Chapter 157: The Quantized Heart

In the study of the mage tower, the shadows from the firelight in the fireplace cast themselves upon the deep red carpet, weaving an intimate silhouette.

Outside the window was the brilliant night view of the Royal Capital, while inside, the air was as thick as honey that wouldn’t thin.

Lia was currently in a very awkward, or rather, a very dangerous position.

She was sitting on Klein’s lap.

It wasn’t of her own volition.

Originally, she had only wanted to go over and look at the newly delivered journal, but her waist was suddenly gripped tight, and she was scooped up by a certain “rigorous” mentor and secured in his embrace.

He held the journal, which exuded the scent of fresh ink, while his other hand rested quite naturally on the side of her waist, his fingertips tapping idly on the soft flesh there.

“Don’t move.”

Klein’s voice came from above her head, the vibration of his chest transmitting to Lia through her back, which was pressed against him. “I’m reading a paper.”

“Do you need to hold me to read a paper?” Lia protested cautiously, trying to shift her position, only to be met with a tighter hold on her waist.

“It’s more efficient this way,” Klein said, his expression unchanging. “I can ask you questions at any time, which saves the time it would take to turn my head.”

Lia rolled her eyes.

Even a single-celled organism like Adèle wouldn’t be fooled by that excuse.

She gave up struggling and turned her gaze to the journal in Klein’s hands.

On the cover, several large, gold-stamped words were emblazoned—”On the Correlation Between Internal Atomic Structure and Spectral Phenomena.”

Evidently, after their inspiration that morning, the couple had not only forgotten to eat but had immediately organized their pile of drafts into a revolutionary manifesto for the magical world.

“Here.”

Klein’s long fingers tapped on the third page of the paper. “Horace mentions in his hypothesis that when an electron is in a specific orbit, it does not radiate energy outwards, even while accelerating. This completely violates Maxwell’s equations.”

His tone carried a hint of confusion, an emotion this Eighth Circle Archmage rarely showed.

“If an electron is a charged particle, its acceleration must produce a change in the magnetic field, which in turn generates electromagnetic waves.

This is the foundation of electromagnetism. On what grounds does Horace say it ‘does not radiate’? Just because the atom would collapse if he didn’t make that assumption?”

Klein lowered his head, his chin resting on Lia’s shoulder, his warm breath fanning her ear. “Lia, this sounds like forcibly changing the question to fit the answer.”

Lia shrank back, her earlobe starting to itch.

“Because Maxwell’s equations don’t apply there.”

Lia leaned back, finding a more comfortable position. “Klein, if you were to shrink a person down to the size of an ant, do you think they could still lift an object that was originally very heavy for them?”

“Their strength would shrink proportionally.”

“No, I mean the rules.”

Lia pointed to the diagram of the atomic model in the paper.

“The laws of the macroscopic world may not apply in the microscopic world.

Horace and Eleonora are describing a fact—the atom does not collapse. Since the fact conflicts with the theory, it must be that the old theory is invalid at that scale.”

Klein was silent for a moment.

“Invalid…”

He mulled over the word, and his fingers, which were resting on Lia’s waist, paused. “Like that day in the tailor shop, when my rationality became invalid?”

Lia’s face flushed, and she jabbed her elbow back into his chest. “Be serious!”

Klein chuckled softly, the rise and fall of his chest sending a numbing sensation through Lia’s back.

“Alright, back to business.”

He looked at the paper again.

“The concept of the quantum.”

“Correct.”

Lia looked at the familiar yet strange formulas.

“An electron doesn’t just fly wherever it pleases. It’s been forcibly held in a few fixed orbits by an invisible hand. Everywhere else is a forbidden zone.”

Just as she was speaking, there was a knock on the study door.

“Come in.”

Klein’s voice instantly returned to its usual coldness.

The door opened a crack, and Adèle poked her head in, her curly hair a mess.

She was also clutching a copy of the same journal, her face a mask of “who am I, where am I” confusion.

“Mentor, Lia.”

Adèle froze for a moment when she saw their position, then immediately looked down, pretending to be blind.

“Um, I have a question. I really don’t understand the mathematical derivation in this paper, especially this part about the energy transition…”

Lia’s face grew hot. She wanted to stand up, but Klein held her in place.

