The quantum numbers of six electrons are given below. Arrange them in order of increasing energies. If any of these combination(s) has/have the same energy list them: 1. n = 4, l = 2, $m_l$ = -2, $m_s$ = $-\tfrac{1}{2}$; 2. n = 3, l = 2, $m_l$ = 1, $m_s$ = $+\tfrac{1}{2}$; 3. n = 4, l = 1, $m_l$ = 0, $m_s$ = $+\tfrac{1}{2}$; 4. n = 3, l = 2, $m_l$ = -2, $m_s$ = $-\tfrac{1}{2}$; 5. n = 3, l = 1, $m_l$ = -1, $m_s$ = $+\tfrac{1}{2}$; 6. n = 4, l = 1, $m_l$ = 0, $m_s$ = $+\tfrac{1}{2}$.

The quantum numbers of six electrons are given below. Arrange them in order of increasing energies. If any of

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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 energy of an electron in a multi-electron atom depends primarily on the principal quantum number $n$ and the azimuthal quantum number $l$ via the $(n+l)$ rule. For equal $(n+l)$ , the lower $n$ gives lower energy. Applying this rule, the increasing order of energies is: 5 < 2 = 4 < 3 = 6 < 1. Electrons 2 and 4 have the same energy; electrons 3 and 6 have the same energy.

The key idea here is that in multi-electron atoms, the simple hydrogen-like energy ordering (where energy depends only on $n$ ) breaks down due to electron-electron repulsion and shielding. The $(n+l)$ rule (also called the Aufbau principle or Madelung rule) gives the correct ordering for filling orbitals: lower $(n+l)$ means lower energy; if $(n+l)$ is equal, lower $n$ means lower energy.

Let’s apply this step by step.

Identify the $(n+l)$ value for each electron.

For electron 1: $n=4$ , $l=2$ → $n+l = 4+2 = 6$ .

For electron 2: $n=3$ , $l=2$ → $n+l = 3+2 = 5$ .

For electron 3: $n=4$ , $l=1$ → $n+l = 4+1 = 5$ .

For electron 4: $n=3$ , $l=2$ → $n+l = 3+2 = 5$ .

For electron 5: $n=3$ , $l=1$ → $n+l = 3+1 = 4$ .

For electron 6: $n=4$ , $l=1$ → $n+l = 4+1 = 5$ .
Order by $(n+l)$ first.

The smallest $(n+l)$ is 4 (electron 5). Next comes $(n+l)=5$ (electrons 2, 3, 4, 6). The largest is $(n+l)=6$ (electron 1). So electron 5 has the lowest energy, and electron 1 has the highest.
Resolve ties using $n$ .

Among electrons with $(n+l)=5$ , we compare $n$ :
- Electrons 2 and 4 have $n=3$ .
- Electrons 3 and 6 have $n=4$ . Since lower $n$ gives lower energy when $(n+l)$ is equal, electrons 2 and 4 are lower in energy than electrons 3 and 6.
Check for identical energies. …

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