Question

The spin-orbit interaction splits the hydrogen 4f state into many.

(a)Identity these states and rank them in order of increasing energy.

(b)If a weak external magnetic field were now introduced (weak enough that it does not disturb the spin-orbit coupling). Into how many different energies would each of these states be split?

Short answer

$$\begin{gathered} a)4f_{\frac{5}{2}},4f_{\frac{7}{2}} \\ b)4f_{\frac{5}{2}}\to 6\mathrm{states},f_{\frac{7}{2}}\to 8\mathrm{states} \end{gathered}$$

Step-by-step solution

Given data

We are considering the 4f state.

Concept of the quantum number

The quantum number $\mathit{j}\mathbf{=}\mathit{l}\mathbf{\pm }\mathit{s}$

Possible States of 4f52 and 4f72and.        

Since the state is 4f, we conclude that the quantum number l is . Thus, the quantum number can have the following values

$j=l\pm s=3\pm \frac{1}{2}=\frac{5}{2},\frac{7}{2}$

We write the possible states as

$4f_{\frac{5}{2}},4f_{\frac{7}{2}}$

The states are ordered starting from the lower energy ones: this is due to the fact that higher j corresponds to higher energy.

Influence of a Weak Magnetic Field.

(b)

For $j=\frac{5}{2}$, the number of the possible values for ${\mathrm{m}}_{\mathrm{j}}$is,

$$\begin{gathered} 2j+1=2\times \frac{5}{2}+1 \\ =6 \end{gathered}$$

The number of states that this state would break into under the influence of a weak magnetic field is 6.

For $j=\frac{7}{2}$, the number of possible values for ${\mathrm{m}}_{\mathrm{j}}$is,

$$\begin{gathered} 2j+1=2\times \frac{7}{2}+1 \\ =8 \end{gathered}$$

The number of states that this state would break into under the influence of a weak magnetic field is 8.