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I see many cases where electrons jump from various energy levels of some $L$ value to various levels of other $L$ value. But is there any transition in same $L$ level between its different orientations? $EDIT $ (How to calculate energy difference?)

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They can, following selection rules. As an example the line of hydrogen $\lambda=21 cm$ corresponds to a transition with $\Delta L=0, \Delta S=1$

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