Match the quantum numbers with the information provided by these. Quantum number (i) Principal quantum number

NCERT Exemplar · Q43

Q. Match the quantum numbers with the information provided by these. Quantum number

(i) Principal quantum number

(ii) Azimuthal quantum number

(iii) Magnetic quantum number

(iv) Spin quantum number
Information provided

(a) orientation of the orbital

(b) energy and size of orbital

(d) shape of the orbital

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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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Each quantum number encodes a specific physical property of an orbital or electron. The correct matches are: (i)→(b), (ii)→(d), (iii)→(a), (iv)→(c).

The four quantum numbers are not arbitrary labels — they emerge directly from the solutions of the Schrödinger equation for the hydrogen atom. Each one restricts a degree of freedom of the electron’s wavefunction, and together they uniquely specify an electron’s state in an atom. Understanding what each number physically means is the key to matching them correctly.

Principal quantum number ( $n$ )

$n = 1, 2, 3, \dots$ determines the energy of the electron and the size of the orbital. For hydrogen-like atoms, energy depends only on $n$ : $E_n \propto -1/n^2$ . Larger $n$ means a larger, higher-energy orbital. This clearly matches (b) energy and size of orbital.
Azimuthal (or angular momentum) quantum number ( $l$ )

For a given $n$ , $l$ can be $0, 1, 2, \dots, n-1$ . This number defines the shape of the orbital: $l=0$ gives an s-orbital (spherical), $l=1$ gives p-orbitals (dumbbell), $l=2$ gives d-orbitals (cloverleaf), and so on. It also quantifies the orbital angular momentum. So it matches (d) shape of the orbital.
Magnetic quantum number ( $m_l$ )

$m_l$ takes integer values from $-l$ to $+l$ , including zero. It describes the orientation of the orbital in space relative to an external magnetic field. For example, a p-subshell ( $l=1$ ) has three orbitals: $m_l = -1, 0, +1$ , pointing along the $x$ , $z$ , and $y$ axes respectively. This matches (a) orientation of the orbital.
Spin quantum number ( $m_s$ ) …

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Q.Match the quantum numbers with the information provided by these. Quantum number

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Q.Match the quantum numbers with the information provided by these.
Quantum number

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