Chapter 162: The Wave Nature of Matter

Horace had been miserable lately.

Ever since that paper on the “atomic shell structure” was published, the gate of his manor had not known a moment of peace.

The manor, which should have been a place for the sweet daily life of the couple, had now practically become a marketplace.

Countless letters flew in like snowflakes every day.

Worse still, some had pitched tents directly at his main gate, claiming that if Horace didn’t clearly explain the question of “why an electron moving in an orbit does not radiate energy,” they would never leave.

And the most damnable part was that one of those guys was also a Ninth Circle!

Even more fatally, Maxwell’s equations were like a lingering ghost, constantly hovering over his head.

That perfectly symmetrical set of equations, which governed the macroscopic world of electromagnetism, clearly declared: an accelerating charged particle—such as one orbiting a nucleus—must radiate energy outwards.

According to this classical theory, the electron would rapidly spiral into the nucleus due to energy loss, and the entire world would collapse and be destroyed in an instant.

But the world still existed.

Horace looked at the still-solid oak tree outside the window and viciously grabbed a handful of his own hair.

“This makes no sense at all!”

He roared in a low voice to Eleonora beside him, the dark circles under his eyes making him look like a panda. “If Lia is right, then Maxwell’s equations are wrong!

No! Maxwell can’t be wrong! This set of equations perfectly explains the wireless telegraph, lightning, and optics. But Lia’s model can explain spectra.

These two damned theories are like two drunkards hacking at each other with knives, neither willing to give an inch!”

Eleonora sympathetically handed him a cup of calming tea. “My dear, perhaps it’s as Lia said, there’s a boundary between the microscopic and the macroscopic…”

“Where is the boundary? In that damned Planck’s constant h?”

Horace downed the tea in one gulp, grimacing from the heat.

“Now some people are even saying that both theories are wrong! In the latest issue of ‘Arcane’ published by the Magic Association yesterday, a few freshly graduated brats actually dared to question Lia’s basic assumptions!

They asked, on what grounds must energy levels be quantized? Ha! On what grounds? I’d like to know that too!”

***

Meanwhile, inside Klein’s mage tower, the eye of the storm, the atmosphere was strangely peaceful, even carrying a hint of sweet, caramelized aroma.

In the private laboratory on the fourth floor, the fireplace was burning brightly.

Lia was curled up in a sofa that belonged exclusively to Klein, a thick woolen blanket covering her, with only half her head showing.

She was holding a newly delivered journal, which contained the very article that questioned the foundation of her theory.

“‘On Several Questions Regarding the Quantization Hypothesis and the Hydrogen Atom Dilemma’…”

Lia read the title aloud, a playful smile on her lips. “Authored by… Victor? Never heard of him.”

“A newly promoted Third Circle Mage. His foundation in mathematics and physics is very solid.”

Klein sat at the desk opposite her, holding a small silver spoon, carefully drizzling a spoonful of amber-colored honey onto a freshly baked muffin.

“I have to say, the question is very well-posed.”

Lia casually tossed the journal aside, her eyes fixed on the muffin. “I only told them ‘what’, but I didn’t tell them ‘why’. It’s like I told them there was a staircase, but I didn’t tell them why there had to be a staircase instead of a ramp.”

“Because you haven’t explained it clearly yourself.”

Klein put down the spoon and walked over with the plate.

He didn’t hand the plate to Lia, but instead sat on the armrest of the sofa himself, picked up a small piece of muffin, and brought it to Lia’s lips.

“Open your mouth.”

Lia very naturally opened her mouth and took a bite, mumbling indistinctly, “How can something like this be explained so easily? I haven’t even figured it out myself yet.”

Klein watched her puffing cheeks, his gaze softening.

He reached out to wipe a crumb from the corner of her mouth and casually put it in his own mouth to taste.

“Too sweet,” he commented.

“That’s because you added too much honey,” Lia retorted.

“You said yesterday you wanted something sweeter.”

Klein’s tone was flat, yet it carried an undeniable indulgence.

“Also, regarding the second point mentioned in that paper—the Helium atom dilemma. How do you plan to solve that?”

Lia swallowed the muffin and sighed.

“It can’t be solved.”

She even tucked her legs into the blanket, curling up into a ball.

