Which of the following is responsible to rule out the existence of definite paths or trajectories of electrons

NCERT Exemplar · Q10

Q. Which of the following is responsible to rule out the existence of definite paths or trajectories of electrons?

(i) Pauli's exclusion principle.

(ii) Heisenberg's uncertainty principle.

(iii) Hund's rule of maximum multiplicity.

(iv) Aufbau principle.

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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 Heisenberg's Uncertainty Principle states that it is impossible to simultaneously know both the exact position and exact momentum of an electron, thereby ruling out the concept of definite paths or trajectories. The correct option is (B).

In classical mechanics, we can precisely determine the position and momentum of an object at any given time, allowing us to predict its exact trajectory. However, when we move to the quantum realm, specifically for particles as small as electrons, this classical notion breaks down. The very act of observing or measuring a quantum particle affects its properties. The concept that prevents us from assigning a definite path to an electron is a fundamental principle of quantum mechanics.

Let's examine each option to understand why one of them is responsible for this phenomenon.

Pauli's Exclusion Principle:

This principle states that no two electrons in an atom can have the same set of four quantum numbers ( $n, l, m_l, m_s$ ). This means that each electron in an atom occupies a unique quantum state. It is crucial for explaining the electron configuration of atoms and the stability of matter, as it dictates how electrons fill orbitals. However, it does not directly address the nature of an electron's movement or whether it follows a definite path. It's about the uniqueness of electron states, not the determinacy of their trajectories.
Heisenberg's Uncertainty Principle:

This principle, formulated by Werner Heisenberg, is a cornerstone of quantum mechanics. It states that it is fundamentally impossible to simultaneously determine with perfect accuracy both the position ( $\Delta x$ ) and momentum ( $\Delta p$ ) of a particle. Mathematically, it is expressed as:

\Delta x \cdot \Delta p \ge \frac{h}{4\pi}

where $h$ is Planck's constant.
This means that if we try to measure the position of an electron with high precision (making $\Delta x$ very small), our knowledge of its momentum ($\Delta p$) becomes highly uncertain, and vice-versa. A "definite path" or "trajectory" implies knowing both the exact position and exact momentum (and thus velocity) of the electron at every instant. The Heisenberg's Uncertainty Principle directly forbids this simultaneous precise knowledge, thereby ruling out the existence of definite paths for electrons. Instead, electrons are described by probability distributions (orbitals) rather than fixed orbits.

> [!IMPORTANT]
> The Heisenberg's Uncertainty Principle is a fundamental limitation imposed by nature, not a limitation of our measuring instruments.

3. Hund's Rule of Maximum Multiplicity: …

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Q.Which of the following is responsible to rule out the existence of definite paths or trajectories of electrons?

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