“Speak here.”

Klein said faintly.

“I happen to be listening as well.”

Adèle was on the verge of tears. She felt like a husky that had wandered into a wolf pack, not only forced to watch a public display of affection but also expected to study amidst its fragrant aroma.

She shuffled in, spread the journal open on the desk, and pointed to the derivation formula.

“Master Laplace said that the energy released when an electron jumps from one orbit to another is fixed. But how is that possible? Shouldn’t energy be continuous? Like when I pour water, I can pour as much as I want…”

Lia sighed.

She knew how difficult this was for the people of this world to understand.

After all, in their cognition, “Natura non facit saltus”—nature makes no leaps—was an axiom.

“Adèle, look at me.”

Lia took the quill from Klein’s hand and drew a simple diagram on a blank piece of parchment.

“Let’s do a thought experiment.”

Lia drew a stick figure standing on a high platform.

“Let’s assume this stick figure weighs 100 kilograms. Now, he jumps down from a step that is 1 meter high.”

Lia drew a downward arrow on the paper.

“According to the formula for gravitational potential energy, how much energy will he release when he lands?”

Adèle calculated quickly in her head.

“If we take the acceleration of gravity as 10… 100 times 10 times 1, that’s 1000 joules.”

“Very good.”

Lia nodded.

“This 1000 joules of energy, at the moment he lands, is converted into kinetic energy, or the heat and sound of impacting the ground.”

“Now, the situation changes.”

Lia drew a staircase on the paper, but it was obscured by mist, making it impossible to see how many steps there were or how high each step was.

“We can’t see the specifics of the staircase.

But, through some precise instrument—like Balmer’s spectroscope—we measure that after this person jumps down a certain number of steps, he has released a total of 1000 joules of energy.”

Lia looked up, her gaze fixed on Adèle.

“So, what can we conclude about the height of these steps?”

Adèle bit her lip and thought for a moment.

“If the steps are uniform… then if there’s only one step, it’s 1 meter high. If there are two steps, each is 0.5 meters. If there are three, each is one-third of a meter.”

“Exactly,” Lia pointed out an impossible scenario for this assumption.

“Exactly.” Lia couldn’t contain the smile in her chest, pointing at the calculation on the paper.

“First, the total height of the steps must be divisible by the height of a single step. The total height of a single step must be a divisor of the total height.

Or in other words, the total height must be an integer multiple of the single step’s height.”

“But!” Lia emphasized her tone, tapping the tip of the pen hard on the paper. “Regardless, can you conclude that the height of each step is 0.6 meters?”

Adèle was stunned.

She began to calculate furiously in her head.

If each step was 0.6 meters, one step is 0.6 meters, not enough for 1 meter.

Two steps would be 1.2 meters, which is more than 1 meter.

“No.”

Adèle shook her head. “If it’s 0.6 meters, then for the last step he shouldn’t be jumping down, but… suspended in mid-air? Or burrowing into the ground?”

“Yes, that’s exactly it.”

Lia smiled. “In order to match our observation that a total of 1000 joules were released, the height of the steps cannot be arbitrary. If there are two steps, each must be 0.5 meters. It can’t possibly be 0.51 meters, nor can it be 0.49 meters.”

“This is quantization.”

Lia pushed the paper with the staircase drawing in front of Adèle.

“In the macroscopic world, you can build steps of any height. But inside the atom, the Creator is an obsessive-compulsive. He has decreed that electrons can only stand on specific steps.”

She then pulled out Eleonora and Horace’s previous paper on the planetary model and spread it open in front of Adèle.

“Look, in this model, the electrons orbit the nucleus like planets. The innermost orbit is the ground floor. The outer orbits are the steps of the staircase.”

Lia’s finger traced over the concentric circles.

“When an electron gains energy, it jumps up to a higher step. But it cannot stop between two steps. There is no foothold there. It’s either on the second floor or the third floor, it can never hang from the window of floor two-and-a-half.”

“And when it jumps down from a high place, back to a lower orbit, it must spit out the energy it previously consumed.”

“How does it spit it out?”

Lia pointed to the illustration of the Balmer spectrum in the journal—those few distinctly colored bright lines.

