33: Chapter 33 Quantum Notebook

In the spring of the twenty-first year, when Jiang Lin opened the Quantum Mechanics folder for the first time, he did not feel too much fear in his heart.

He thought he was already used to difficulty.

Advanced Mathematics was hard, General Physics was hard, Linear Algebra was hard...

At the beginning, every course was like a wall, cold, straight, with no door.

But over the years, he had figured out a little method.

Do not rush to crash into it; first walk along the wall, find a crack, find a seam, find a place where concepts could be dragged into the Wasteland.

Limits were dragged by him into potato yields, derivatives into Stone House cooling, integrals into solar power curves, matrices into farmland states, and Joseph-Louis Lagrange equations into wind-powered water-lifting devices...

Therefore, Quantum Mechanics should be the same.

No matter how hard it was, there was always an entrance.

At least at the very beginning, Jiang Lin thought this way.

He sat in the fifteen-square-meter Stone House, with a cup of warm water beside his hand.

The water had just been boiled in the morning, carrying a slight taste left by filtered charcoal.

The fire in the stove had been lowered, the solar panels were still outside catching the cold light after the rain, and the input power on the controller screen was not high, but it was enough to support the computer being on for a while.

On the computer screen was the first chapter of Quantum Mechanics.

Blackbody radiation.

Photoelectric effect.

Wave-particle duality.

Jiang Lin slowly read downward.

At the very beginning, he felt okay about himself.

Blackbody radiation could not be explained by classical theory, so Planck introduced energy quanta.

The photoelectric effect was explained by Einstein that light comes in discrete packets, E = h ν.

Jiang Lin did not have too much trouble with these.

The solar panels were right outside, light turned into electricity, and he had relied on it to live for more than twenty years. The fact that light could strike semiconductors packet by packet had a rough footing in his intuition.

He wrote in his concept notebook.

"Photoelectric effect: Like solar panels, light surrenders energy packet by packet."

After writing it down, he felt that he had already understood it.

Continuing downward.

De Broglie matter waves, electrons also have wave properties.

Here he frowned, but did not stop yet.

The first chapter of Quantum Mechanics was thus flipped past by him.

He closed the computer, feeling that while it was difficult, it was not as mystical as legend had it.

The real trouble started from the first time he did exercises.

That was the autumn of the twenty-first year.

He found basic Quantum Mechanics exercises from an offline question bank, preparing to test his understanding.

The first question was to calculate the maximum kinetic energy of electrons escaping in the photoelectric effect.

He picked up his pen, substituted the formulas, and wrote down the answer.

Flipping to the analysis, the answer was correct.

The second question was still about the photoelectric effect.

The question changed the phrasing of frequency and light intensity, asking whether the maximum initial kinetic energy of the escaping electrons would change if the light intensity was increased while the frequency of the incident light remained constant.

Jiang Lin bit the cap of his pen in his mouth, feeling pretty good.

The third question gave a one-dimensional infinite potential well, asking him to write down the wave function and normalization coefficient of the particle.

Jiang Lin flipped back to the textbook and glanced at the example problem.

He recognized the form.

Sine function.

Boundary conditions.

n equals 1, 2, 3.

Following his memory, he wrote down, first writing the wave function as Asin ( n π x / a ).

When writing the normalization coefficient, he stopped.

How much was A?

He remembered it seemed to be a square root of something.

Square root of one-half?

Square root of two divided by a?

Or square root of two divided by the square root of a?

He flipped through the example and learned that it should be square root of 2 / a.

This was only the first place.

The subsequent question asked for the energy difference between n = 1 and n = 2, and he subconsciously assumed that energy increased by double increments.

Flipping to the analysis, he discovered that energy levels were related to n ^ 2.

n = 2 was not twice n = 1.

It was four times.

Jiang Lin bit the pen cap until it turned white.

Clearly every step was in the textbook.

Clearly the example had just been read.

But when turned into exercises, he was like someone who dumped a basket of parts onto a table, knowing they all belonged to the same machine, but not knowing which screw to tighten first.

Finally, facing the analysis, he copied the whole question over again.

When he finished copying, he felt he understood again.

Redoing it the next day, the normalization coefficient was still written wrong.

...

When it came to the fifth question.

