M(l) Quantum Number

[Emma Healy 2J]

Hi! If we were looking at the 2p orbital, then n=2 and l=1, then m(l) could equal -1, 0, or 1, thus being 3 possible quantum values. If we made our own arbitrary scale, like p(x)=-1, p(y)=0, and p(z)=1, we could say that m(l) = -1 if we wanted the quantum number for p(x)? Or does that matter?

I know that if the question said that p(x)=-1, we would only have one quantum number for m(l) since we were given only one value.

[Jiwon_Chae_3L]

m(s) is referring to the spin orientation of an electron, and its only possible values are +1/2 and -1/2 to my understanding. Sorry if this didn't help much.

[Emma Healy 2J]

Yes, okay thank you!

[Juliet Carr 1F]

I think m(l) itself is a quantum number, so I don't know if you could have three different quantum numbers for m(l), but instead you could have three different values for the quantum number m(l). I'm pretty sure if the last electron falls in in p(x) m(l) is -1, 0 for p(y), and 1 for p(z). So, I think the scale p(x)=-1, p(y)=0, and p(z)= 1 is accurate.

[505598869]

m(l) depends on the value of l. If l=1, then m(l) can equal -1,0, or 1.
If l=2, then m(l) can equal -2,-1,0,1,2.

So if the problem says that n=2, l=1, m(l)= 1
then it would be referring specifically to the "2pz" state.

[Chem_Mod]

I think your question is asking about the assignment of directionality with regards to the p_x, p_y, and p_z orbitals. This is a good question because we use the magnetic quantum number, m_l, to distinguish between differences in the orientation of orbitals within the same subshell, in this case the p subshell. Since the orbital angular momentum (or azimuthal) quantum number, l, for the p subshell is l=1, indeed electrons in this shell can take on m_l = -1, 0, or +1.

To start, there's a level of breakdown in the correspondence between the mathematical model and actually assigning/determining orientations of orbitals in reality. It's indistinguishable to us if an electron is in either three of the p orbitals; between degenerate orbitals, labelling orbitals is arbitrary insofar as direction is arbitrary.

When it comes to our mathematical model, the wavefunctions themselves that describe the p orbitals are mathematically distinguishable for values of m_l, but the wavefunctions are constructed using spherical coordinates with complex (or imaginary) numbers. Doing the quantum mechanics shows us that the p_z orbital corresponds to m_l = 0 directly, and this is because the angular components in spherical coordinates are defined by the z axis.

Concerning the p_x and p_y orbitals, these two are actually linear combinations of the angular components of the wavefunctions for m_l = -1 and +1. Having a linear combination of the angular components brings them from complex space (with imaginary numbers) into real space (no imaginary numbers), although the problem now is that these two orbitals are indistinguishable with respect to the x and y axes -- we only know, in the model, that they lie in the x-y plane perpendicularly to each other and at 45 degrees with respect to the x and y axes.