“It spits it out in the form of light.”

“Because the height difference between the steps is fixed, the energy it releases when it jumps is also fixed. A specific energy corresponds to a specific frequency, and a specific frequency corresponds to a specific color.”

“This is why the hydrogen spectrum we see has only those few lines, instead of being continuous like a rainbow.”

The study fell silent.

Only the occasional crackle of the firewood in the fireplace could be heard.

Adèle stared at the crude stick-figure diagram, then at the complex atomic model diagram beside it.

Her pupils trembled slightly, as if her entire worldview was experiencing a violent earthquake.

“So…” Adèle’s voice was dry. “The world, at its core, is really made of discrete units?”

“From everything we know so far, yes.”

Lia snapped her fingers.

“Although inside the atom, the heights of these steps are not uniform—they get denser the further out you go—the principle is the same.”

Klein, who had been silent all this time, suddenly spoke.

“A brilliant analogy.”

He looked at Lia in his arms, a hint of deep appreciation flashing in his eyes.

This ability to instantly make an extremely abstract theory concrete was even more fascinating to him than her Seventh Circle magic.

“If Horace heard this analogy, he would probably crumple up your draft paper and hang it on the wall,” Klein said faintly.

“He would only complain that my drawing is too ugly.”

Lia tossed the pen back onto the desk and leaned back lazily into Klein’s embrace.

“Alright, Adèle, do you understand now?”

Adèle nodded vigorously, then shook her head.

“I understand the principle… but…”

She looked at the paper, her expression on the verge of tears. “This means…”

“It means I have to learn math all over again? All those integrals, those derivatives… they all have to change in the face of the quantum?”

“It’s not just math.”

Klein delivered the merciless final blow.

“The ‘Elementary Theory of Elements,’ the ‘Foundations of Magical Field Construction,’ and even the ‘Energy Optimization of Fireball’ that you learned before—in the face of this new theory, they might all be wrong. Or rather, imprecise.”

Adèle’s vision went black, and she nearly fainted.

“Alright, stop scaring her.”

Lia patted the back of Klein’s hand. “Classical theory is still very useful on the macroscopic level. You don’t need to calculate electron transitions to conjure a fireball.”

She turned to Adèle. “This paper is just the beginning. It tells us that our previous understanding of the world was too crude. Now, a door has been opened a crack. Although it’s dark inside, at least we know where the way out is.”

Adèle clutched the heavy journal as if it were a booby-trapped bomb.

“I… I’ll go back and digest this first.”

She turned and walked out as if in a trance. “I think I need to eat something sweet, or just dunk my head in cold water.”

Only two people remained in the study.

Klein rested his chin on Lia’s shoulder again, his arms tightening, completely enclosing her in his territory.

“You just said that in the microscopic world, an electron cannot stop between two steps.”

“Mmm.”

“Then what about feelings?”

Klein’s voice suddenly dropped, carrying a barely perceptible hoarseness. “Are they also like quanta, either zero or everything? With no ambiguous middle ground?”

Lia’s heart skipped a beat.

This man, how could he relate everything back to this?

“Feelings are macroscopic.”

Lia forced herself to remain calm, trying to bluff her way through with academic terms. “They are continuous, complex, chaotic…”

“Is that so?”

Klein’s fingers slid down her arm, capturing her hand and intertwining their fingers.

“But for me.”

He turned his head, his lips almost touching Lia’s earlobe, the warm breath making her whole body tremble.

“It is quantized.”

“Once a transition is made, there is no going back.”

Lia could feel her face rapidly heating up, probably already as red as a ripe tomato.

She opened her mouth, wanting to retort, wanting to counter with something like “wave-particle duality” or the “uncertainty principle.”

But her mind was a complete blank, with only the strong, steady heartbeat of the person behind her remaining.

Thump, thump, thump.

Each beat was like the announcement of some irreversible collapse.

Outside the window, the night was deep.

And the storm in the magical world was just beginning.

The old mages who clung to the old theories were destined for a sleepless night, for this paper was like a giant stone thrown into a calm lake, and the ripples it created would completely shatter their illusions of “continuity” and “perfection.”

If they wanted to understand the truth of the world, they could no longer be ostriches with their heads in the sand.