Structure of Atom · Lesson 8 of 12

The Quantum Model and Quantum Numbers

“Describe electron states with wave functions, probability, and four quantum numbers.”

Learning Objectives

• Explain the roles of the Schrödinger equation and orbital wave function at a conceptual level. • Distinguish an orbit from an orbital and probability from probability density. • Interpret each of the four quantum numbers and its permitted values. • Count orbitals in a subshell and shell and check proposed quantum-number sets.

A different kind of description

Classical mechanics gives an excellent account of a falling stone or an orbiting planet. For electrons in atoms, their observable wave behaviour and the uncertainty principle require quantum mechanics. The new theory keeps the idea that physical predictions must agree with experiments, while changing what can be predicted about individual electron outcomes.

Schrödinger developed a wave equation whose solutions describe allowed states. For a time-independent energy state, it relates an energy operator, a wave function, and the state’s energy. At this level we need its meaning and consequences, not the advanced mathematics required to solve it.

Stationary-state Schrödinger equationLaTeX

\hat{H} ψ = E ψ

Ĥ is the Hamiltonian, the operator representing total energy; ψ is the wave function; E is the state’s energy. Solving the equation requires the physical system and appropriate boundary conditions.

For an atom, the energy includes electron kinetic energy and electrostatic interactions. Electrons are attracted to the nucleus; electrons repel one another; a complete many-particle treatment also accounts for nuclear motion and interactions as appropriate. The hydrogen one-electron problem can be solved exactly within the usual non-relativistic model, whereas multi-electron atoms generally require approximations.

Allowed bound-state energies emerge from the wave-equation solutions, rather than from an extra rule that an electron must run around a circle. The solutions for hydrogen also produce the allowed spatial quantum numbers. Quantum mechanics is therefore a framework that links energy values with spatial probability patterns.

What an orbital means

An atomic orbital is a one-electron spatial wave function used to describe an electron in an atom. Its value depends on position. The value of the wave function itself is not an ordinary measurable density; it can have sign or, more generally, complex phase. Max Born’s interpretation connects the squared magnitude with the physically useful probability density.

Probability density tells us how concentrated probability is near a point. To find an actual probability in a small region, multiply the local density by the region’s volume. For a larger region, combine contributions across it. A normalised orbital gives total probability one when all space is included.

Probability densityLaTeX

ρ (r) = ∣ ψ (r) ∣^{2}, P (small volume) \approx ∣ ψ ∣^{2} Δ V

ρ is non-negative. ΔV must be small enough that the density is approximately uniform within it. Probability density is probability per unit volume.

A Bohr orbit is a prescribed path. An orbital is not a path, a shell of material, or a track traced by the electron. Saying an electron occupies an orbital means that this wave function describes its state. For hydrogen-like species the familiar orbital patterns are exact spatial solutions within the model; for many-electron atoms related one-electron orbitals are useful approximations.

Shells, subshells, and orbitals— The third shell contains three subshells and nine spatial orbitals, not three orbitals.

The principal and azimuthal quantum numbers

The principal quantum number, $n$ , identifies the shell and takes positive integer values 1, 2, 3, and higher. Shells are also named K, L, M, N, and so on. Increasing $n$ generally increases spatial extent; in hydrogen-like species it also raises the energy towards zero. In multi-electron atoms, subshell type matters as well, so $n$ alone does not give a complete energy order.

The azimuthal quantum number, $l$ , identifies a subshell and describes its orbital angular-momentum type and shape. For a fixed $n$ , its values run from zero to $n - 1$ . We use the letters s, p, d, f, g, and h for $l = 0, 1, 2, 3, 4, 5$ . A shell with principal number n therefore has n subshells.

Allowed shell and subshell labelsLaTeX

n = 1, 2, 3, \dots, l = 0, 1, \dots, n - 1

An orbital written 3p has n = 3 and l = 1. The label 2d is impossible because n = 2 permits only l = 0 or 1.

Shell n	Allowed l	Subshell labels
1	0	1s
2	0, 1	2s, 2p
3	0, 1, 2	3s, 3p, 3d
4	0, 1, 2, 3	4s, 4p, 4d, 4f

Orientation and electron spin

For each subshell, the magnetic orbital quantum number $m_{l}$ distinguishes possible spatial angular states relative to a chosen axis. Its integer values range from $- l$ to $+ l$ , including zero. This gives one s orbital, three p orbitals, five d orbitals, and seven f orbitals per subshell.

A spatial orbital is specified by the three numbers $n, l, m_{l}$ . Describing an electron additionally requires its spin state. Spin is an intrinsic quantum angular momentum, not the literal rotation of a miniature solid sphere. Relative to a chosen axis, its projection is labelled by $m_{s} = + 1/2$ or $- 1/2$ , often drawn as up and down arrows.

