Foundations — Quantum numbers — n (principal), l (azimuthal), mₗ (magnetic), mₛ (spin)

This page assumes nothing. Before you meet $n$, $l$, $m_l$, $m_s$ on the parent note, we build every ingredient from the ground up, in the order they lean on each other.

1. Integers, and "positive integers"

A positive integer is $1,2,3,\dots$ — the counting numbers, starting at one, skipping zero. When the parent note says $n=1,2,3,\dots$ and "never zero", it is picking exactly the positive integers off this line.

2. The $\pm$ sign and half-integers

A half-integer is a whole number plus one half: $\frac{1}{2},\frac{3}{2},\dots$ or their negatives. The electron's spin value $\frac{1}{2}$ is a half-integer — that is why spin never behaves like the "counting" quantum numbers; it lives on the half-way ticks.

3. Angles going around a circle: $\varphi$ and $2\pi$

Before we can understand why orbitals come in orientations, we need the language of angles.

4. The exponential $e^{ix}$ — a point riding a circle

The parent note writes $\Phi (\varphi )\propto e^{i{m}_l\varphi }$. This looks scary. It is really just a point walking around a circle.

Concretely: $e^{i2\pi m_l}=1$ (back to the start) only when $m_l=0,\pm 1,\pm 2,\dots$

5. Vectors and the arrow $\vec{L}$

6. Sums and the $\sum$ symbol

7. Wavefunction $\psi$ and "separation of variables"

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer out loud.

Connections

• Schrödinger equation for hydrogen atom — the equation these symbols were built to solve
• Shapes of atomic orbitals (s, p, d) — what the fog $|\psi {|}^2$ actually looks like
• Pauli exclusion principle — why the four-number address must be unique
• Zeeman effect — where the $z$-axis projection $L_z$ becomes physically visible