[mantis shrimp]

For p and d orbitals, can anyone tell me to which orientation each magnetic quantum number relates?
In other words, for p orbitals, when m_l = -1, 0, +1, does the -1 describe a p_x, p_y or p_z and how do you know?

Follow up question, is there any filling order when it comes to orientation or is it random - i.e. does p_x fill before p_y?
Thanks

[Corribus]

First question - the assignment is arbitrary, because the x, y, and z axes are mathematical abstractions. There are frequently conventions that assist in the standardized naming of axes (e.g., in benzene the z axis p orbitals are often described as those perpendicular to the molecular plane) but there's no absolute reason that you have to pick one or the other.

Second question - in the absence of a magnetic field, the p orbitals are degenerate (have the same energy) and so there is no preference. The energy of hydrogenic atom orbitals depends only on the principle quantum number n. This changes of course when electrons are filled in, as the presence of an electron modulates the energies of the remaining orbitals to different degrees.

The situation is a little complicated because the x- and y-oriented orbitals in real space are actually linear combinations of the (imaginary) components of the hydrogen atom wavefunctions. But in any case, in a magnetic field the orbital energies split because of different vectoral interactions between the p-orbital electrons and the static magnetic field. This splitting is responsible for the Zeeman effect and it is why the z-axis projection quantum number is often called the magnetic quantum number.

[Enthalpy]

Oh, Corribus has left a bit for me!

[...] when m_l = -1, 0, +1, does the -1 describe a p_x, p_y or p_z [...]?

A nice illustration of 2p orbitals there where m_l=0
http://winter.group.shef.ac.uk/orbitron/AOs/2p/index.html
only their orientation differs.

These peacock-shaped 2p have zero angular momentum around their symmetry axis because the phase of the wavefunction is constant over a turn around the axis.

But you can define equally well three 2p orbitals where the phase makes +1 or -1 time 360° over a turn around the axis. These would have an angular momentum of +1 or -1. More than 1 needs a d orbital or more, which begins at 3d (more illustrations at the linked website).

The second type is doughnut-shaped. Each doughnut 2p is a linear combination of two peacock 2p: 2p_z doughnut is (2p_x+2p_y)/sqrt(2) peacock, where one peacock is +90° or -90° out-of-phase. That's the same math story as the sum of two linearly polarized electromagnetic waves, perpendicular to an other and with ±90° phase, making a circularly polarized one.

If you take the exp(iEt/ħ) term too and observe the locations of constant phase at a doughnut, they turn around the nucleus, since the phase evolves both with position and time. This makes the angular and magnetic momenta. In this sense, there is movement. But the envelope of the wavefunction is constant over time: in that other sense, there is no movement, the wavefunction is "stationary", and the electron radiates no light.

Peacock 2p are linear combinations of two doughnut 2p too: for instance peacock 2p_z is the sum of +1 and -1 doughnut 2p_y or 2p_z with adequate sqrt(2) and a relative phase that orients the peacock direction. So doughnuts and peacocks are just equally good - each set of three defines a base of all 2p wavefunctions. Even elliptic 2p would be as fundamental as peacocks and doughnuts, just less convenient.

Chemistry uses peacock 2p because they are easier to imagine in chemical bonds. On the other hand, an external magnetic field would separate the energies of +1 and -1 magnetic moment, so their linear combination as a peacock isn't a stationary wavefunction more: the doughnuts are a better description then.

If you take a peacock p_z, it has 0 momentum around z, but one lobe has 0° phase and the other 180°. Now, in a turn around x or y, the phase does change from 0 through 180° to 360°, so there is a momentum - just uncertain because +180° can be -180° too. This is what the sum of two doughnuts around x or y tells as well: the angular momentum around x or y is uncertain, it can be +1 or -1. 2p orbitals are a nice illustration that a definite momentum around z makes an uncertain momentum around x and y. I use 2p orbitals as an imperfect mental image for the intrinsic spin.

As peacocks and doughnuts are linear combinations of an other, they don't add places for the electrons - the wavefunctions must be orthogonal. So you can count the places with peacocks or doughnuts, but not both, and have at most 2+2+2 electrons as 2p orbitals.

[mantis shrimp]

Wow, thanks for the great answers! That is what I was thinking, but I haven't been able to find that in any of my texts or online. Thanks!