Do not confuse the symbol s in “s subshell” with electron spin. The former is the letter for $l = 0$ ; spin is a separate property present for every electron. Opposite spin states are needed when two electrons occupy the same spatial orbital.

Orientation and spin labelsLaTeX

m_{l} = - l, - l + 1, \dots, 0, \dots, l; m_{s} = + \frac{1}{2} or - \frac{1}{2}

There are 2l + 1 spatial orbitals per subshell. Spin supplies two possible spin labels for each spatial orbital.

Quantum number	What it labels	Allowed values
n	Shell and general spatial extent	Positive integers
l	Subshell and orbital shape type	0 through n − 1
m_l	Spatial angular state / orientation label	−l through +l
m_s	Electron spin projection	+1/2 or −1/2

Counting and checking states

Counting orbitals is different from counting subshells or electrons. The third shell has three subshells, but those subshells contain one, three, and five orbitals respectively. Each orbital can later accommodate up to two electrons under the Pauli principle; this does not make two spatial orbitals.

To check a proposed set, begin with $n$ , then check whether $l$ fits that shell, whether $m_{l}$ fits that subshell, and whether $m_{s}$ is one of the two permitted values. This order mirrors how the allowed labels are nested.

Spatial orbitals in a shellLaTeX

N_{orbitals} = l = 0 \sum n - 1 (2 l + 1) = n^{2}

The sum of the first n odd numbers is n². This counts all spatial orbitals in shell n, including unoccupied ones.

Example — Counting the third shell

Problem
How many spatial orbitals are associated with $n = 3$ ?

1.The allowed subshells are 3s, 3p, and 3d, with $l = 0, 1, 2$ .
2.They contain $1$ , $3$ , and $5$ orbitals, giving $1 + 3 + 5 = 9$ .
3.The direct rule gives the same result: $n^{2} = 3^{2} = 9$ . A subshell count of three would answer a different question.

Example — Translating quantum numbers

Problem
Name the subshell labels for $(n, l) = (2, 1), (4, 0), (5, 3), (3, 2)$ .

1.Translate $l = 0, 1, 2, 3$ into s, p, d, f respectively, keeping n in front.
2.The labels are 2p, 4s, 5f, and 3d.
3.These pairs specify subshells, not individual orientations. To specify a spatial orbital within one of them, an allowed magnetic quantum number is also required.

Example — Checking proposed electron states

Problem
Compare $(n, l, m_{l}, m_{s}) = (3, 2, - 2, + 1/2)$ and $(2, 2, 0, - 1/2)$ .

1.In the first set, n = 3 allows l = 2, and l = 2 allows magnetic values from −2 to +2. Its spin value is also allowed.
2.In the second set, n = 2 allows l = 0 or 1 only. Therefore l = 2 makes the set invalid even though its other individual values look familiar.
3.A list of valid-looking numbers is not enough: their relationships must also be valid.

Common mistake

An orbital is a wave function, not a circular orbit. The three spatial quantum numbers label an orbital; the fourth labels the spin state of an electron. Spin arrows are a notation for intrinsic angular momentum, not a drawing of an electron physically spinning like Earth.

Quick check

Which quantity has direct meaning as a probability density?

Quick check

How many subshells are permitted for n = 4?

Quick check

Which spatial orbital label is impossible?

Quick check

For l = 2, how many magnetic orbital values are possible?

Quick check

What distinguishes two electrons in the same spatial orbital?

Practice Problems

List every allowed l and magnetic orbital value for n = 3.
Find the smallest n for which a g subshell is possible.
Explain the difference between probability density at a point and probability in a small volume.
Count the spatial orbitals in the fourth shell by adding the subshell counts.
Test these pairs: (n,l) = (1,1), (3,0), (4,3), and (2,−1).
Explain why spin must not be interpreted as the physical spinning of a small charged ball.

Key Takeaways

The Schrödinger equation gives allowed wave functions and their energies. An orbital is a spatial wave function; its squared magnitude is probability density. n, l, and m_l label a spatial orbital, while m_s labels electron spin projection. Allowed values are nested: n restricts l, and l restricts m_l. A subshell has 2l + 1 orbitals and a shell has n² spatial orbitals.

Previous · Lesson 7

Matter Waves and the Limits of Orbits

Next · Lesson 9

Orbital Shapes and Nodes

A different kind of description

What an orbital means

The principal and azimuthal quantum numbers

Orientation and electron spin

Counting and checking states

Quiz

Practice Problems

Key Takeaways