Match the following (i) Photon (ii) Electron (iii) ψ² (iv) Principal quantum number n (a) Value is 4 for N she

NCERT Exemplar · Q46

Q. Match the following

(i) Photon

(ii) Electron

(iii) ψ^2

(iv) Principal quantum number n

(a) Value is 4 for N shell

(b) Probability density

(d) Exhibits both momentum and wavelength

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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 question asks to match physical quantities/particles with their correct descriptions. The key is to recall the fundamental properties of photons, electrons, the wavefunction squared, and the principal quantum number. The correct matches are: (i)→(d), (ii)→(c), (iii)→(b), (iv)→(a).

Let's think about what each term means, so the matching becomes natural rather than memorised.

Photon — a particle of light. Light is an electromagnetic wave, so a photon has a wavelength. But it also carries momentum (given by $p = h/\lambda$ ). So a photon exhibits both wave-like and particle-like behaviour — that is, both momentum and wavelength. This directly points to option (d).

Electron — a fundamental particle with negative charge. Its mass is fixed and positive. But more importantly, the question asks about a property that is "always positive". An electron's charge is negative, its mass is positive, but the key here is that the energy of a free electron can be positive, and in atomic orbitals, the probability interpretation never gives negative values. However, the simplest and most direct match: an electron itself is a particle with a definite, always-positive rest mass. Option (c) fits.

ψ² — this is the square of the wavefunction in quantum mechanics. The wavefunction ψ can be positive, negative, or even complex. But ψ² (or more precisely, |ψ|²) is always non-negative — it represents the probability density of finding a particle at a given point. That's exactly option (b).

Principal quantum number n — this number defines the main energy shell in an atom. For the N shell, n = 4 (K=1, L=2, M=3, N=4). So option (a) is correct.

Now let's match them systematically:

Photon → (d) Exhibits both momentum and wavelength.

A photon has zero rest mass but carries momentum $p = h/\lambda$ and has wavelength $\lambda = c/\nu$ . So it truly exhibits both properties.
Electron → (c) Always positive value. …

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