The question asked: A particle is in a superposition state; when measuring energy, what is the probability of obtaining the first excited state?

Jiang Lin stared at this question for a while, feeling a bit fuzzy, but seeming to know what to do.

He started writing, wrote halfway, found it did not match, and crossed it out.

Rewriting it, it was still wrong.

He flipped back to the textbook and read the section on superposition states once.

After reading, he felt he understood, and came back to do the problem.

Still wrong.

He did not know where exactly he went wrong.

The concept seemed to be there, and the steps seemed to be there too, but he just could not write out the answer.

Finally, he pulled out the analysis, compared it, and copied the steps over again, feeling: "Oh, so that is how it is."

Then he went on to do the next question.

This cycle of sudden realization when looking at the analysis, but still not knowing how to do similar questions a few days later, ran through his first three years of studying Quantum Mechanics.

In the twenty-second year, Jiang Lin began to seriously look at superposition states.

He read it many times.

Every time he finished reading, he felt he understood.

A superposition state means particles are in multiple states simultaneously.

He wrote in his concept notebook.

"Superposition state: Particles are in several places at the same time, collapsing to one place after measurement."

After writing, he closed the notebook with satisfaction.

He felt this understanding was very intuitive.

Particles being in several places at the same time was like, he thought about it, like a seed that had not yet decided which direction to sprout, and only when you went to look at it would it become fixed.

He felt this analogy was not bad.

So he also wrote this analogy into the concept notebook, and drew a sprouting seed beside it.

However, this understanding was wrong.

Where it was wrong, he had no idea at the time.

He even used this image of being in several places at the same time to successfully explain several multiple-choice questions, making him even more convinced that he understood.

Until the twenty-fourth year, when he did a calculation question about quantum state superposition coefficients that would not match no matter what, forcing him to relearn superposition states from scratch, did he faintly feel something was not right.

It was not that particles were in several places at the same time.

That was not what it meant.

But he could not articulate what it actually meant.

He took out the expressions from different textbooks to compare them.

One said linear superposition of quantum states.

One said each eigenstate had a corresponding probability amplitude.

One said measurement results were random, but probabilities could be precisely predicted.

He could understand every single word of these expressions, but put together, there was always a subtle sense of disconnect.

It was like clearly knowing all the parts, yet being unable to assemble the whole device.

He crossed out the sprouting seed in his concept notebook and wrote a sentence.

"Superposition state is not being in several places at the same time, but what it is, I do not know yet."

After writing this sentence, he felt very uncomfortable.

Not because of not understanding itself.

But because he had always thought he understood before.

From the twenty-fourth year to the twenty-sixth year, Jiang Lin repeatedly got stuck in one place.

What on earth was a wave function?

He knew the wave function was a mathematical object, knew its modulus squared gave probability density, and knew the Schrödinger equation described its evolution.

He could recite all of these.

But there was a feeling, like being separated by a thin membrane, just unable to pierce through it.

He tried to drag the wave function into the Wasteland.

The first time, he compared the wave function to the water level distribution in a water storage pit.

Where there was more water, the particle was more likely to appear there.

He felt this analogy was quite good.

But after a while, when doing problems, he found this analogy did not help much, and sometimes even led him astray.

He crossed out this analogy.

The second time, he tried to compare the wave function to the rainfall probability distribution in a weather forecast.

Where the rainfall probability in a certain area was high, the particle was more likely to appear there.

He again felt this analogy was not bad, and wrote half a page in his concept notebook.

A few months later, he got stuck again on a question about interference fringes.

Rainfall probabilities could not superimpose to produce interference, but wave functions could, because wave functions were complex numbers with phases.

He crossed out this half page.

The third time, he simply stopped looking for analogies.

He just stared at the mathematical form of the wave function.

Complex numbers, modulus, phase, superposition, inner product, expectation value...

He had learned all these things, but Quantum Mechanics used them together in a way he was unfamiliar with.

Every time he thought he was about to grasp something, doing a problem revealed he had not grasped it after all.

What was this feeling like?

Like trying to pinch a cluster of water with your fingers.

When the fingers touched it, the water scattered, and there was nothing in the hand.

He wrote in his concept notebook.

"I do not know what a wave function is."

