The transition line, as shown in the figure, arises between \( 2D_{3/2} \) and \( 2P_{1/2} \) states without any external magnetic field. The number of lines that will appear in the presence of a weak magnetic field (in integer) is \(\underline{\hspace{2cm}}\).
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In the presence of a weak magnetic field, the energy levels split due to the Zeeman effect, resulting in multiple transition lines.
In the absence of an external magnetic field, the transition between the \( 2D_{3/2} \) and \( 2P_{1/2} \) states would result in a single transition line. However, in the presence of a weak magnetic field, the degeneracy of the energy levels is lifted, leading to the splitting of the transition line.
For the \( 2D_{3/2} \) state, the total angular momentum \( J = 3/2 \), and for the \( 2P_{1/2} \) state, \( J = 1/2 \). The magnetic field causes the splitting of these levels according to the Zeeman effect.
The total number of possible lines that can appear is determined by the number of different transitions between the \( m_j \) values of the \( 2D_{3/2} \) and \( 2P_{1/2} \) states. For \( 2D_{3/2} \), there are 4 possible \( m_j \) values (\( m_j = -3/2, -1/2, 1/2, 3/2 \)), and for \( 2P_{1/2} \), there are 2 possible \( m_j \) values (\( m_j = -1/2, 1/2 \)).
Therefore, the total number of lines that can appear is:
\[
3 \times 2 = 6.
\]
Thus, the number of lines that will appear in the presence of a weak magnetic field is 6.
