Quantum Numbers

Overview

Quantum numbers are a set of four numbers that completely describe the state of an electron in an atom. Each electron in an atom has a unique set of quantum numbers.

The Four Quantum Numbers

Quantum Number	Symbol	Values	Describes
Principal	$n$	1, 2, 3, ...	Energy level (shell)
Angular Momentum	$\ell$	0 to $(n-1)$	Orbital shape (subshell)
Magnetic	$m_\ell$	$-\ell$ to $+\ell$	Orbital orientation
Spin	$m_s$	$+\frac{1}{2}$ or $-\frac{1}{2}$	Electron spin direction

Principal Quantum Number ($n$)

• Determines the energy level and size of the orbital
• Larger $n$ = higher energy, larger orbital
• Maximum electrons in shell $n$:

$$\text{Max electrons} = 2n^2$$

$n$	Shell	Max Electrons
1	K	2
2	L	8
3	M	18
4	N	32

Angular Momentum Quantum Number ($\ell$)

• Determines the shape of the orbital
• Values: $0, 1, 2, \ldots, (n-1)$

$\ell$	Subshell	Shape	Orbitals
0	s	Spherical	1
1	p	Dumbbell	3
2	d	Cloverleaf	5
3	f	Complex	7

Magnetic Quantum Number ($m_\ell$)

• Determines the orientation of the orbital in space
• Values: $-\ell, \ldots, -1, 0, +1, \ldots, +\ell$
• Total orbitals in subshell:

$$\text{Number of orbitals} = 2\ell + 1$$

Examples

For p orbitals ($\ell = 1$):

$$m_\ell = -1, 0, +1 \quad \rightarrow \quad p_x, p_y, p_z$$

For d orbitals ($\ell = 2$):

$$m_\ell = -2, -1, 0, +1, +2 \quad \rightarrow \quad \text{5 d orbitals}$$

Spin Quantum Number ($m_s$)

• Describes the spin of the electron
• Only two values: $+\frac{1}{2}$ (spin up ↑) or $-\frac{1}{2}$ (spin down ↓)
• Each orbital can hold maximum 2 electrons with opposite spins

Allowed Combinations

For $n = 3$:

$\ell$	Subshell	$m_\ell$ values	Orbitals
0	3s	0	1
1	3p	-1, 0, +1	3
2	3d	-2, -1, 0, +1, +2	5

Valid vs Invalid Sets

Valid

• $n=2, \ell=1, m_\ell=0, m_s=+\frac{1}{2}$ ✓ (2p orbital)
• $n=3, \ell=2, m_\ell=-1, m_s=-\frac{1}{2}$ ✓ (3d orbital)

Invalid

• $n=2, \ell=2, m_\ell=0, m_s=+\frac{1}{2}$ ✗ ($\ell$ cannot equal $n$)
• $n=1, \ell=0, m_\ell=1, m_s=+\frac{1}{2}$ ✗ ($m_\ell$ must be between $-\ell$ and $+\ell$)
• $n=3, \ell=1, m_\ell=0, m_s=0$ ✗ ($m_s$ must be $+\frac{1}{2}$ or $-\frac{1}{2}$)

Pauli Exclusion Principle

No two electrons in an atom can have the same set of four quantum numbers.

This means:

• Each orbital holds at most 2 electrons
• Electrons in the same orbital must have opposite spins

Summary Formulas

$$\text{Number of subshells in shell } n = n$$ $$\text{Number of orbitals in subshell } \ell = 2\ell + 1$$ $$\text{Number of orbitals in shell } n = n^2$$ $$\text{Maximum electrons in subshell} = 2(2\ell + 1)$$ $$\text{Maximum electrons in shell} = 2n^2$$