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Fresh out of a flare up and never one to learn from his past skirtings of the sun, Kaveh was once again working on quantum homework at un-archon-ly hours of the night on an empty stomach and too much tea in his system. The cyan and yellow tinted lights of the House of Daena were reflecting in a quite unflattering way on his bags on his eyes while providing ample illumination on the empty sheet of paper before him. The problem that they – his beloved, cursed study group – had set their sights to had a one-dimensional particle with a time-independent Hamiltonian and a passing wave, a Delta function both time and space dependent - the problem their professor mentioned in the lecture before last. Asking no one in particular, he wondered about the time dependent perturbation theory needed for this undertaking. Something about the transition coefficient… “How does the derivation go again? You scale by lambda, series expand, and like you’re sprinkling sumac on dumplings, the lambdas just fall out. Then… you count all those powers?”
~~~~~
On the other side of the main chamber of the library, a grey-haired figure entered with no rush in his steps. While not exactly ahead on this week’s problem set, Haitham was confident in his ability to understand the material on his own time. He preferred to self-study from the mostly adequate textbook, choosing to attend the lectures only as a second review. His gaze swept over the dome for the routine scan of his environments. Layla was tiredly drawing charts on her comically large papers in one tucked-away corner, looking stressed as always. And, it was hard to miss, if not from the visual spectacle of partly sleep deprived students congregating around one chalkboard, then from the truly grating amount of noise coming from it, his study group. He continued his stride towards the table and board, electing to skip a greeting. Instead he quietly asked Kaveh what problem he was working on. Hm. Passing wave? That sounded familiar, a traveling –
A not-so-quiet sigh. “You are a traveling perturbation.” Kaveh hissed.
“We start with setting up that at the infinite past the particle is at eigenstate psi_i of the unperturbed Hamiltonian. Here,” he pointedly tapped a ridiculously fancy quill on the textbook page, “part a asks for probability to find it in a different final state psi_f in the infinite future. It specifically wants the answer as an integral over time and not over space.”
Haitham interceded, “And we assume we know psi_i and psi_f, and integrate over space to explicitly get the matrix element, got it.”
Recalling the reading assignment he had just reviewed earlier today, he continued: “Starting with the general time dependent perturbation theory setup, we get a Kronecker delta for the first term, 0th order, in the perturbation theory expansion when we consider that we start from an eigenstate of the unperturbed system. They have this derivation in section three of the chapter, I believe. Then, the second term, 1st order, has a time integral from the differential equation we solved in the derivation, and the matrix element that’s in it is a space integral. Since the perturbation is a Dirac delta, we can just take out the space integral.”
“… And we can set the upper bound of the time integral to infinity and lower bound to negative infinity because of the definition of when we had or reached these initial or final states, respectively,” a voice chimed in. Tighnari flashed a toothy grin at the late arrival. He enjoyed Haitham’s presence as they both tended to entertain themselves by coming up with edge case questions in review sessions.
Sethos, on the other hand, seemed troubled. “In this problem’s case, replacing by negative infinity for when the initial state was reached makes sense, but why is the original lower bound set to 0 in the time dependent perturbation theory setup? … I forgot, I should review soon. Hey, we should review soon!”
The conversation having drifted off to the perturbation theory derivation, Haitham returned his attention to the next steps. The resulting simplified integral and coefficients, which together are the coefficient of order 1 term – order 0 term was 0 since we know in this case the final state is different – when modulo squared, give the probability of this transition event.
Little known to him, over at the other side of the table, Cyno was echoing his current streamlined cogitations, scribbling concise annotations next to his scratch work. Haitham would be hard pressed to guess this as the thing he could register the most was how his friend was concurrently taking small bites of chalk. As if it was the most natural snack in all of Teyvat. Hm.
~~~~~
“Part b: We are asked to reinterpret results of part a via a Fourier mindset on the Dirac delta function – that is, a superposition of periodic perturbations of different frequencies. And I believe we know how to deal with periodic perturbations from earlier.” Tighnari announced.
