Quantum numbers

A single electron inside an atom is fully characterised by a set of four quantum numbers: the principal quantum number $n$, the azimuthal (subsidiary) quantum number $l$, the magnetic quantum number $m$, and the spin quantum number $s$. It's worth noting where each of these actually comes from mathematically: solving the Schrödinger equation for a wavefunction $\Psi$ naturally produces the first three - $n$, $l$ and $m$ - as the parameters needed to specify a valid solution. The fourth quantum number, spin, does not come out of that same spatial wave equation at all; it arises separately, from the fact that the electron itself spins about its own axis. (That said, picturing this literally as a tiny sphere physically spinning like a top is misleading and not to be taken as a literal classical picture - "spin" is a genuinely quantum property that simply behaves, in some ways, analogously to classical spinning.)

Principal quantum number ($n$). This number specifies which energy shell the electron occupies - the shell it "revolves" in - and is denoted by the letter $n$.

• $n$ can take positive integer values $1,2,3,\dots$. Shells are also given letter names: $n=1$ is called the K shell, $n=2$ is the L shell, and $n=3,4,5$ are the M, N and O shells respectively.
• The maximum number of electrons a given shell can hold is $2n^2$.
• $n$ alone fixes both the electron's energy, $E_n=-\left(1312.8\,Z^2/n^2\right)$ kJ mol$^{-1}$, and its average distance from the nucleus, $r_n=\left(0.529\,n^2/Z\right)$ Å - the same formulas already met in the Bohr model, now reinterpreted as quantum-mechanical results rather than as classical orbit properties.

Azimuthal (subsidiary) quantum number ($l$).

• Denoted by the letter $l$, it can take any integer value from $0$ up to $n-1$ for a given shell.
• Each distinct value of $l$ defines a subshell (equivalently, describes the shape of the orbitals in that subshell): $l=0,1,2,3,4$ correspond to the s, p, d, f and g subshells respectively.
• The maximum number of electrons a given subshell can hold is $2(2l+1)$.
• $l$ also fixes the size of the electron's orbital angular momentum, through

Magnetic quantum number ($m_l$).

1. Denoted $m_l$, it takes integer values running from $-l$ to $+l$ passing through 0 - so for example if $l=1$, then $m_l=-1,\,0,\,+1$.
2. For a fixed $l$, each different value of $m_l$ represents a different spatial orientation of that subshell's orbitals.
3. The experimental evidence for this quantum number is the Zeeman effect - the splitting of spectral lines when an atom is placed in a magnetic field, which happens precisely because the different $m_l$ orientations, ordinarily all of equal energy, are pulled apart in energy once an external field singles out a preferred direction in space.
4. To put it another way: $l$ fixes the magnitude of the orbital angular momentum, while $m_l$ fixes its direction.

Spin quantum number ($m_s$).

1. This quantum number represents the electron's own intrinsic spin and is denoted $m_s$.
2. Loosely, it is described as the electron spinning about its own axis either clockwise or anticlockwise - but as noted above, this literal picture is not physically accurate; spin is better understood simply as an intrinsic property that reveals itself experimentally through the electron's behaviour in a magnetic field.
3. Correspondingly, only two values of this quantum number are possible, matching the two senses of "clockwise" and "anticlockwise." …