Chapter Text
The physics building was almost empty by the time Lumine finished her research work for the day. Most of the classrooms lining the hallway were dark now, their doors propped open to reveal abandoned chalkboards and rows of empty desks. The only sounds were the distant hum of fluorescent lights and the clacking of a keyboard from another room. She noticed a bit of movement inside the study lounge when she approached the door.
Through the window on the door, she could see Ajax, exactly where Lumine expected him to be. The lounge had become something of his second home over the past two semesters. He occupied the same corner table nearly every evening, surrounded by a mess of notebooks, reference books, and color-coded dry erase markers. One side of the whiteboard was filled with neatly written equations she recognized from their lecture this morning. The other side had been left blank.
She punched in the code written on the top left corner of the whiteboard into a keypad, then opened the door.
"Oh, hey Lumine," Ajax said, turning to greet her. He pushed his glasses up his nose and raked a hand through his already tousled copper curls. "I, uh, I didn't expect you to be here."
"You know," she said, dropping into the chair across from him with a sigh, "I'm starting to think you actually live here."
The corner of his mouth lifted. "I was wondering how long it'd take someone to figure it out."
"So the rumors are true."
"They're greatly exaggerated."
"You don't sleep in here?"
"Only during finals."
She laughed, the knot that had been sitting in her chest loosening just a little. "Can I interrupt your... whatever this is?"
Ajax glanced at the pages spread around him. "Yep. Just attempting to convince myself I understand, uh... perturbation theory."
"We're not learning that this year, are we?"
"Nope."
"Then why study it now?"
He chuckled. "Because I can."
She crossed her arms. "How's that going for you anyway, smarty pants?"
"Not well."
"Oh."
"Enough about me. So." He folded his hands on the table. "What's up?"
Lumine hesitated, before pulling a packet of papers from her backpack and setting it on the table between them. The corners were bent from being shoved into her bag. Pencil marks crowded the margins, many of them rubbed away so aggressively that the paper had begun to wear thin.
"I see you've been fighting with a problem."
"It's a losing battle, I fear."
He glanced at the title. Infinite Square Well. "Ah."
"That's all you're going to say?" She let out a dramatic sigh before dropping into the chair across from him. "I've read the chapter, I've read the lecture notes three times, I watched two videos on it..."
"Mhm."
"I even read the chapter again."
Ajax rested his chin in one hand, pinky finger brushing near his lip piercing. "And?"
"I think I understood it for all of thirty seconds."
"What happened?"
"I don't know."
He chuckled. "Very helpful."
"Hey! I don't know what it is, but I keep feeling like I'm missing something obvious. Everyone in lecture was nodding along, asking good questions, and I was just... there."
Ajax's expression softened. "Yeah, that's true, but also, I want you to know I've tutored classmates on this chapter four times today already."
"Wait... you have?"
He nodded. "Mhm."
"Did people really struggle this much?"
He didn't answer immediately, opting instead to reach over and turn the homework packet around so it faced him before flipping through the pages. "Every single one of them."
"Seriously?"
"Seriously. Quantum mechanics has a way of making smart people suddenly feel very average."
For the first time all day, the knot in her stomach loosened another notch. "That's oddly comforting."
Ajax picked up one of the black dry-erase markers scattered on the table and pulled the cap off with a pop. "I'll let you in on a secret. The name scares people more than the actual problem."
"Well, to be fair, it sounds like a medieval torture device."
"It really does." He stood, walked over to the whiteboard, and erased the equations he'd been working on without a second thought. Lumine watched him blink.
"Weren't you using those?"
"Nah, I wasn't. It's fine, I can derive them again."
"You remember all that?"
He looked over his shoulder. "I remember enough."
She shook her head. "Show-off."
"I'll pretend that was a compliment." He gave her a cheeky grin before drawing two thick vertical lines on the board. "Okay. For the next five minutes, forget everything the textbook told you."
Lumine raised an eyebrow. He tapped the two lines.
"Let's start with a hallway. A hallway with perfectly solid walls."
"Okay."
"And now imagine there's a tiny marble rolling around inside."
She nabbed a fresh sheet of paper and a pen from her backpack. "So it just bounces back and forth forever?"
"Exactly. Except instead of a marble, it's an electron."
Lumine frowned. "But electrons don't roll."
"Yeah, they don't," Ajax laughed. "But this is just an analogy. The walls are so strong that the electron can never leave. That's what 'infinite' means here—the walls are impossible to get through."
"So it's trapped."
"Right. What do you know? Let's start there."