"I know its mathematical form, I know how to use it, but I just do not know what it is."

After writing this sentence, he remembered when studying high school Physics, the teacher said what a field is, do not worry about it for now, just knowing how to use it is enough.

He thought back then he did not care what an electric field actually was, and ended up using it just fine for many years.

Perhaps the wave function was the same.

Perhaps not everything required him to figure out what it was.

He wrote down this thought and felt a little relieved.

Then he flipped out the next question, continued to do it, and got it wrong again.

In the winter of the twenty-sixth year, Jiang Lin paused Quantum Mechanics for a full two months for the first time.

It was not a planned rest, but a forced stop.

Because he found himself trapped in a very strange state; he had a feeling while reading, but nothing remained when he put it down.

He could not summarize what the reason was, nor could he analyze where his problem lay.

He just opened Quantum Mechanics one day, sat blankly for an hour, then turned off the computer and went to fix the drainage ditch on the west side of the Stone House.

For the next two months, he did other things.

It was not that he was deliberately avoiding it.

When he thought of it, there was only a vague, inexplicable discomfort.

Like getting a question wrong, but not knowing where the mistake was.

In the spring of the twenty-seventh year, one night, Jiang Lin was awakened by a dream.

In the dream, he returned to the age of eighteen.

Not the Wasteland.

It was early morning in the Real World.

The sky was not yet completely bright, and there was a bit of gray-blue light in the room.

He sat by the bed and saw the inexplicably appearing countdown in front of his eyes.

The numbers ticked down second by second.

He did not know what that was.

He did not know what would happen after it reached zero.

He did not know where he would go.

When he woke up, the stove had already gone out.

It was very cold in the Stone House.

The wind curtain hung behind the door, and the small window on the south wall was sealed with a night shutter.

The wind outside had stopped, and there was a nearly empty quietness.

Jiang Lin lay in the sleeping bag, with his eyes open, looking at the ceiling for a long time.

Then, like sitting up suddenly in dying illness, he suddenly crawled up and reopened Quantum Mechanics.

He could not explain why.

Maybe he just could not sleep and needed to find something to do.

He started reading from the section on superposition states, felt nothing at all, read halfway, turned off the computer, and continued to sleep.

But ever since then, he started over again.

From the twenty-seventh year to the twenty-ninth year, Jiang Lin did something he had never done before.

He began to redo the problems, finding all the questions he had gotten wrong before and tackling them one by one.

While tackling them, he kept chasing down a question: Why on earth did I get it wrong last time?

This matter was much harder than he had imagined.

Some problems, he thought at the time were just calculation errors; upon closer inspection, he realized he had set up the entire framework completely wrong from the start.

Some problems, he found that he had applied the correct steps, but couldn't explain what each step was actually doing.

Some problems, he found he had used an analogy that led him astray, but he hadn't realized this beforehand.

The analogy of the germinating seed—he found its shadow in his mistake notebook.

That analogy made him mistake the coefficients of the superposition state for the probabilities at various positions, rather than the probability amplitudes of the respective eigenstates.

These two things differed by just one character, but were vastly different in essence.

He wrote a line of text in his mistake notebook.

[ This problem was wrong because of the germinating seed. ]

He felt it was a bit absurd, but that was indeed how it was.

It was this seed drawn in his concept notebook two years ago that had quietly led him astray for two years.

In his twenty-eighth year, Jiang Lin began to doubt the feeling of "I understand it."

Previously, after finishing a section and feeling like he understood it, he would close the computer and go do something else.

Now, every time he felt he understood something, he would force himself to do another problem.

Usually, this problem would tell him that he didn't understand it at all.

He gradually figured out a pattern.

He had two kinds of understanding.

One was the kind of understanding where he could restate the textbook sentences in different words.

The other was the kind of understanding where, when encountering a new problem, he knew where to start and could solve it correctly.

These two kinds of understanding were worlds apart.

He wrote in his concept notebook.

[ You cannot completely trust problems either, but if you cannot solve them, you basically do not understand. ]

This sentence looked simple, but it took Jiang Lin nearly seven years to truly treat it as a principle.

In the summer of his twenty-eighth year, he finally straightened out the basics of the wave function.

It wasn't because he suddenly had a breakthrough on a certain day.

