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Atomic orbitals are usually labeled $1s$, $2p_x$, $2p_x$, $2p_z$ and so on. These wave functions are defined to be real valued. The original wave functions are complex valued. The $2p_x$ orbital is for example a combination of the complex valued wave functions having quantum numbers $(l,m=1,-1)$ and $(l,m=1,1)$. Is it common to refer to the complex valued wave functions also by $2p_x$ and $2p_y$ for $(l=1,m=1)$ and $(l=1,m=-1)$ (or vice versa), respectively?

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No.

The $p_x$ and $p_y$ orbitals are always real-valued, and the complex-valued $|m|=1$ orbitals are always denoted $p_{1}$ and $p_{-1}$. Depending on what scheme you're using, the third $l=1$ orbital can be denoted either $p_z$ or $p_0$, with both notations completely equivalent. Just because the $p_x$ and $p_y$ orbitals are linear combinations of $p_1$ and $p_{-1}$ does not mean that you can use the symbols interchangeably.

(Is it common, though? Who knows, but it doesn't matter - it's incorrect use of the notation regardless of how common it is.)

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You have stumbled upon a difference between how chemists and physicist denote orbitals. The $p_x, p_y, p_z$ notation is common in chemistry because the resulting orbitals are real. Physicist use $p_{-1}, p_0, p_1$ where the subscripts are the values of m and embrace the complexity of the resulting wave function. It is purely a matter of choice since all calculations yield the same result.

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