She nodded slowly. "Okay... I remember the textbook said the electron can't just have any energy it wants."
"Right. It can't. Energy has to be quantized. Think about a guitar string. When you pluck it, it doesn't vibrate in every possible shape. Only certain patterns work."
"So, like different musical notes."
"Exactly." He sketched a wavy line between the walls with a blue marker. "Electrons can be particles and waves, but in this case, they're waves. Since the wave has to fit perfectly between the walls, only certain patterns are allowed."
Lumine nodded. "So if the wave doesn't fit..."
"It can't exist."
"Right." She stared at the drawing for a moment. "So then, the allowed wave patterns are what give us the allowed energy levels."
"You got it."
"Then, the size of the box is forcing the wave to fit in certain ways."
"That's the whole idea."
Lumine laughed. "The textbook spent six pages saying what you explained in two minutes."
Ajax shook his head. "Physics textbook writers like making things sound harder than they actually are."
She picked up her pen and pushed her glasses up on her face. "So the shortest wave that fits is the lowest energy."
"Yep."
"And longer, more complicated wave patterns mean higher energies."
"Exactly."
She wrote that down in her notebook, pen scritching against the paper. "I think I understand this now."
Ajax grinned and pushed his glasses further up his nose. "Physics is weird."
"You're telling me."
"But sometimes," he said, tapping the drawing of the little box, "it's weird in a surprisingly logical way."
Lumine looked down at the two simple lines in her notebook that represented the box. "I guess that's not so bad." She flipped back to the homework assignment. "Okay, then..." She pointed to a diagram of several horizontal lines stacked one above the other. "These are the energy levels."
"Yeah."
"And they're not evenly spaced."
"Good catch." He wrote a few numbers in the margin. "The jump from the first level to the second is smaller than the jump from the second to the third."
Lumine blinked. "That's kinda weird."
"It seems that way at first, but it'll make sense in a moment. Remember how we said the electron is acting like a wave? Each new wave has to squeeze in an extra wiggle." He drew the lowest-energy wave first—a single smooth hump stretching across the box. "That's the simplest pattern." Then he drew another line with two humps. "And this one has to bend more." Finally, he added a third with three humps. "And this one bends even more."
Lumine tilted her head. "So more bends means more energy."
"Exactly. Making a wave bend more takes more energy."
"Kind of like stretching a rubber band farther."
He pondered her analogy for a moment. "That's actually a pretty decent comparison."
She smiled proudly. "I have my moments."
"You do."
She looked back at the page. "So the electron always starts at the bottom?"
"No. It can be in the lowest energy state—that's called the ground state—but if something gives it energy, like light, it can jump to one of the higher levels."
She frowned thoughtfully. "But only those levels."
"Right. It can't stop halfway."
"So..." She hesitated. "If it needs exactly enough energy to reach the next level, what happens if it gets the wrong amount?"
Ajax shrugged. "Nothing."
"Nothing?"
"If the energy doesn't match one of the allowed jumps, the electron can't absorb it."
Lumine stared. "So it's like trying to climb stairs."
He raised a brow. "How so?"
"Well if you think about it, if each stair is a foot high, you can't stand halfway between two steps."
He pointed his pencil at her. "That's one of the best analogies I've heard."
"You mean it?"
"Absolutely. I'm stealing it."
She grinned. "The stairs are the allowed energies. The electron has to land on a step. If someone only lifts me six inches... I don't reach the next step. So I stay where I am."
"Bingo."
She sat back with a satisfied sigh. "I finally feel like I better understand why they're called quantized energies."
Ajax nodded. "And that right there, is one of the biggest ideas in quantum mechanics."
She stared at the sketch of the box again. "It's funny."
"What is?"
"We started with a box, and somehow ended up talking about things like waves, stairs, and guitar strings."
He chuckled. "That's how physics usually goes. To explain it well, you have to spend half the time pretending one thing is another until your brain suddenly clicks."
Lumine capped her pen. "You know..."
"What?"
"I don't think the concept scares me anymore."
"It was never the scary part of all this. The scary part is believing that an electron isn't behaving the way your everyday intuition says it should. And of course, the math."
"Ah, yes. The math. Joy."
"You're scared of it?"
He pushed his glasses up his nose. "At first, sure. I can be. Once I get the hang of what to do, it's not so bad."
"So..." She slid the homework packet toward him. "Can we do problem one before my confidence evaporates?"
Ajax laughed. "Sure thing." He pulled the paper stack between them. "All right. Problem one."