“This current theoretical model is indeed half-baked. It can only handle single-electron systems.

Once a second electron is introduced, the repulsive force between the two electrons, the nucleus’s attraction to them… it becomes a ‘three-body problem,’ which is a mathematical nightmare.”

“This is the tragedy of the old quantum theory. It’s like a patched-up leaky boat. While it can barely explain hydrogen, the moment it encounters a slightly more complex situation, it immediately takes on water and sinks.”

“So, this is the limit?”

Klein tasted another piece of muffin, this time with a blueberry on top.

“For this theory, this is the limit of thought.”

Lia opened her mouth to accept the offering. “But I always feel that there must be something… more fundamental.”

Of course, she knew the answer. The Schrödinger equation, Heisenberg’s matrix mechanics.

Those were the two keys to the new world. But these keys were too heavy, and the current magical world was simply not ready to receive them.

Klein looked at her thoughtful profile, illuminated by the firelight, her eyelashes trembling like golden butterfly wings.

“Since it can’t be solved now, let’s put it aside for a while.”

Klein set the plate aside, leaned forward slightly, and propped his hands on either side of her, trapping her between the sofa and himself.

“In any case, the louder those people outside argue, the more it proves they’re thinking. Compared to that…”

His voice dropped low, carrying a dangerous magnetism.

“You’ve been cooped up here with me for three days without going out. You do nothing but eat and sleep, not even willing to lift a finger.”

“That’s to conserve energy!”

Lia shrank back into the blanket with righteous justification, revealing only a pair of wary eyes. “Thinking is exhausting! And it’s so cold outside…”

“Is it really just because it’s cold? Or is it because someone doesn’t want to face her fanatical followers? I hear your name is now carved on the toilet stall doors at the Magic Academy.”

Lia’s face flushed crimson. “They’re perverts! Who carves their professor’s name on a toilet stall door?!”

“Not only that,” Klein leaned even closer, his nose almost touching hers.

“Someone has also founded the ‘Lia Farrien Fan Club.’ Their slogan is ‘Even a jumping electron will eventually fall into the arms of the goddess.’ Sounds like someone wants to be that electron?”

Lia felt like she was suffocating.

This man had become more and more audacious lately. Ever since that “Exclusion Principle” joke, it was as if he had unlocked some strange new skill, teasing her with practiced ease.

“That’s a metaphor! A metaphor!”

Lia, angry and embarrassed, pushed against his chest. “Get away from me! I’m going to the lab to see what Adèle is doing!”

“Adèle took the day off today.”

Klein didn’t budge an inch. Instead, he caught her wrist, his thumb gently caressing the delicate skin on the inside. “She said she was going on a blind date. So, it’s just us in the tower today.”

Just us.

Those two words echoed in the quiet room with a certain ambiguous resonance.

Lia’s heart instantly accelerated.

She looked at Klein’s face, so close to her own. His eyes reflected a tiny version of herself, so focused it was as if she were the only person in the world.

“Klein…”

Her voice was a little weak.

“Mmm?”

“I’m thirsty.”

Klein chuckled softly, the vibration of his chest traveling through their touching hands.

He didn’t call her out on it, just leaned down and placed a light kiss on her forehead.

“Wait here.”

He got up to get her some water.

Lia took the opportunity to gasp for air, using the back of her hand to cool her burning face.

This wasn’t a mage tower, it was a spider’s den!

Had this man been secretly studying enchantment magic behind her back?

Could he even learn such magic?

A few minutes later, Klein returned with a glass of warm water, only to find that Lia was no longer on the sofa.

She was leaning over the desk, a quill in her hand, frantically drawing something on a piece of parchment.

“Inspiration struck again?”

Klein placed the glass of water by her hand.

“Not inspiration, a response.” Lia didn’t look up, the tip of her pen scratching against the paper.

“Since Victor and the others asked, why can’t I solve the problem completely instead of just giving them a direction?

Since they have so much energy, why don’t I give them a clear direction, so they can find the answer directly instead of just arguing?”

Klein leaned over to take a look.

On the paper was drawn a string of a musical instrument.

To be precise, a string fixed at both ends.

“A standing wave?”

Klein recognized it at a glance. It was a basic concept in acoustics.

“Correct.”

