Quantum numbers — n (principal), l (azimuthal), mₗ (magnetic), mₛ (spin): Question bank

Spot the error

"The set

(n, l, m_{l}, m_{s}) = (2, 1, + 2, + \frac{1}{2})

describes a valid

2 p

electron."

Error is in

m_{l} = + 2

. For

l = 1

m_{l}

can only be

- 1, 0, + 1

, so

+ 2

exceeds the allowed range. A projection cannot beat its own magnitude.

"Since

n

counts shells starting at 1, and

l

depends on

n

, the smallest

l

1

Error: the smallest

l

0

, not

1

. The rule is

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

, and

0

is always included — that is the

s

subshell present in every shell.

(3, 0, 0, - \frac{1}{2})

is invalid because

m_{l}

and

m_{s}

can't both be part-negative and zero at once."

No error — this set is perfectly valid.

l = 0

forces

m_{l} = 0

(only one orientation), and

m_{s} = - \frac{1}{2}

is an allowed spin. It describes a

3 s

electron.

d

subshell holds 5 electrons because there are 5

d

-orbitals."

Error: each orbital holds two electrons (opposite spins), so

5

orbitals hold

5 \times 2 = 10

. Confusing orbital count with electron capacity is the classic slip.

"Because energy is

- 13.6 Z^{2} / n^{2}

, an electron with larger

n

is more tightly bound."

Error: larger

n

makes

E_{n}

less negative (closer to zero), meaning less tightly bound and easier to remove. The magnitude of binding shrinks with

n

"The

n = 3

shell can hold

2 \times 3 = 6

electrons."

Error: capacity is

2 n^{2} = 2 \times 9 = 18

, not

2 n

. You must sum

2 l + 1

over all

l

(

1 + 3 + 5 = 9

orbitals) before doubling for spin.

"Spin-up and spin-down electrons differ only in energy, so they occupy different subshells."

Error:

m_{s}

does not change energy in a field-free atom, and it never changes the subshell (which is fixed by

n, l

). The two spins share the same orbital; they differ only in the fourth address slot.

Why questions

Why is a fourth quantum number (

m_{s}

) needed when the Schrödinger equation gives only three?

The 3D equation quantizes one number per spatial dimension. Spin is not a spatial coordinate — it was added to explain fine spectral splitting and to let two electrons share an orbital under Pauli exclusion principle.

Why must

m_{l}

be an integer rather than any real number?

The angular part

Φ (ϕ) \propto e^{i m_{l} ϕ}

must return the same value after a full

2 π

turn (single-valuedness). That forces

e^{i 2 π m_{l}} = 1

, which holds only when

m_{l}

is a whole number.

Why is

m_{l}

called the magnetic quantum number if it doesn't change energy?

In a field-free atom the

2 l + 1

orientations are degenerate. Only when a magnetic field is applied do these orientations split into distinct energies — the Zeeman effect — earning

m_{l}

its name.

Why can't

∣ m_{l} ∣

exceed

l

L_{z} = m_{l} ℏ

is the projection of the angular-momentum vector onto one axis, and a projection can never be longer than the whole vector

∣ L ∣ = l (l + 1) ℏ

. Hence

∣ m_{l} ∣ \leq l

Why does a larger

l

"use up" more of the shell's energy budget, limiting

l \leq n - 1

Higher

l

means more angular motion, which the radial equation can only support if the wavefunction stays finite and normalizable. That balance caps

l

n - 1

; beyond it, no well-behaved solution exists.

Why is the maximum number of electrons in a shell exactly

2 n^{2}

Summing the

2 l + 1

orbitals over

l = 0

n - 1

gives

n^{2}

(the sum of the first

n

odd numbers), and each orbital takes

2

electrons for the two spins — hence

2 n^{2}

Why do the

s, p, d, f

block widths of the periodic table match

2, 6, 10, 14

Those are the electron capacities

2 (2 l + 1)

for

l = 0, 1, 2, 3

. The table's structure directly mirrors how many electrons each subshell type can hold, which is set by

m_{l}

and

m_{s}

Edge cases

What is the only allowed

m_{l}

when

l = 0

, and why?

Only

m_{l} = 0

. An

s

orbital is spherically symmetric — it has no distinct orientation — so there is just

2 l + 1 = 1

value.

What happens to

E_{n}

n \to \infty

, and what does it physically mean?

E_{n} = - 13.6 Z^{2} / n^{2} \to 0

. The electron becomes infinitely spread out and infinitely weakly bound — the boundary of ionization, where it barely belongs to the atom anymore.

n = 0

ever allowed, even for a hypothetical exotic atom?

Never —

n = 0

divides by zero in

E_{n}

and leaves no valid

l

(since

l \leq n - 1 = - 1

is empty). Shells always start at

n = 1

For the very first shell

n = 1

, list the complete set of allowed quantum-number addresses.

Only two:

(1, 0, 0, + \frac{1}{2})

and

(1, 0, 0, - \frac{1}{2})

. With

l = 0

forced and

m_{l} = 0

forced, spin is the sole degree of freedom, giving capacity

2 n^{2} = 2

Can two electrons in the same atom ever share all four quantum numbers?

No — the Pauli exclusion principle forbids it. At least one of

n, l, m_{l}, m_{s}

must differ, which is why electrons stack into distinct orbitals and spins (Aufbau principle and electron configuration).

For a half-filled

p

subshell in a free atom, why do the three electrons occupy separate orbitals with parallel spins?

Because Hund's rule minimises repulsion by spreading electrons across all three

m_{l}

orbitals before pairing, and parallel spins (

m_{s}

all

+ \frac{1}{2}

) are allowed since the electrons already differ in

m_{l}

What is the smallest shell in which a

d

subshell can first appear, and why?

n = 3

, because

d

means

l = 2

and the rule

l \leq n - 1

needs

n \geq 3

. That is why

3 d

exists but

2 d

does not.

Recall One-sentence self-check

Say aloud: "Each quantum number's range is decided by the one to its left, the projection $m_{l}$ can never beat the magnitude $l$ , spin is always $\pm \frac{1}{2}$ , and no two electrons share all four." If you can justify every clause, you've cleared the traps.

Connections

Parent topic (Hinglish) — the rules these traps test
Pauli exclusion principle — the no-shared-address law behind many items
Zeeman effect — why $m_{l}$ is "magnetic"
Schrödinger equation for hydrogen atom — source of the $l \leq n - 1$ and integer- $m_{l}$ rules
Aufbau principle and electron configuration — filling that respects these addresses
Hund's rule — the spin/orientation edge case
Shapes of atomic orbitals (s, p, d) — why shape follows $l$

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

True or false — justify

Spot the error

Why questions

Edge cases

Connections