Sethos mused that they needed to keep the coefficient c_f as parametrized with t, instead of substituting in t=infinity yet, in order for their various algebra and calculus shenanigans to work. He thought for a split second that he should make a pun about the various shenanigans with variables. Cyno would surely appreciate it – but he should at least attempt to keep the study session on the rails.
Ever the questioning mind, Haitham was pondering why they started with the bounds of the time integral being negative to positive infinity, only to then simplify it to a symmetric-over-the-origin time parametrization? This symmetry only made sense when they knew that they would substitute t=infinity. Was it really ok to generalize like this? As if reading his mind, Tighnari quite expressively circled his integral’s lower and upper bounds on the chalkboard. He and Tighnari seemed to be on the same frequency quite often.
“... I don’t really know, but in this case it’s infinite in a helpful way. Besides, the 0 bound is malleable and kind of arbitrary anyway,” Nari mumbled.
After a beat of blissfully silent work time, as harmonious as they could manage, Tighnari spoke up in a clearer voice this time. “For the Dirac delta we substitute its Fourier expansion, and evaluate, reshape, sculpt the results of the integral. You will get something that looks a bit like a sinc function.” Periodic treatment should bring out the variations on the trigonometric functions, like the brassica oleracea; the variations could be traced back to the shared complex exponential form similarities. Tighnari thought longingly with a rumbling stomach, to the cauliflower dinner he had prepared but had to delay indefinitely for this problem set.
There was another kind of periodic perturbation going on in the peripherals of his vision, Sethos and Wanderer tucked behind a bookcase. For a moment he imagined having Haitham’s headphones to dampen the awareness instead of his own low amplitude signal detecting ears. Just how quickly had Sethos slinked away from his search for more chalk around the board?
Tighnari continued on with the archons-damned integral, drawing another smooth curve before the integrand on a new line. He noticed a pair of eyes on him lingering longer than quite a few seconds. Turning his head slightly to have his gaze fall on Kaveh, who was staring directly at him with an unreadable expression, Nari’s heart skipped a beat: He realized that his complex exponent was not unitless... Kaveh’s intense zoning out had to be put to the side as he traced his steps back four two lines prior. This would be yet another headache. Luckily, Tighnari was quick enough to notice he had forgotten to multiply in the omega for energy difference in the brackets.
Catching everyone up to speed with the integral, Cyno was narrating and directing the next step: “We change the remaining spatial variable x to vt from the constant speed perturbation, then replace bounds and all that. Get it? Because x is a variable for space, for position, for placement, and replace includes the word pl–” A collective, loud sigh from the table cut Cyno’s beautifully executed pun short, though if one looked closely they might have caught a flicker of the smallest smile on Haitham’s face. In their integral, their communally cared for, shared integral, now everything was expressed with the integration variable and bounds as time. From this version of the coefficient c_f they could easily get the probability of transition from the initial to final state again, and see that the answers matched to the first part!
Cyno was almost steamrolling past Tighnari’s efforts with the now-time-parametrized integral. How could the bounds be infinite? An improper integral? What was truly improper was Sethos’ current extracurricular activity but the matter at hand for this coefficient still required addressing. Infinite bounds did not make sense for this t’ variable of integration. Ah – he forgot to change the bounds in his change of variables. The position infinities corresponded to the time parametrization t, -t and t to be specific, for the bounds of the integration variable t’. Attention to detail was hard to pay when working with his friends, especially when he felt compelled to watch Cyno with the corner of his eye to try to convince him to eat less of the chalk, if – no, once – he passed a certain volume threshold of the toxic material. It was not purely calcite, especially the fancy variety of chalk for slate writing that the Akademiya provided in the library. What was it about the chalk? He had heard Cyno mention Albedo in relation to this, perhaps Nari should ask either of them more on the topic: He was intrigued, if not concerned.
