Quantum Numbers n, l, m and s: Allowed Values and What Each Controls
Quantum numbers are the "address" of an electron in an atom. Every Class 11 chapter on atomic structure uses them, and board papers regularly ask which sets of quantum numbers are allowed and which are not. The rules are short. Once you understand where each rule comes from, you do not need to memorise lists of sets.
The four quantum numbers at a glance
l = 0, 1, 2, … (n − 1)
ml = −l, …, 0, …, +l
ms = +½ or −½
| Symbol | Name | What it controls | Allowed values |
|---|---|---|---|
| n | Principal quantum number | Shell, size and main energy of the orbital | 1, 2, 3, … (whole numbers, never 0) |
| l | Azimuthal (subsidiary) quantum number | Subshell and the shape of the orbital | 0 to (n − 1) |
| ml | Magnetic quantum number | Orientation of the orbital in space | −l to +l, including 0 |
| ms | Spin quantum number | Spin direction of the electron | +½ or −½ |
Reading each number
n tells you the shell. Larger n means the orbital is larger and the electron is, on average, farther from the nucleus. n = 1 is the K shell, n = 2 is the L shell, and so on.
l tells you the subshell. We usually write it as a letter: l = 0 is s, l = 1 is p, l = 2 is d and l = 3 is f. The shape follows from l: s is spherical, p is dumbbell-shaped. Because l can only go up to n − 1, the first shell (n = 1) has only an s subshell. The second shell has s and p. The third has s, p and d.
ml counts the orbitals inside a subshell. For a given l there are (2l + 1) values of ml, so there are (2l + 1) orbitals. An s subshell has 1 orbital, p has 3, d has 5 and f has 7.
ms has only two values. It does not depend on n, l or ml. Each orbital can therefore hold at most two electrons, and those two must have opposite spins. This is the Pauli exclusion principle: no two electrons in one atom can have all four quantum numbers the same.
Counting orbitals and electrons in a shell
Maximum electrons in shell n = 2n²
This follows directly from the rules above. Add (2l + 1) for l = 0 to n − 1, and the sum is n². Each orbital holds two electrons, so the shell holds 2n².
Step 1: For n = 3, l can be 0, 1 or 2.
Step 2: Orbitals: l = 0 gives 1, l = 1 gives 3, l = 2 gives 5. Total = 1 + 3 + 5 = 9, and n² = 3² = 9. ✓
Step 3: Electrons: 9 orbitals × 2 = 18, and 2n² = 2 × 9 = 18. ✓
Answer: 9 orbitals, up to 18 electrons.
(a) n = 2, l = 1, ml = 0, ms = +½
(b) n = 3, l = 3, ml = 0, ms = −½
(c) n = 1, l = 0, ml = 0, ms = −½
(d) n = 4, l = 2, ml = −3, ms = +½
Step 1, set (a): l = 1 is less than n = 2, so it is fine. ml = 0 lies between −1 and +1. Allowed (this is a 2p electron).
Step 2, set (b): l must be at most n − 1 = 2, but l = 3. Not allowed. (A 3f subshell does not exist.)
Step 3, set (c): l = 0 is allowed for n = 1, and ml = 0 is the only choice. Allowed (a 1s electron).
Step 4, set (d): l = 2 is allowed for n = 4, but ml must lie between −2 and +2. ml = −3 is outside that range. Not allowed.
Answer: sets (b) and (d) are not allowed.
Step 1: "4d" means n = 4, and d means l = 2.
Step 2: ml can be −2, −1, 0, +1 or +2. That is 2l + 1 = 5 values, so there are 5 orbitals in the 4d subshell.
Step 3: Each orbital has two spin states, so 5 × 2 = 10 different sets of four quantum numbers.
Answer: n = 4, l = 2, ml = −2 to +2, ms = ±½. There are 5 orbitals and a maximum of 10 electrons in 4d.
Step 1: Orbitals in n = 4 are n² = 16.
Step 2: Every orbital can have one electron with ms = +½ (the other electron in that orbital has −½).
Answer: 16 electrons.
Common mistakes
- Starting n at 0. The lowest value of n is 1. There is no n = 0 shell.
- Letting l equal n. The largest l is n − 1. That is why 1p, 2d and 3f do not exist.
- Forgetting the zero in ml. For l = 1 the values are −1, 0, +1, which is three values, not two.
- Writing ml larger than l. The magnitude of ml can never exceed l.
- Mixing up ml and ms. ms is only ±½. It is never 0 or 1.
- Saying a p subshell holds 3 electrons. It has 3 orbitals, so it holds 6 electrons.
Subshell summary table
| Subshell | l | Number of orbitals (2l + 1) | Maximum electrons | First appears in shell |
|---|---|---|---|---|
| s | 0 | 1 | 2 | n = 1 |
| p | 1 | 3 | 6 | n = 2 |
| d | 2 | 5 | 10 | n = 3 |
| f | 3 | 7 | 14 | n = 4 |
Where this is used in exams
| Question type | What to use |
|---|---|
| Is this set of quantum numbers valid? | Check n ≥ 1, then 0 ≤ l ≤ n − 1, then |ml| ≤ l, then ms = ±½ |
| Number of orbitals or electrons in a shell | n² orbitals, 2n² electrons |
| Name a subshell from n and l | Write the number n and the letter for l |
| Writing electron configurations | Use the subshell capacities in the table above together with the filling rules |
If your question is about the order in which subshells fill, see the related article on electron configuration rules below. Always check the exact wording and marking scheme of your own board or school paper.
Want to check a set of quantum numbers quickly? Try the calculator after you have worked the examples by hand.
Open the Quantum Numbers Calculator →Preparing for Class 11–12 chemistry? ABC Chemistry runs Class 11–12 coaching online, and online classes for students across India.