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- TL;DR Summary
- The electron's wavefunction is usually expressed in the standard basis {n, l, m_l, s, m_s}, but how to express it in the basis {n, l, m_l, s, m_j} ? (Note that m_s is replaced with m_j.) Or is it that certain combinations of quantum numbers are forbidden?

I've seen the hydrogen electron's wavefunction expressed in the basis ##\ket{n l s m_l m_s}## or ##\ket{n l s j m_j}##, but so far, never in ##\ket{n l s m_l m_j}##. My question is, are certain combinations of quantum numbers, eg, ##\ket{n l s m_l m_j}##, forbidden?

If ##\ket{n l s m_l m_j}## is not forbidden, how do we get it from the standard basis ##\ket{n l s m_l m_s}##?

I know how to get ##\ket{n l s j m_j}## from ##\ket{n l s m_l m_s}## using Clebsch-Gordan coefficients:

where ##J=L+S##.

##J## is the total angular momentum.

But other than that, I do not know how to express the wavefunction in other bases.

If ##\ket{n l s m_l m_j}## is not forbidden, how do we get it from the standard basis ##\ket{n l s m_l m_s}##?

I know how to get ##\ket{n l s j m_j}## from ##\ket{n l s m_l m_s}## using Clebsch-Gordan coefficients:

where ##J=L+S##.

##J## is the total angular momentum.

But other than that, I do not know how to express the wavefunction in other bases.

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