(i) An atomic orbital has n = 3. What are the possible values of l and ml? (ii) List the quantum numbers (ml a

Exercises · 2.28

m_l

(ii) List the quantum numbers (

m_l

and l) of electrons for 3d orbital.

(iii) Which of the following orbitals are possible? 1p, 2s, 2p and 3f

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Concept understanding — Energy Level Quantization

Energy Level Quantization: From Intuition to Precision

Imagine you're climbing a smooth ramp. You can stop at any height — 1 metre, 1.5 metres, 2.1 metres — anywhere you like. That's how we intuitively think about energy in everyday life: continuous, like a slide.

Now imagine a staircase. You can stand on step 1, step 2, or step 3 — but you cannot stand halfway between step 2 and step 3. There's no such place. The steps are discrete, not continuous.

Energy level quantization is the idea that in the microscopic world of atoms and molecules, energy behaves like a staircase, not a ramp. Electrons in an atom cannot have just any energy — they can only occupy specific, allowed energy levels. Everything else is forbidden.

Why does this happen? The core intuition

In the classical world, an electron orbiting a nucleus would continuously radiate energy, spiral inward, and crash — atoms would be unstable. But atoms are stable. Nature solved this problem by imposing a rule: the electron's angular momentum (and therefore its energy) can only take certain discrete values.

Think of a guitar string. It can only vibrate at specific frequencies — its fundamental and harmonics. You can't pluck it to produce a frequency halfway between two harmonics. The string's vibration is quantized by its boundaries. Similarly, an electron bound to a nucleus is confined in space, and that confinement forces its energy to be quantized.

Note

Quantization is not a mysterious extra rule — it emerges naturally whenever a wave (like an electron's matter wave) is confined. Confinement creates standing waves, and standing waves only exist at specific frequencies.

The precise statement

For a bound system (like an electron in an atom), the total energy $E$ of the system can only take certain discrete values:

E = E_1, E_2, E_3, \dots

where each $E_n$ is a specific, fixed number. The integer $n$ (1, 2, 3, …) is called the principal quantum number. The lowest energy level ( $n=1$ ) is the ground state; higher levels ( $n>1$ ) are excited states.

For the hydrogen atom, the allowed energies are given by:

E_n = -\frac{13.6 \text{ eV}}{n^2}

So:

$n=1$ : $E_1 = -13.6$ eV (ground state)
$n=2$ : $E_2 = -3.4$ eV
$n=3$ : $E_3 = -1.51$ eV
and so on, approaching 0 eV as $n \to \infty$ (the ionization limit)

Important

The negative sign means the electron is bound to the nucleus. Zero energy corresponds to the electron being free (ionized). The more negative the energy, the more tightly bound the electron.

How do we know this is real?

The most direct evidence comes from atomic spectra. When an electron jumps from a higher energy level to a lower one, it emits a photon of light with energy exactly equal to the difference:

\Delta E = E_{\text{higher}} - E_{\text{lower}} = hf

where $h$ is Planck's constant and $f$ is the frequency of the emitted light. …

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The quantum numbers $l$ and $m_l$ are determined by the principal quantum number $n$ and the orbital shape. For $n=3$ , $l$ can be 0, 1, or 2, and $m_l$ ranges from $-l$ to $+l$ . The 3d orbital has $l=2$ and $m_l = -2, -1, 0, +1, +2$ . Among 1p, 2s, 2p, and 3f, only 2s and 2p are possible.

The Core Idea: Why Quantum Numbers Are Not Arbitrary

In quantum mechanics, an electron in an atom is described by a set of four quantum numbers. The first three — $n$ , $l$ , and $m_l$ — come directly from solving the Schrödinger equation for the hydrogen atom. They are not random; they are linked by strict rules.

The principal quantum number $n$ (1, 2, 3, …) sets the energy level and roughly the size of the orbital. The azimuthal quantum number $l$ defines the shape of the orbital (s, p, d, f, …) and can only take integer values from 0 up to $n-1$ . The magnetic quantum number $m_l$ describes the orientation of the orbital in space, and it ranges from $-l$ to $+l$ in integer steps.

So the moment you know $n$ , you know the possible $l$ values. And once you know $l$ , you know the possible $m_l$ values. This is the chain of dependence.

(i) For $n = 3$ , what are the possible values of $l$ and $m_l$ ?

Step 1: Determine possible $l$ values.

The rule: $l$ can be 0, 1, 2, …, up to $n-1$ .

For $n = 3$ , $n-1 = 2$ . So $l$ can be:

l = 0, 1, 2

These correspond to the 3s, 3p, and 3d orbitals respectively.

Step 2: For each $l$ , find possible $m_l$ values.

$m_l$ takes integer values from $-l$ to $+l$ , including zero.

If $l = 0$ : $m_l = 0$ (only one orientation — the s orbital is spherical).
If $l = 1$ : $m_l = -1, 0, +1$ (three orientations — the three p orbitals).
If $l = 2$ : $m_l = -2, -1, 0, +1, +2$ (five orientations — the five d orbitals).

Watch out

A common mistake is to think $m_l$ can be any integer between $-l$ and $+l$ including values like $\pm 3$ for $l=2$ . It cannot — the range is strictly $-l$ to $+l$ . For $l=2$ , the maximum $|m_l|$ is 2.

So the complete set for $n=3$ is:

$l$	$m_l$ values
0	0
1	-1, 0, +1
2	-2, -1, 0, +1, +2

(ii) Quantum numbers ( $m_l$ and $l$ ) for the 3d orbital

The notation "3d" tells you two things directly:

The number "3" is $n = 3$ .
The letter "d" means $l = 2$ (since s=0, p=1, d=2, f=3).

So for a 3d orbital, $l$ is fixed at 2. The possible $m_l$ values are therefore:

$m_l = -2, -1, 0, +1, +2$ …

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Q.(i) An atomic orbital has n = 3. What are the possible values of l and $m_l$ ?

Concept understanding — Energy Level Quantization

Energy Level Quantization: From Intuition to Precision

Why does this happen? The core intuition

The precise statement

How do we know this is real?

Key Idea: Energy Level Quantization

Reasoning

The Core Idea: Why Quantum Numbers Are Not Arbitrary

(i) For $n = 3$ , what are the possible values of $l$ and $m_l$ ?

(ii) Quantum numbers ( $m_l$ and $l$ ) for the 3d orbital

Q.(i) An atomic orbital has n = 3. What are the possible values of l and mlm_lml​?

The Core Idea: Why Quantum Numbers Are Not Arbitrary

(i) For n=3n = 3n=3, what are the possible values of lll and mlm_lml​?

(ii) Quantum numbers (mlm_lml​ and lll) for the 3d orbital

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Q.(i) An atomic orbital has n = 3. What are the possible values of l and $m_l$ ?

(i) For $n = 3$ , what are the possible values of $l$ and $m_l$ ?

(ii) Quantum numbers ( $m_l$ and $l$ ) for the 3d orbital

Unlock everything free for 14 days