Lia wrote a formula next to the string. “If we stop thinking of an electron as a little ball, but as a kind of wave… a wave oscillating around the nucleus.”

She drew a closed circle around the nucleus.

“If this wave is to exist stably, its head must connect to its tail. In other words, the circumference of the orbit must be an integer multiple of the wavelength.

Just like this string, only vibrations of specific frequencies can form stable standing waves. All others will cancel themselves out.”

Lia looked up, her eyes shining brightly.

“See, this is why it must be an integer. It wasn’t decreed by a god; it was determined by its wave nature. Only integer multiples of the wavelength can survive.”

Klein stared at the simple diagram, his pupils contracting slightly.

To think of an electron… as a wave?

This was a complete subversion of the traditional particle concept.

But he had to admit, the explanation was so elegant it was thrilling.

It didn’t require the forced introduction of some “quantization hypothesis”; everything flowed naturally from the properties of a wave.

“De Broglie would thank me,” Lia muttered under her breath, then pushed the parchment towards Klein.

“This can be sent out as a response in the next issue of ‘Arcane.’ Even if it can’t completely solve the hydrogen atom problem, it can stir the waters even more.”

Klein stared at the parchment for a moment. He suddenly spoke, “Sometimes I wonder, just what kind of crazy theories are packed inside your head?”

“Only one.”

Lia yawned and curled back up on the sofa like a lazy cat. “And this head is very hungry right now and wants to eat meat. I want beef stew for dinner tonight, the kind with lots of potatoes.”

The deep meaning in Klein’s eyes dissipated, replaced by a helpless indulgence.

“As you command, my esteemed colleague.”

***

The next day, this short article titled “Preliminary Thoughts on the Wave Nature of the Electron and the Standing Wave Model” once again detonated a deep-water bomb in the magical world.

Victor stared at the response, which was only a few hundred words long, and spilled coffee all over his pants.

“A wave? The electron is a wave?”

He muttered to himself. “If the electron is a wave, then what’s the point of particles? And what are we? A pile of waves?”

And in Horace’s manor, Horace looked at the diagram of the standing wave and suddenly burst out laughing, laughing so hard that tears streamed down his face.

“Hahaha! So that’s how it is! So that’s how it is!”

He spun around the lab like a madman. “It’s not particles dancing, it’s waves resonating! All things in this world… perhaps they are all waves!”

He rushed to the window and shouted at the crowd of protestors outside:

“All of you, go back and study acoustics! Don’t bother me if you don’t understand standing waves!”

On that day, the instrument shops near the Tower of Truth did exceptionally good business.

Countless mages rushed to buy strings, trying to understand the mysteries of the atom by plucking them.

And the instigator, Lia, was currently sitting at the dining table, contentedly eating the potato and beef stew that Klein had made himself.

“How is it?”

Klein poured her a glass of juice.

“The potatoes are soft, and the beef is flavorful.”

Lia gave a thumbs-up. “A million times better than that auntie in the school cafeteria who only knows how to make dark cuisine. Klein, if you ever lose your job, you could definitely be a five-star chef.”

“I won’t lose my job.”

Klein methodically cut the meat on his own plate. “As long as you’re still making trouble, I’ll always have work.”

“Am I making trouble? I’m advancing physics!”

“Yes, advancing progress.”

Klein put down his knife and fork, his gaze darkening as he looked at a bit of soup on the corner of her mouth.

He didn’t use a handkerchief. Instead, he leaned over directly, his tongue gently flicking away that drop of soup.

The action was so fast that Lia had no time to react.

“You…”

Lia was instantly petrified, the fork in her hand clattering onto the plate.

“It does taste pretty good.”

Klein sat back in his seat as if nothing had happened, but the faint, almost imperceptible blush at the tips of his ears betrayed him.

“You can add more tomatoes next time.”

Lia covered her mouth, feeling as if her heart was about to leap out of her throat.

‘This wasn’t eating beef, this was clearly eating her!’

Outside the window, the wind and snow began to pick up.

The storm in the magical world continued, but in this small world of theirs, there was only the crackling of the fireplace and the sound of two hearts beating out of rhythm.

In this world that was just beginning to study quantization, some feelings were continuous, were indivisible, and were collapsing with irreversible momentum into a single reality.

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