# Does the angular momentum quantum number l designate the shape of the orbital?

Feb 7, 2016

Basically yes. The angular momentum quantum number $l$ corresponds to the shape of the orbital sublevel, and the magnetic quantum number ${m}_{l}$ basically "builds" the shape when the $p$ electron is subjected to a magnetic field.

But for the most part you can say that $l$ indicates the shape because different values of $l$ correspond to different orbital shapes.

If we take the $2 {p}_{z}$ orbital as an example and subject it to a magnetic field oriented in the $z$ direction, the orbital angular momentum in the $z$ direction sweeps out cones like this:

(Ignore the "|uarr>>" and "|darr>>"; it is not relevant, though it is called bra-ket notation if you look it up and bother to read more on it! Also, you would not have to explain this diagram on a General Chemistry test. In case you were curious though, the diagram shows ℏ"/"2, which is the height of the cones in units of ℏ. Anyways...)

Looks like a $2 {p}_{z}$ orbital, right? It should.

The orbital angular momentum in the $z$ direction is ${L}_{z}$. In the following equation (which you don't need to use on a General Chemistry test):

color(blue)(hatL_z)Y_l^(m_l)(theta,phi) = color(blue)(m_lℏ)Y_l^m(theta,phi)

What you should notice here is that ${\hat{L}}_{z}$, the orbital angular momentum operator for the z-direction, corresponds to m_lℏ.

If $n = 2$, $l = 1$ corresponds to a $2 p$ atomic orbital, so ${m}_{l} = 0 , \pm 1$.

• The $- 1$ refers to the lower cone on the above image.
• The $0$ refers to a dot at the origin in the above image. It is also where the node is.
• The $+ 1$ refers to the upper cone in the above image.

This tells us that ${m}_{l}$ is also known as the vector projection of $l$.

Therefore, what you should notice is that ${m}_{l}$ "builds" the shape of the orbital, while the number of ${m}_{l}$ values corresponds to how many unique, orthogonal (perpendicular) orientations there are for any orbital in that sublevel.

(The orthogonality is not crucial knowledge for General Chemistry, but it matters because a quantum mechanics postulate states that any orbital must be orthogonal to every other orbital in its sublevel.)

Ultimately, since there are three values for ${m}_{l}$, there are three $2 p$ orbitals: $2 {p}_{x}$, $2 {p}_{y}$, and $2 {p}_{z}$.