}$ in the equation (1).

$\begin{array}{rll}\Delta m_l& =& m_{l,\text{f}}-m_{l,\text{i}}\\ & =& 0-\left(-1\right)\\ & =& +1\end{array}$

The value of $\Delta m_l$ is $+1$.

The selection will allow this transition. Therefore, the transition $\left(1,0,0\right)\to \left(2,1,-1\right)$ is allowed.

Conclusion

The transition $\left(1,0,0\right)\to \left(2,1,-1\right)$ is allowed.

Interpretation Introduction

(d)

Interpretation:

Whether the transition $\left(4,3,2\right)\to \left(6,3,1\right)$ is allowed or not is to be stated.

Concept introduction:

Electronic spectra are formed by the excitation and relaxation process of electrons. The excitation process is due to the transition of an electron from a lower energy level to a higher energy level. The relaxation process is due to the transition of an electron from a higher energy level to a lower energy level.

Answer to Problem 15.8E

The transition $\left(4,3,2\right)\to \left(6,3,1\right)$ is not allowed.

Explanation of Solution

The selection rules are shown below.

$\begin{array}{l}\Delta l=\pm 1\\ \Delta m_l=0,\pm 1\end{array}$

Where,

• $l$ represents the azimuthal quantum number.

• $m_l$ represents the magnetic quantum number.

The change in principal quantum number $\left(\Delta n\right)$ can have any value.

The electronic transition is $\left(4,3,2\right)\to \left(6,3,1\right)$.

The azimuthal quantum number for the initial energy level is $3$.

The azimuthal quantum number for the final energy level is $3$.

The value of $\Delta l$ is calculated as shown below.

$\Delta l=l_{\text{f}}-l_{\text{i}}$ …(1)

Where,

• $l_{\text{i}}$ represents the azimuthal quantum number for the initial energy level.

• $l_{\text{f}}$ represents the azimuthal quantum number for the final energy level.

Substitute the values of $l_{\text{i}}$ and $l_{\text{f}}$ in the equation (1).

$\begin{array}{rll}\Delta l& =& 3-3\\ & =& 0\end{array}$

The value of $\Delta l$ is $0$.

The magnetic quantum number for the initial energy level is $2$.

The magnetic quantum number for the final energy level is $1$.

The value of $\Delta l$ is calculated as shown below.

$\Delta m_l=m_{l,\text{f}}-m_{l,\text{i}}$ …(2)

Where,

• $m_{l,\text{f}}$ represents the magnetic quantum number for the initial energy level.

• $m_{l,\text{i}}$ represents the magnetic quantum number for the final energy level.

Substitute the values of $m_{l,\text{f}}$ and $m_{l,\text{i}}$ in the equation (1).

$\begin{array}{rll}\Delta m_l& =& m_{l,\text{f}}-m_{l,\text{i}}\\ & =& 2-1\\ & =& +1\end{array}$

The value of $\Delta m_l$ is $+1$.

The selection will not allow this transition. Therefore, the transition $\left(4,3,2\right)\to \left(6,3,1\right)$ is not allowed.

Conclusion

The transition $\left(4,3,2\right)\to \left(6,3,1\right)$ is not allowed.