Question

e) Sketch the n = 1, and n = 2 energy levels for a hydrogen atom in a magnetic field. Indicate three possible transitions with solid lines. Indicate three forbidden (or at least highly suppressed) transitions with dotted lines, and state why they are forbidden. (Ignore fine and hyperfine structure.)

Answer 1

The selection rule is given by

t1=ܐܠ

and

$\Delta m_j =0,\pm 1$,

where is the orbital quantum number , $m_j$ is the total magnetic quantum number and $\Delta x$ is the change in the quantity
T
for a transition from initial to final state of energy $(i \rightarrow f)$ .

Also keep in mind that $l=0$ gives an orbital and $l=1$ gives a orbital.

All possible energy levels for and $n=2(l=0,1)$ in a magnetic field B are shown in the following diagram.

.

2 - 0 В + 0 т. 312 арза 1 (2. 1/2 у -32 у ze @P / 2 1/2 - 112 25 / 2 2 - 1/2 15112 12 n= | \/ - 112

In the above diagram, 3 allowed transitions  are shown with red solid lines and 3 forbidden transitions with violet dotted lines. They are forbidden because they do not obey the selection rule.

From the left, the 1st dotted transition is forbidden as $\Delta m_j =-2$ in this case.

The 2nd dotted transition is forbidden as $\Delta l =0$ in this case.

The 3rd dotted transition is forbidden as $\Delta m_j =2$ in this case.