It was after finishing a problem calculating eigenvalues that he found he had gotten it right, and without that vague unease while doing it.

He knew what he was calculating, why each step was written that way, and what the result meant.

This feeling was different from plugging into formulas to get the answer as he did before.

He couldn't quite explain how it was different, just that it felt steadier.

It was like standing in the mud before, but standing on a stone now.

It wasn't flying, it was just steadier.

He opened his concept notebook and found the line "I don't know what a wave function is."

He added a line underneath.

[ Twenty-eighth year: I still can't clearly say what it is, but I know how to use it. This is probably the extent of what I can achieve for now. ]

In his thirtieth year, he truly began to tackle the double-slit experiment and the problem of measurement.

He had looked at this part before, but couldn't get into it.

It wasn't that the content was too difficult; it was that his foundational tools were not yet in place, and even if he read it, it would be a fake understanding.

Now that he had a slight grasp of the tools, he discovered how hard of a nut this content was to crack.

He got stuck on the double-slit experiment for nearly a year.

It wasn't stuck on the mathematics.

He could follow along with the math calculations.

It was stuck on a question: Which slit did the electron actually pass through?

He knew the correct answer—before measurement, you cannot ask this question like that; this is not a classical problem.

He knew this sentence.

He could write this sentence out.

But he didn't truly accept it.

There was a voice in his heart always saying, it definitely took one of them, I just don't know which one.

He knew this voice was wrong.

Yet he couldn't suppress it.

He had tried using the wording in the textbook to persuade himself, tried explanations from different teachers, and tried maneuvering back from the mathematics.

None of it worked.

That voice was still there.

It took one, I just don't know.

He wrote in his concept notebook.

[ I know thinking this way is wrong, but I don't know how to stop thinking this way. ]

After writing, he looked at it for a moment and didn't continue writing further.

In his thirtieth year, he did something a bit foolish.

He derived the entire mathematical part of the measurement section from scratch.

It wasn't the first time deriving it; it was the seventeenth or eighteenth time, he couldn't count anymore.

This time, he didn't rush to find a conceptual foothold; he just walked down line by line, seeing what the mathematics itself was saying.

Operators, eigenstates, eigenvalues, probability amplitudes, projections...

Walking halfway through, he suddenly remembered when he was studying matrices before.

Matrix multiplication—when he first learned it, he had completely no intuition about multiplying rows by columns, relying purely on memorized rules.

Later, when he learned that matrices represent linear transformations and connected matrices with stretching and rotation in space, he slowly developed a bit of intuition.

But that feeling didn't come all at once either; it was after calculating the same thing many times that during some calculation, it suddenly felt a bit smoother.

He didn't know why it became smooth, and that smoothness itself didn't last long, but it had indeed existed.

Measurement in Quantum Mechanics perhaps was also like this.

Perhaps it wasn't by figuring it out through thinking, but by calculating to a certain extent, until one day it felt a bit smoother.

He wrote this down and then continued calculating.

From the autumn of his thirtieth year to his thirty-first year, Jiang Lin was doing something he himself felt was a bit foolish.

He reclassified all the problems he had done in the Quantum Mechanics basic problem bank according to the type of error.

They were divided into five categories.

Category 1: Applying the wrong formula.

Category 2: Using the correct concepts, but copying calculations incorrectly.

Category 3: Using an analogy that sounded reasonable but was actually incorrect.

Category 4: Knowing how to calculate the steps, but not knowing what the steps were doing.

Category 5: Having no idea where to start at all.

Categories 3 and 4 were the most numerous.

He stared at these two categories for a long time.

Category 3 was very easy to understand; he had already suffered from the germinating seed.

Category 4 was more troublesome.

Problems where he could calculate the steps but didn't know what they were doing—he used to treat these as correct because the answer actually came out.

But these kinds of problems were the most dangerous.

Change the problem, change the steps, and he wouldn't know how to do it again.

He began to target Category 4 problems, forcing himself to write a sentence next to every step explaining what that step was doing.

Not writing the name of the formula, but writing what it was doing.

For example, for the normalization step, he wrote: make the sum of the probabilities of finding the particle everywhere equal to one, otherwise a probability is not a probability.

