Quantum numbers n, l, mₗ, mₛ: Question bank — Physics deep dive

First — the words and symbols this page leans on

Before you attempt a single trap, make sure these are not mysteries. Everything below is earned here so the questions stay clean.

Definition Symbols used everywhere on this page

$ℏ$ (read "h-bar") is the reduced Planck constant, $ℏ = h /2 π$ . It is a fixed constant of nature (a specific measured number, not a unit we chose) that sets the scale of angular momentum in quantum mechanics — every angular momentum turns out to be some plain number times $ℏ$ . So $ℏ$ is not like "a metre"; it is more like the fixed speed of light $c$ : a built-in physical constant that everything is measured against.
$L$ is the orbital angular-momentum vector of the electron — an arrow whose length says how much rotational motion the orbital carries and whose direction says which way the "rotation axis" points.
$∣ L ∣$ is the length of that arrow (a single positive number), while $L_{z}$ is its shadow on the vertical $z$ -axis (its projection — can be positive, zero, or negative).
$θ$ (theta) and $ϕ$ (phi) are the two angles used to point anywhere on a sphere: $θ$ is the polar angle measured down from the top ( $z$ ) axis (like latitude, $0$ at the north pole, $π$ at the south), and $ϕ$ is the azimuthal angle measured around the equator (like longitude, $0$ to $2 π$ ). See the sphere in the figure below.
$m_{s}$ is the spin magnetic quantum number, the label for the electron's intrinsic (built-in) spin direction. It takes only two values, $+ \frac{1}{2}$ and $- \frac{1}{2}$ ("spin up" and "spin down"), and unlike $n, l, m_{l}$ it does not come from the shape of the wave in space — it was discovered by experiment (Stern–Gerlach).

True or false — justify

Every item below is a statement. Decide true or false, and — this is the whole point — why.

TRUE or FALSE: The quantum numbers are labels physicists chose by convention, like naming particles.

False. They fall out of boundary conditions (single-valuedness, non-divergence, decay at infinity) when solving Schrödinger's equation; the integers are forced by the mathematics, not assigned by choice.

TRUE or FALSE: For

l = 0

the electron has zero orbital angular momentum.

True.

∣ L ∣ = l (l + 1) ℏ = 0 ℏ = 0

; an s-electron's cloud is spherically symmetric and does not "circulate."

TRUE or FALSE: The magnitude of orbital angular momentum can equal

l ℏ

False. It equals

l (l + 1) ℏ

, which is strictly larger than

l ℏ

for every

l \geq 1

(e.g.

l = 1

gives

2 ℏ > 1ℏ

);

l ℏ

is the old Bohr guess.

TRUE or FALSE:

m_{l}

can be zero even when

l

is large.

True.

m_{l}

runs

- l, \dots, 0, \dots, + l

, so

0

is always in the list; it means

L

lies in the

x y

-plane with no

z

-projection.

TRUE or FALSE: A shell with

n = 4

can hold at most 32 electrons.

True.

2 n^{2} = 2 (4)^{2} = 32

, counting all

(l, m_{l}, m_{s})

combinations allowed once

n = 4

TRUE or FALSE: Two electrons in the same atom may share

n

l

, and

m_{l}

if their spins differ.

True. Pauli only forbids sharing all four;

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

and

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

make the full sets distinct, so one orbital holds exactly two electrons.

TRUE or FALSE: The spin quantum number

m_{s}

comes out of the Schrödinger equation.

False. The basic (non-relativistic) Schrödinger equation gives only

n, l, m_{l}

; spin was forced by the Stern-Gerlach Experiment and later derived by Dirac's relativistic equation.

TRUE or FALSE: Increasing

n

always increases the number of possible

l

values.

True.

l

ranges

0

n - 1

, so there are exactly

n

possible

l

values; a larger

n

means one more subshell type becomes available.

TRUE or FALSE: The vector

L

can point exactly along the

z

-axis.

False. That would require

L_{z} = ∣ L ∣

, i.e.

m_{l} ℏ = l (l + 1) ℏ

, but the largest allowed

m_{l}

l

and

l < l (l + 1)

for

l \geq 1

; the eigenvalues simply don't allow it, so

L

is always tilted (space quantization — see the cones figure).

Spot the error

Each line contains a claim with a flaw. Reveal names the flaw and fixes it.

"For

n = 2

, the allowed

l

values are

1

and

2

Wrong upper limit —

l

stops at

n - 1 = 1

, so

l = 0, 1

only;

l = 2

would need

n \geq 3

m_{l}

counts orbitals, so it runs

0, 1, \dots, l

m_{l}

is a signed projection, not a count; it runs

- l

+ l

, giving

2 l + 1

values including the negatives.

"An electron in a 1p orbital has

n = 1, l = 1

No 1p exists —

l \leq n - 1 = 0

for

n = 1

, so the only

n = 1

subshell is 1s (

l = 0

"Spin means the electron physically rotates about an axis."

Spin is intrinsic angular momentum with no classical rotation; a spinning charged sphere would need its surface to exceed light speed, so the picture is only a mnemonic.

