Question

What values of m are permitted for an electron with `l = 2`?

Answer 1

Possible `m_l` values are `-2,-1,0,1,2`. See below.

Explanation:

There are four quantum numbers: the principle quantum number, `n`, the angular momentum quantum number, `l`, the magnetic quantum number, `m_l`, and the electron spin quantum number, `m_s`. For this question we are concerned with `l` and `m_l`.

The angular momentum quantum number, `l`, describes the shape of the subshell and its orbitals, where `l=0,1,2,3...` corresponds to `s, p, d,` and `f` subshells (containing `s, p, d, f` orbitals), respectively. Each shell has up to `n-1` types of subshells/orbitals.

An angular momentum quantum number of `l=2` describes a `d` subshell.

The magnetic quantum number, `m_l`, describes the orientation of the orbitals (within the subshells) in space. The possible values for `m_l` of any type of orbital (`s,p,d,f...`) is given by any integer value from `-l` to `l`.

Therefore, given `l=2`, the possible `m_l` values are `-2,-1,0,1,2`. This tells us that the `d` subshell contains five `d` orbitals, each with a different orientation (`d_(yz)`, `d_(xy)`, `d_(xz)`, `d_(x^2-y^2)`, and `d_(z^2)`).