What Are the Selection Rules for the Zeeman Effect?

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The Zeeman effect is influenced by four selection rules that dictate the allowed transitions of atomic states in the presence of a magnetic field. For a transition to be permitted, it must satisfy all four selection rules, not just one. This means that if any of the rules are violated, the transition will not occur. Understanding these rules is crucial for predicting spectral lines in magnetic fields. Therefore, all four selection rules are essential for determining allowed transitions in the Zeeman effect.
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I am reading some online article on Zeeman effect
http://math.ucr.edu/home/baez/spin/node10.html
In the article, the author mentions 4 selection rules, I wonder the corresponding transition is allowed by satisfying one of these selection rules or the corresponding line will be allowed only when all rules are satisfied?

Thanks.
 
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All four selection rules must be satisfied.
 
Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?

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