Lumine read the question aloud. "An electron is confined to a one-dimensional infinite square well of width L. Which of the following statements is true?" She groaned. "It's a multi-step problem, with so many parts broken down into multiple-choice question. Those somehow make me more nervous."
"They're dangerous because with the way Professor Marionette designs them, she makes every answer sound reasonable."
She scanned the options. "Option A says the electron can have any positive energy."
"We can eliminate that immediately."
"Yup, because we just talked about quantized energies."
"Exactly. If it could have any energy, there wouldn't be separate energy levels."
She crossed out A. "Option B says the electron is most likely to be found outside the box."
Ajax raised an eyebrow. "What do our stubborn walls say?"
"They say absolutely not."
"Right." She crossed that one out too. "So outside the box is impossible."
"In this model, yes."
She looked at the remaining answers. "Option C says the electron's lowest possible energy is zero." She hesitated. "This one feels... tricky."
"It's meant to stump you, yes."
"I mean, if it's sitting still..."
"Is it sitting still?"
She paused.
"Remember, we're describing it as a wave."
"Oh."
"If the electron had zero energy, what would the wave look like?"
Lumine frowned at her notebook. "I guess..." She drew a perfectly straight line. "Something like this? With no wiggles."
"Does that fit the rules of the box?"
The drawing of the walls was where her gaze flicked to. "No. The wave has to be zero at the walls and still exist in between." She erased the straight line. "So within our boundary conditions, there has to be at least one hump."
"Exactly."
"And making that hump means it has some energy."
"You've got it."
"So even the lowest energy isn't zero."
"Right."
She crossed out option C. "Which leaves... Option D. The electron can occupy only certain allowed energy levels."
Ajax nodded. "And that there's our winner."
Lumine circled the answer with far more confidence than she'd had ten minutes earlier. "One down." She flipped the page. "Twelve more to go. Why does she do this to us?"
"These homework sets have so many problems for a reason. Setting up equations, converting to a different type of math... or maybe Professor Marionette wants us to show her our calculus skills, I guess?"
"I don't think it's that third one, Ajax."
He chuckled. "Definitely not that, then."
The next problem was longer. "Suppose the width of the box is doubled. What happens to the electron's allowed energies?" She stared at it for a moment, mulling over what to do. She knew to widen the box to two times its size, but she was unsure of what to do afterward. "Okay, I don't know this one."
"That's fine." He uncapped a whiteboard marker and drew two rectangles. The first was short, and the second was twice as long. "Imagine you're shaking a jump rope."
"I've never been very good at that sort of thing."
"You don't have to be. If the rope is short, it's harder to make long, gentle waves."
Lumine nodded. "Okay."
"But if someone hands you a much longer rope? The waves spread out more." He tapped the longer rectangle. "The bigger the box, the easier it is for the electron's wave to fit. So... it doesn't have to bend as sharply."
"And less bending means lower energy."
He pushed his glasses further up his nose and pointed at her with a grin. "There you go."
"Making the box bigger lowers the allowed energies."
"Exactly."
She scribbled a note beside the problem. Bigger box → gentler waves → lower energies.
"I like that," Ajax said.
"The note?"
"The fact that you're translating the math into plain language."
"My high school chemistry teacher always told us to do that if at all possible, which is... all the time."
"Good advice."
"She also said if I only memorize equations," Lumine said, "I'll forget them after the exam. But if I remember the story they're telling... then I can rebuild the equation if I need to."
Ajax smiled. "Now that's thinking like a physicist."
She looked at the next page of the assignment, where the equations finally began in earnest. The equation she saw first was quite bulky, with Hyperborean letters, fractions, and square roots filling the page. Her confidence wavered. "So... the equations."
"The equations don't create the idea, right?" He tapped the sketches they had made—the box, the waves, the staircase of energy levels. "They just put numbers to the story."
Lumine looked down at the doodles crowding the margins of her notebook. There were hardly any equations at all—just little waves, boxes, stairs, and a guitar string with a few musical notes floating above it. "Quantum mechanics feels less like a foreign language and more like... a weird fairy tale with really strict rules."
Ajax chuckled. "I've heard people call it many things."
"But not that?"
"What you said was a first for me." He closed the textbook with a soft thump. "I think that's my favorite description so far. Better than black magic, and better than runic script, hah." He glanced at the clock on the wall. "Oh. We've got about half an hour before the physics building closes."
Lumine looked at the growing pile of worked examples. "We're making good progress."
"Don't jinx it."
She laughed. "I'll try not to, then."
