Magnetic Quantum Numbers and Spatial Orientation

As discussed here, p orbitals are created when an orbital has one angular node. We used the example of the xy plane to show this, and that resulted in what is called the p_z orbital. However, we could just as easily have chosen the xz plane or the yz plane. In fact, Schrödinger’s math allows for all three of these possibilities, as shown here. 

Which of these orbitals is generated by the math depends on the value of the variable m. It is not possible to say definitively which value of m gives which orbital, because atoms don’t actually have axes, but by convention (in other words, by agreement) we say that m=-1 is the p_x orbital, m=0 is the p_y and m=1 is the p_z orbital.

We know that there are three possibilities because Schrödinger’s equation puts a numerical restriction on the possible values of m. 

Specifically, m can be any whole number from -ℓ to 0 to ℓ. 

So, if ℓ=1 (for a p orbital) then m can be -1, 0 or 1. If ℓ=3, then m has values of -3, -2, -1, 0, 1, 2, and 3. That means that on a given energy level, there can only be 1 s orbital (ℓ=0, m=0), there can be 3 p orbitals (ℓ=1, m=-1,0,1), 5 d orbitals (ℓ=2, m=-2,-1,0,1,2), etc.