"Since

L_{z} = m_{l} ℏ

and

m_{l}

can be large,

L_{z}

can exceed

∣ L ∣

Impossible — the largest allowed value is

m_{l} = l

, and

l ℏ < l (l + 1) ℏ

, so

∣ L_{z} ∣

(the shadow) is always strictly less than

∣ L ∣

(the full arrow).

"The set

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

is a valid state."

Invalid —

∣ m_{l} ∣ \leq l = 1

, but

m_{l} = - 2

violates that cap; the allowed

m_{l}

are only

- 1, 0, + 1

m_{s} = 0

is allowed for an electron with no preferred spin direction."

For a spin-

\frac{1}{2}

particle only

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

exist; there is no

m_{s} = 0

state — the spin projection is always half-integer.

"Energy depends on all four quantum numbers."

For a pure hydrogen atom, energy depends only on

n

(

E_{n} = - 13.6/ n^{2}

eV);

l, m_{l}, m_{s}

leave it unchanged (degeneracy), broken only by finer effects.

Why questions

WHY does the single-valuedness condition

Φ (ϕ + 2 π) = Φ (ϕ)

force

m_{l}

to be an integer?

Because

Φ = e^{i m_{l} ϕ}

must return to itself after a full loop around the azimuthal angle

ϕ

, requiring

e^{i m_{l} 2 π} = 1

, which holds only when

m_{l}

is a whole number — a non-integer would make the wave multi-valued at the same physical point.

WHY does

l

cap

m_{l}

\pm l

rather than allowing any value?

The polar (Legendre) solutions in the angle

θ

stay finite at the poles

θ = 0, π

only when

∣ m_{l} ∣ \leq l

; larger

∣ m_{l} ∣

makes the wavefunction blow up there, which is unphysical.

WHY can't the largest

L_{z}

ever equal

∣ L ∣

The eigenvalues fix

L_{z} = m_{l} ℏ

with maximum

m_{l} = l

, but

∣ L ∣ = l (l + 1) ℏ

, and

l < l (l + 1)

for every

l \geq 1

; so even the tallest allowed shadow

l ℏ

falls short of the arrow's true length — the vector must stay tilted.

WHY does an s-orbital (

l = 0

) have a spherical shape?

Zero orbital angular momentum means no preferred direction of "circulation," so the probability cloud has no angular pattern and looks the same in every direction.

WHY is the maximum number of electrons in shell

n

equal to

2 n^{2}

and not just

2 n

Each subshell

l

holds

2 (2 l + 1)

electrons (

2 l + 1

orbitals times 2 spins), so you sum

\sum_{l = 0}^{n - 1} 2 (2 l + 1) = 4 \sum_{l = 0}^{n - 1} l + 2 n = 4 \cdot \frac{( n - 1 ) n}{2} + 2 n = 2 n^{2} - 2 n + 2 n = 2 n^{2}

; the arithmetic-series sum

\sum_{l = 0}^{n - 1} l = \frac{( n - 1 ) n}{2}

is what makes it collapse.

WHY did the Stern–Gerlach experiment demand a fourth quantum number?

A beam of silver atoms split into exactly two spots, revealing a two-valued magnetic degree of freedom not accounted for by

n, l, m_{l}

— this is intrinsic spin, labelled by

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

WHY does

l

stop one short of

n

instead of reaching

n

The radial wave must have a valid bound-state form that decays at infinity, and the "budget" of radial structure requires

l \leq n - 1

; setting

l = n

leaves no admissible square-integrable radial solution.

Edge cases

WHAT are all the quantum-number possibilities when

n = 1

Only

l = 0

m_{l} = 0

, and

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

— exactly two states (1s²); no p, d, or f subshells can exist here.

WHAT is

∣ L ∣

for an electron in any s-orbital, regardless of

n

Exactly zero, since

l = 0

gives

∣ L ∣ = 0 (0 + 1) ℏ = 0

; the value of

n

changes energy and size but not this angular momentum.

WHAT happens to the number of allowed

m_{l}

values as

l \to 0

It shrinks to

2 (0) + 1 = 1

; there is a single

m_{l} = 0

state, meaning an s-subshell holds just one orbital (two electrons with spin).

WHAT is the smallest

n

for which a d-electron (

l = 2

) can exist?

n = 3

, because

l = 2

requires

l \leq n - 1

, i.e.

n \geq 3

; there is no 1d or 2d.

WHAT does

m_{l} = 0

mean geometrically for a nonzero-

l

orbital?

The angular-momentum vector has no

z

-component (

L_{z} = 0

), so

L

lies entirely in the plane perpendicular to

z

while still having full magnitude

l (l + 1) ℏ

WHAT is the total number of distinct

(n, l, m_{l}, m_{s})

states for

n = 3

2 n^{2} = 2 (3)^{2} = 18

states — the 3s (2), 3p (6), and 3d (10) capacities added together.

WHAT is special about the case

l = n - 1

(the maximum

l

The orbital has the most angular structure that shell allows and the fewest radial nodes; it is the "most circular" orbit in the Bohr analogy, closest to a classical orbit.