He reopened the textbook. "Let's do one more concept. This one confuses almost everyone the first time."
"Oh, joy."
"It asks where the electron is most likely to be."
Lumine frowned. "I thought we already answered that."
"We answered where it can't be."
"Ah."
He turned the book so they could both see a graph. A smooth curve rose from zero at one wall, peaked in the middle, then fell back to zero at the other wall. "So... this is related to the electron's wave."
Lumine studied the curve. "It touches zero at both walls."
"Yes, because the wave has to fit inside the box." He pointed to the highest part of the curve. "Now, imagine this graph is telling you how likely you are to find the electron if you measured where it was."
She repositioned her glasses. "So the taller the graph... the more likely the electron is there." She blinked. "Wait... I thought the electron was moving back and forth."
"In a classical picture, it is, yes. But this isn't classical."
She leaned back. "So before we measure it... it's not following one tiny path?"
"That's a good way to think about it. Instead, we should shift our thinking to probabilities."
"Okay, that means the graph isn't showing a path. It's showing where we're more or less likely to find it."
"Exactly."
Lumine traced the curve with her finger. "So for this lowest-energy state..." She tapped the center with a manicured nail. "It's most likely to be near the middle, and impossible to find exactly at the walls."
"Yup, you got it."
She looked genuinely delighted. "That's... actually kind of beautiful."
He nodded. "It is."
"The wave isn't just some mathematical trick, then. It actually tells us something meaningful." She stared at the graph for another moment. "So if we measured the electron over and over again..."
"We'd get different positions."
"But after thousands of measurements, the pattern would match this curve—oh!"
Ajax looked eager to hear what she had to say. "What?"
"I finally understand why people say quantum mechanics is probabilistic."
"You've been hearing that word for the past few weeks, haven't you?"
"Constantly."
"And Professor Marionette explained it in plain language. Remember?"
"She did." She smiled sheepishly. "But I might have been jotting some whiteboard notes down. That, or I was too busy panicking about the equations."
Ajax laughed. "Fair enough."
She flipped to the end of the chapter. "Can I ask something that's probably dumb?"
"There are no dumb questions."
"If the infinite walls don't actually exist... then why spend so much time studying them?"
Ajax rested his elbows on the table. "Because they're simple."
"Simple?"
"Well. As simple as quantum mechanics gets." He shrugged. "If you want to understand how waves behave, it's easier to start with a perfect box than with messy real-world material. It teaches you the basic concepts of quantum mechanics, which we already covered. Energy quantization, wave function nodes, and symmetry of the wave patterns without heavy mathematical complexity."
"Which we will be using later in more complicated situations."
"Yes." She thought for a moment. "So eventually the boxes become atoms."
"Or semiconductor devices."
"Or lasers."
"Or quantum dots."
She raised an eyebrow. "Quantum dots sound like something from a superhero movie."
"They're real, trust me."
"And they use ideas like this?"
"They do."
She looked back at the simple sketch they'd started with—two walls and a wave trapped between them. "It's funny."
"What?"
"Well... some imaginary box that could never exist helps explain real things."
Ajax smiled. "And that's one of the nicest parts of physics. Sometimes the simplest imaginary models are the best teachers."
Lumine closed her notebook. "I think my brain's about full."
"You've earned a break. You're plenty ahead too, as the homework isn't due until a week from now."
She slid her notes into her backpack, pausing to look at the first page. It was covered not in dense equations, but in little sketches: a guitar string, a staircase, a hallway with impossible walls, and waves of different shapes fitting neatly inside a box. At the top of the page she'd written, in large letters: The math tells the story. Understand the story first.
Ajax caught a glimpse of it as she zipped her bag. "I like that."
"I think future me will appreciate the reminder."
He stood and slung his backpack over one shoulder. "So... I expect you will be here next week?"
She groaned theatrically. "What are we learning next week?"
He grinned. "Harmonic oscillators."
Lumine froze. "Should I be scared?"
He considered it for a moment, before he shook his head. "No. Scared isn't how I'd word it."
"Concerned?"
"Yeah, that's a better word for it."
She sighed dramatically as they headed toward the library doors. "I knew quantum mechanics was going to humble me."
Ajax held the door open for her. "It humbles everyone, even me. But don't worry," he said. "If you need help, well... you know exactly where I'll be."
Outside, the evening air was cool, and the campus buzzed with students heading home after long hours of studying. Lumine felt tired, but for the first time all week, she wasn't intimidated by the problem set waiting for her in her backpack.