Chapter 156: Quantized Orbits

The timeline rewinds slightly by a day or two.

The morning sun passed through the finely polished glass dome, spilling onto the long table in Horace’s manor, a table large enough to seat twenty for dinner.

There was no food on the table, only mountains of calculation paper and one frantic old man.

“Wrong! Still wrong!”

Horace forcefully crumpled a parchment full of formulas into a ball and slammed it onto the ground.

A thick carpet of paper balls already covered the floor, deep enough to bury an adult up to their ankles.

“If the electron is orbiting the nucleus, it’s a damned accelerating charge! According to Maxwell’s flawless equations, it must constantly and ceaselessly radiate electromagnetic waves outwards!

Radiation means energy loss, and energy loss means a shrinking orbit—”

Horace shot his head up, his eyes bloodshot, and roared at an electron that wasn’t there:

“You should have plunged into the nucleus within one nanosecond! You should have committed spontaneous suicide! All matter should have collapsed in an instant!

But why is this damned world still stubbornly standing here? Why hasn’t this cup of tea turned into an unrecognizable pile of junk?”

He grabbed the teacup, and a few drops of black tea spilled from his trembling.

A slender, fair hand reached out and steadily took the cup from him.

“Done shouting?”

Eleonora was wearing a loose, casual morning robe, holding a somewhat old issue of the “On Magic” journal.

In her other hand, she held a small spray bottle, its nozzle aimed directly at Horace’s face.

Horace shrank back, a survival instinct honed over many years.

“If you’re done shouting, go water the flowers in the garden.”

Eleonora pressed the spray bottle into his hand, her tone so gentle it was hair-raising.

“If even one of my Blue Moon Grasses withers, you can hug your Maxwell’s equations and sleep on the lab floor tonight.”

“Eleonora, this is a moment of life and death for physics!”

Horace waved his arms in grief and indignation. “Classical electromagnetic theory is about to die! Or the atomic model we proposed is about to die! It’s one or the other! How can you expect me to be in the mood to tend to a few blades of grass?”

“Are you going or not?”

Eleonora raised an eyebrow slightly, and a spark of fire element at her fingertip made the surrounding air instantly hotter.

“Going. Right away.”

Although Horace was a Ninth Circle Archmage, when it came to his status in the family, he was always firmly controlled—dead at the bottom.

Watching her husband trudge dejectedly towards the garden with the spray bottle, Eleonora chuckled softly and sat down in a wicker chair.

Compared to Horace’s frantic rage, she preferred to find clues amidst the chaos.

The atom does not collapse. This is a fact.

The electron is in motion. This is a deduction.

Classical physics says that if an electron moves, it dies. This is also a deduction.

Since the fact and the deductions were fighting, it meant that something, somewhere beyond the jurisdiction of classical physics, had quietly established new rules.

Eleonora opened the journal she was holding.

It wasn’t the latest issue, but an old one from a few months ago.

She wanted to relax by flipping through a previous issue.

She casually turned the pages, and the sunlight fell on the paper, illuminating an author’s name: Balmer. A marginal mage from the Optical School.

It was an experimental record of spectra.

Months ago, this Mage Balmer, in order to study a purer form of illumination magic, had frantically heated various elements.

He discovered in his experiments that when a single gaseous element was excited by high temperatures, the light it emitted, when passed through a prism, did not form a continuous seven-colored band like sunlight.

Instead, it presented several discontinuous, isolated bright lines.

Like a few colored ribbons cut against a dark background.

Eleonora’s gaze lingered on the illustrations.

The spectral lines of hydrogen: one red line, one blue-green line, two violet lines.

Beyond that, pitch blackness.

Why?

Why didn’t the atom generously release light of all colors like the sun? Why did it only miserly offer these few specific colors?

Like a clumsy musician who could only press a few fixed notes on the piano keys.

Eleonora continued to read.

Balmer was clearly a data fanatic as well. He didn’t just stop at observation but attempted to find the mathematical pattern behind these messy lines.

He measured the wavelengths of these four spectral lines and, in an almost patchwork fashion, worked out an empirical formula.

Eleonora looked at the formula.

B was a constant. And that n…

Balmer had noted beside it: n can be 3, can be 4, can be 5… it must be a positive integer greater than 2.

A positive integer.

Eleonora murmured to herself.

An extremely faint electric current shot up her spine to the back of her head.

Integers.

Not 3.1, not 4.5, only 3, 4, or 5.

This kind of discontinuous, jumping number reminded her of that world-shaking quantum theory, reminded her of the “quanta” that could only be divided into specific packets of energy.

What if…

Eleonora suddenly sat up straight, the magazine in her hand crinkling.

She recalled a hypothesis she’d seen in an even more obscure corner while flipping through materials a few days ago: the generation of spectra might originate from the movement of electrons between different potential energy positions.

