Allowed Quantum-Number Sets

Introduction

Quantum-number questions become straightforward when their rules are checked in order. Each value constrains the next: n limits l, l limits mₗ, and electron spin has only two choices. A set can look familiar yet be invalid because one value lies just outside its permitted range.

Core explanation

For an atomic electron, n is a positive integer. Given n, l is an integer from 0 through n − 1. Given l, mₗ is an integer from −l through +l. Finally mₛ is either +1/2 or −1/2. These four rules define allowed individual labels in the ordinary atomic-orbital model. The sequence matters because testing mₗ before checking l can hide an earlier error.

Consider (n, l, mₗ, mₛ) = (3, 2, −1, +1/2). It is valid: n is positive, l = 2 is within 0–2, mₗ = −1 is within −2–+2, and spin is one of the two allowed values. The set (3, 3, 0, +1/2) is invalid because l must be no greater than 2 when n = 3. The set (3, 2, +3, −1/2) fails the mₗ range, despite valid n and l.

The Pauli principle adds a population rule: two electrons in the same atom cannot share all four numbers. An allowed set may label one electron, but assigning the identical set to a second electron in that atom is forbidden. Two electrons can share n, l, and mₗ if their mₛ values are opposite. Electrons in separate atoms can of course have the same numerical labels; the uniqueness statement concerns electrons within one atom.

Counting possible sets also checks shell capacities. For fixed n, there are n² spatial orbital triples and two spin choices each, giving 2n² complete sets. At n = 2 there are four spatial orbitals and eight possible electron labels. This is a capacity count, not a prediction that every one is occupied in every atom.

Notation may vary: some texts write m instead of mₗ and use arrows in place of ±1/2. The allowed ranges remain unchanged. A value of +1 or −1 is not an allowed electron mₛ, even though it might look like a plausible sign or a quantum number used elsewhere.

Step-by-step reasoning

1. Check n ∈ {1, 2, 3, ...}. 2. Check integer l satisfies 0 ≤ l ≤ n − 1. 3. Check integer mₗ satisfies −l ≤ mₗ ≤ +l. 4. Check mₛ = ±1/2 and then apply Pauli if comparing electrons.

Visual explanation

Draw nested gates: n first, then an l gate whose upper bound is n − 1, then an mₗ gate bounded by l, and finally a two-way spin branch. An invalid value stops at its first failed gate.

Real-world analogy

A building address may specify floor, section, room, and seat. An impossible section on a floor invalidates the address even if the seat number itself sounds reasonable.

Real-world example

Checking configuration diagrams against allowed quantum-number sets prevents impossible labels such as a 2d electron or a p orbital with mₗ = +2 in an atomic structure problem.

Why?

Why check the hierarchy in order? Later allowed ranges depend on earlier values; an invalid l makes a later mₗ assignment meaningless for that proposed atomic state.

Common misconception

“If each number individually appears somewhere in quantum theory, the set is valid.” The values must satisfy the joint nested restrictions for the same electron.

Worked example

Test (2, 1, 0, −1/2) and (2, 0, 1, +1/2). The first is valid: l = 1 is allowed, mₗ = 0 lies within −1 to +1, and spin is valid. The second is invalid because an s subshell has l = 0 and therefore only mₗ = 0. Changing its mₗ to zero would make an individually allowed set.

Quick check

1. Is (n, l, mₗ, mₛ) = (4, 3, −3, +1/2) allowed? Answer: Yes. l = 3 is allowed for n = 4, and mₗ = −3 is at the permitted endpoint.

Exam focus

State the specific failed rule rather than merely writing “invalid.” If two electrons are compared, check whether their complete four-number sets duplicate.

Advanced insight

The ranges of n, l, and mₗ arise from physically acceptable wavefunctions; the two mₛ values reflect electron spin. Pauli's uniqueness condition applies to identical fermions and underlies atomic electron configurations.

Summary

An allowed electron label must pass nested n, l, mₗ, and mₛ restrictions. Pauli further forbids duplicate complete sets for two electrons in the same atom.

Practice questions

1. Why is (2, 2, 0, +1/2) invalid? Answer: For n = 2, l can only be 0 or 1, so l = 2 is forbidden. 2. Why is (3, 1, −2, −1/2) invalid? Answer: For l = 1, mₗ can only be −1, 0, or +1. 3. Can two electrons share (3, 1, 0) as spatial labels? Answer: Yes, if one has mₛ = +1/2 and the other −1/2.