Then came the logistics: The House of Daena would close at midnight, only one of the other smaller library rooms in the Akademiya building open two hours past it, and after all that they would need to choose someone’s apartment to grace with their presence. Luckily, most of their roommates were also in this sleep deprived, archons-forsaken study group. So it was that finally unentangled from Wanderer due to external observation and measurement – much like the system of fermions they saw last term – Sethos led their brief walk to Tighnari’s cramped apartment. The lack of space there was partly due to the architecture but most visibly from the horde of herbs he had dried hanging from almost all available ceiling space.
~~~~~
Once they settled into a well packed tiling of two dimensional space in their sitting arrangement, Cyno led the next part of the problem. “Part c says to apply the result of part a to a one dimensional infinite square well with walls at x=0 and x=a. Wants us to evaluate the transition probability for ground to first excited state, state one to two, and express results in dimensionless parameters alpha – hm, it relates perturbation strength scaling, velocity of perturbation – and beta, which relates energy difference, velocity of perturbation, well width. Then… we find transition amplitude c_f, and after that the probability P_if is its square modulus.”
While he narrated the problem statement, Cyno paced in front of their magnificent chalkboard as if he was lecturing. Two terms ago they had all pooled together some Mora to purchase the old chalkboard and carried it to Nari’s already crowded place.
Kaveh, who had been waiting for people to get to this part to confirm his intermediate answer after some initial attempts, rushed into a summary. “You substitute the eigenfunctions of the infinite square well. Ok. You get a messy expression that simplifies somewhat. Sure. Oh and is that a loose imaginary number i in the numerator? Oh? It’s changed to a 1 in the next line? Is that my handwriting? A mistake? On purpose? Archons, what was I thinking while writing that line? I can’t tell if it’s an error or not.” His usually shiny hair too seemed wilted. “Oh well. Truly a messy integral with the sin(x)sin(2x)exp(ix) style integrand…”
Despite Kaveh’s disheartening commentary, Tighnari took a stab at the problem with unexpected energy. “Cool. I think, first, I define an omega_fi to simplify. I’m not writing all that for the energy difference in every line. Then, the spatial components of the wavefunctions are actually functions of vt from our answer to part b. Hm… Wow, that’s a good amount of coefficients, if anyone else also wants to compare intermediate answers?” The amount of symbols in front of him on the chalkboard had kept accumulating.
In his focused ramblings, his tail had started swishing back and forth rapidly, raising a fog of chalk dust up from the ground around the board. Haitham understood the frustration with the sheer volume of algebra that allowed many openings to simple errors, though just like he knew Nari actually did, he also delighted in the efficient solving of busywork equations, derivatives, simplifications. Like eating sunflower seeds: rapid, instinctual, somewhat mindless.
Haitham spotted a suspicious lack of powers of the well width parameter in the line of expression Tighnari was midway through scrawling, and reached to point at the sine function whose antiderivative it should have come from. In his haste, his hands brushed against Nari’s that was holding the chalk. He quickly drew his fingers away – he had to get the eraser for when Nari would have to rewrite the line. He did notice Nari’s gaze flitting to his eyes, and Haitham himself started coughing. Probably all that chalkdust still unsettled in the air.
This integral was quite bulky, never a lack of terms from expanding the complex exponential representations of their beloved sine functions. With so many compounding mistake opportunities, he was not quite sure how long this whole problem would take. Unfortunately, it was due tomorrow.
Kaveh echoed Haitham’s thoughts, complaining out loud about needing a new bottle of ink for all the work he had to scratch in the course of this integral. Where’s the integration bee when you need him? Haitham wondered. Wanderer – that’s what he asked to be called – was a Vahumana student and had not been required to solve an integral by hand in years, yet he was the most knowledgeable in the topic, and more than willing enough to show off when the opportunity breezed past him. After all, that was how he had won the annual Akademiya integration bee two years ago with the largest recorded point difference in the lead, and how Sethos had earned his “busy little bee” nickname competing against Wanderer. Sethos had not seemed bothered by losing so spectacularly at the final round; to the contrary, he had a more vibrant joy in his eyes than usual, if that was possible. What sportsmanship could do to the spirits of a humble student was astonishing.