What if these two ideas were pieced together?

Then the atom in the microscopic world was no longer a continuous, grand slope.

It was more like a staircase.

A staircase made up of many flat platforms. Orbits with a strict hierarchical system.

Eleonora felt her breathing quicken.

She closed her eyes, and the planetary model that had given her sleepless nights appeared in her mind.

The nucleus in the center.

The electrons flying outside.

If physics said, “You will fall no matter where you fly,” then did the new rule say—

“It is only legal to fly in specific orbits?”

In these specific orbits, which followed the rule of integers, the electrons seemed to have obtained some kind of “immunity.” Maxwell’s curse was ineffective against them. They could orbit stably without radiating any energy.

Only when they didn’t want to stay on one level and wanted to jump to another.

For example, from a higher level to a lower one.

Only then would they throw out the excess energy.

And that thrown-out energy becomes light, flies into Balmer’s spectrometer, and becomes those few lonely spectral lines!

Because the orbits are fixed, the height difference between the steps is fixed, so the energy released during a jump is also fixed, and the corresponding wavelength of light is naturally also fixed!

This is why there is no continuous spectrum!

Because an electron cannot hover between two steps! It is either on the third floor or the second floor, never on floor two-and-a-half!

This is… quantization.

“HORACE!!!!!”

A high-pitched scream, powerful enough to pierce the defensive barrier of a mage tower, exploded over the manor.

Horace, who was having a staring contest with a sunflower in the garden, flinched, and half the water from his bottle poured onto his own boots.

“What is it? What happened? I didn’t kill it with the water…”

Before he could explain, he saw his wife, like a red whirlwind, leap directly from the terrace. She didn’t even bother to use magic to break her fall, simply shattering two flagstones upon landing.

Eleonora rushed up to Horace and grabbed his well-maintained white beard.

“Ow, ow, ow! Eleonora, this is a real beard!”

“Never mind your beard!” Eleonora’s eyes, which were always smiling, were now frighteningly bright. “Calculate! Calculate for me now!”

She shoved the rolled-up “On Magic” into Horace’s arms, pointing at the Balmer formula inside.

“Assume! Assume that Planck’s constant Lia mentioned is correct! Energy equals frequency times h!”

“Assume! Electrons can only operate in specific orbits! The angular momentum in these orbits must be an integer multiple of h divided by 2π!”

“Use these conditions to reverse-engineer Balmer’s formula! To calculate that constant B!”

Although he was grimacing in pain, the instincts of a top mathematician allowed him to instantly grasp the terrifying meaning behind these words.

“Specific orbits?”

“Is this putting a leash on that lawless electron?”

“You’re saying…” Horace, forgetting his wet feet, his gaze sharpened instantly, “at this scale, nature is… discontinuous?”

“We’ll know if it is after you calculate it!”

Eleonora let go, pulled a pen directly from her spatial ring, and without another word, wrote a string of initial conditions on Horace’s white shirt.

“Calculate it right here! If you get a single decimal point wrong, I’ll pluck out your beard and make a writing brush out of it!”

Horace looked down at his expensive silk shirt being turned into scrap paper, feeling no heartache at all. Instead, he took the pen with a trembling hand.

He squatted down on the muddy ground of the garden, the tip of the pen flying across the hem of his shirt.

Coulomb force provides centripetal force…

Conservation of energy…

Introduction of the quantization condition…

Symbols danced under his pen, like an ancient and mysterious incantation, unsealing the most fundamental layer of the material world.

Eleonora stood by, holding her breath, watching the lines of derivation.

This was a challenge to the authority of classical physics.

Ten minutes later.

Horace’s hand stopped moving.

He stared blankly at the final formula derived at the very bottom corner of his shirt.

It was an expression for the constant related to wavelength.

Its form was identical to the empirical formula that the optical mage named Balmer had stumbled upon by blind luck months ago.

Even the value of the constant B, after being calculated from a pile of fundamental physical constants, had an error margin that was small even to the fourth decimal place.

“My heavens…”

Horace sat down hard in the mud, all semblance of dignity gone.

He looked at the pen in his hand, then up at the glaring sun above.

“Eleonora, do you know what this means?”

His voice was trembling.

“It means our previous confusion is solved. The electron didn’t crash because it was ‘stipulated’ to only take these paths.”

Eleonora took a deep breath, suppressing the rising and falling of her chest. “It’s more than just that.”

Horace held up the corner of his shirt filled with formulas, looking at it through the sunlight.

“It means that this world… is essentially made of discrete portions, not a mosaic pieced together from countless tiny pixels.”

“Continuity, at this level, is a lie.”

A breeze blew through the garden, rustling a few fallen leaves.

The world remained quiet, not exploding just because two mages had peeked at its hidden cards.

But at that moment, both Horace and Eleonora knew that the gate of the old era had been thoroughly kicked open.