I Are certain combinations of quantum numbers (basis vectors) forbidden?

[Happiness]

TL;DR

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.

 

[Isaac0427]

$|nlsm_lm_s\rangle$ is equivalent to $|nlsm_lm_j\rangle$ due to the relation $m_j=m_l+m_s$. This relation does render some $|nlsm_lm_j\rangle$ states invalid, though, eg., for a spin-1/2 particle we have the requirement $m_j = m_l \pm 1/2$. I think this just makes $|nlsm_lm_j\rangle$ a less-desirable way to write out the state, though it is equivalent.

In general, with the addition of angular momentum, if you know all three angular momentum quantum numbers (i.e. l, s, and j), you may know at most one magnetic quantum number. If you know all three magnetic quantum numbers, you may know at most two angular momentum quantum numbers. You can know less information, but not more (with the exception of the "top" or "bottom" states, where $j=l+s$ and $m_j = \pm j$).

Also, in the case of adding orbital angular momentum and spin, we always know the s quantum number, because it is an intrinsic property of the particle, so that also effectively limits what combination of quantum numbers you can have.