Cyno’s voice drew him away from the memory of that joyous tournament. “Even after collecting the four terms in the integral the result is nearly uninterpretable… How would one substitute the given dimensionless parameters and call it simplification?” After a beat of silence Haitham registered that Cyno was staring at him. Why him? Surely he could ask Nari who also liked integrals’ busywork.
“You want me to hold your hand through this whole part of the problem, then? There’s still the last question I want to discuss.” Haitham paused to think for a moment and adjust his tone. ”Well, I’ve given you the major hint, I think you can guess how the rest of the expression falls out, regardless of how meticulous the process is. You’re probably on the right path.”
Exasperated, his friend complained, “I’ve been guessing, and checking, and then educated guessing for the past half hour. Maybe you should hold my hand.” Cyno extended his arm in an offering with something restless in how his hands and voice were both shaking. Curious – pay no mind to that now, we have this due before tomorrow’s morning lecture. With no time to waste, Haitham grabbed his hand and switched to writing with his other hand. When he was anticipating in what situation training for ambidextrousness would come in handy (he imagined Cyno would be proud of him if he voiced his pun, but Cyno was probably also running too low on time for that), he had not thought of the case of a friend irritable from hours of subjecting oneself to a problem set. Such were the unforeseen benefits to being prepared for anything.
~~~~~
Kaveh was pleasantly staring at Cyno and Haitham’s tangled hands with a smile when the silence alerted him to the rush he was in and how much the tea was wearing off. Quickly returning to the last part of the problem – just this problem, he had two more to go through… He read aloud the problem statement for part d: “We should show the states’ transition amplitude is small for both beta much greater than and much less than one. beta is from earlier in the problem. Part c, no? Then we explain all that in terms of the natural timescale of the well system in contrast to the time it takes the perturbation to cross the well. Sounds doable!” He was trying to add a bit of motivation to his voice to will it into existence for what was seeming to be an all-nighter.
The mechanical scratch of pencil and pen on paper was their chorus for a little longer of a stretch this time. It seemed like everyone was more than a bit worn down. Haitham finally broke the extended lull in chatter, stating his result: “Leading behavior at both large beta and small beta is small, tends to zero. The natural timescale of this specific infinite square well system depends on the variables of the energy difference between the initial and final states. Well, yes, the time it takes the perturbation to cross the well is the width of the well over the velocity of the perturbation traveling.”
Cyno had arrived at the same conclusion, but he felt his specifics were off. After rubbing his eyes, looking for moral support at his latest rare Genius Invokation TCG card he had gotten thanks to Kaveh, which he kept around for moments just like this, and lastly drinking a cold glass of water, he realized that he was writing the fractions inverted and not getting the inequality directions right as a result of that. Fine. It was satisfying when he finally pinpointed the mistakes, and he did enjoy the theory. Algebra mistakes et al could not stop him. When he got the inverses right, Cyno announced, “For large beta the natural timescale is much smaller than the traveling time, for small beta the natural timescale is much larger than the traveling time. Either the perturbation is too slow – the adiabatic limit – or too sudden for the system to respond. Done!” He smiled bright with determination, almost ready for his next problem. He needed a short break, so he let his gaze wander around the room, basking in the warmth of his friends for a few minutes, away from his notebook. His eyes fell on the textbook Haitham was reading. It was familiar, the one for this course, but he was on a different chapter…? Photon absorption? It was certainly not their current homework, so what was his reasoning behind spending precious time on it? Haitham sure seemed drawn to optics. Ground states excited by particles, frequencies, energies, oscillations. It was certainly fun to get lost in.
Thinking about initial states, he knew he had to access ground state and sleep soon, but another quick glance at his notebook told him he had an optional problem to go through. They were not really optional to him: He would learn this subject until it was second nature like his fighting forms, until he remembered it as well as his favorite TCG decks. He should wield the eraser, chalk, polearm, and cards all as best as he could. Soon he would pass this class… Though first, a nap could be optimal.