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$$a|m_J=-1>+b|m_J=0>+c|m_J=1>$$

i.e. a coherent superposition of different magnetic sublevels? And will different atoms (usually I generate several in the ##J=1## state) have different a, b and c values?

I know that on average, I expect 1/3 of the atoms in the ##J=1## state in each magnetic substate. So if I am to rotate my frame of reference such that the wavefunction above becomes, say, ##|m_J=0>## will in this frame of reference all the atoms be exactly in one of the states ##|m_J=-1>##, ##|m_J=0>## or ##|m_J=1>## or can there still be some of them in linear superpositions of these basis?

Also, if I am in the initial frame, where I have a linear superposition of substates and I apply a magnetic field (adiabatically such that I don't transfer population) along the current defined z-axis, will the atom still have the same population in each of the (now non-degenerate) magnetic substates? Thank you!