# What are the four quantum numbers?

Dec 21, 2016

See below.

#### Explanation:

The four quantum numbers are the principle quantum number, $n$, the angular momentum quantum number, $l$, the magnetic quantum number, ${m}_{l}$, and the electron spin quantum number, ${m}_{s}$.

The principle quantum number , $n$, describes the energy and distance from the nucleus, and represents the shell.

For example, the $3 d$ subshell is in the $n = 3$ shell, the $2 s$ subshell is in the $n = 2$ shell, etc.

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.

For example, the $n = 3$ shell has subshells of $l = 0 , 1 , 2$, which means the $n = 3$ shell contains $s$, $p$, and $d$ subshells (each containing their respective orbitals). The $n = 2$ shell has $l = 0 , 1$, so it contains only $s$ and $p$ subshells. It is worth noting that each shell has up to $n - 1$ types of subshells/orbitals.

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 ($s , p , d , f \ldots$) is given by any integer value from $- l$ to $l$.

So, for a $2 p$ orbital with $n = 2$ and $l = 1$, we can have ${m}_{1} = - 1 , 0 , 1$. This tells us that the $p$ orbital has $3$ possible orientations in space.

If you've learned anything about group theory and symmetry in chemistry, for example, you might remember having to deal with various orientations of orbitals. For the $p$ orbitals, those are ${p}_{x}$, ${p}_{y}$, and ${p}_{z}$. So, we would say that the $2 p$ subshell contains three $2 p$ orbitals (shown below). Finally, the electron spin quantum number, ${m}_{s}$, has only two possible values, $+ \frac{1}{2} \mathmr{and} - \frac{1}{2}$. As the name implies, these values describe the spin of each electron in the orbital.

Remember that there are only two electrons to every orbital, and that they should have opposite spins (think Pauli exclusion principle). This tells us that there are two electrons per orbital, or per ${m}_{l}$ value, one with an ${m}_{s}$ value of $+ \frac{1}{2}$ and one with an ${m}_{s}$ value of $- \frac{1}{2}$.

Therefore, $n$ describes the shell, both $n$ and $l$ describe a subshell, $n$, $l$, and ${m}_{l}$ describe an orbital, and all four quantum numbers ($n$, $l$, ${m}_{l}$, ${m}_{s}$) describe an electron.