For example, for the step of taking the squared modulus, he wrote: turn complex numbers into real numbers, because probabilities cannot be complex numbers.

These words were clumsy to write; sometimes halfway through writing, he felt he was explaining something very silly.

But after finishing and looking back, he found that these clumsy words were much more useful than his previous fancy analogies.

In his thirty-second year, he finished doing that batch of basic problems he had filtered out for the first time, and his accuracy barely passed seventy percent.

He turned to the last problem, filled in the answer, and closed the file.

There was no special feeling.

It wasn't because he didn't care, but because he was too tired.

That afternoon, he went to water the Soybean on the south side, cleared out the silt from the water storage pit, ate a meal, and went to sleep.

The next morning, he opened his concept notebook, wanting to write something.

He sat for a long time, thought for a long time, and forced out a sentence.

[ Quantum Mechanics basic problems, seventy percent now. ]

Then he paused for a moment and added another line.

[ I don't know if this counts as entering the door? It should still be a long way off, but it's stronger than the twenty-first year. ]

After writing, he couldn't clearly describe what kind of taste it was.

It wasn't happiness, or rather not just happiness.

There was something very old that quietly loosened up a bit.

It took him twelve years to go from feeling like he understood to truly being able to solve seventy percent of the problems.

In these twelve years, the most frequent mistake he made was not a lack of effort or the wrong method, but that many times he thought he understood, but actually only understood a superficial layer.

Then he used that layer of skin to survive in the Wasteland for several years without any major problems, which instead made him increasingly convinced that he understood.

Right up until doing problems, right up until that germinating seed.

He flipped to that page from a long time ago in the concept notebook, and found that crossed-out seed.

Drawn in pencil, it had faded a bit, but could still be seen.

A germinating seed, with superposition state written next to it: particles are in several places at the same time, and after measurement, they collapse to one place.

He looked at it for a moment.

This sentence looked very obviously wrong now.

But that him in his twenty-second year, when writing this sentence, truly felt he had figured it out.

This wasn't stupidity.

This was the best he could achieve at that time.

Next to that seed, he added a line of small text.

[ This is the understanding of the twenty-second year; at that time I thought I understood. ]

After writing, he closed the notebook.

In the winter of his thirty-second year, he opened the first file of the advanced section of Quantum Mechanics.

Angular momentum.

Orbital angular momentum operators, quantization, l and m quantum numbers, spherical harmonics...

He read the first paragraph and frowned.

The mathematics here was more complex than the basic part, and there were some things he hadn't seen before.

He opened a blank document and wrote at the top.

[ Quantum Mechanics Advanced, First Round, First Section, Angular Momentum. ]

[ It is expected to take a long time. ]

[ First list all the words I don't understand, then check them one by one. ]

Then he began to list.

The first word, spherical harmonics.

He didn't know what this was.

He opened another file, searched, found the definition of spherical harmonics, read it once, and didn't understand.

He wrote in the blank document.

[ Spherical harmonics: don't understand for now. Leave it aside for now and continue reading forward. ]

He had already learned not to beat himself up over the first word he didn't understand.

Some things you naturally understood after seeing them more, and some things required reading ahead first before coming back to understand the front.

He didn't know which category spherical harmonics belonged to.

So he left it aside for now.

Continuing forward.

The first section on Angular Momentum, read three times, with only about half of it understood.

He underlined all the parts he didn't understand, made them into a list, and saved them in the file.

The list had nine items.

He looked at it once and wrote at the very bottom.

[ Today: nine things I don't understand. ]

[ Tomorrow: strive to make it eight. ]

Then he turned off the computer and finished writing tomorrow's maintenance list.

Item 1: Check the water level of the water storage pit.

Item 2: Clean the surface of the solar panels.

Item 3: Quantum Mechanics Advanced First Section, continue gnawing away at it.

Item 4: Replenish water for the three ridges of Soybean on the south side.

After writing, looking at this list, he suddenly felt something was very steady.

It wasn't that Quantum Mechanics had become simple.

It was that he no longer thought it would become simple.

He already knew that this road would be very long, very slow, and there would be many times he thought he understood but actually didn't.

Next time there would probably still be some new seed quietly drawn in some place he didn't notice, waiting for him to come back and cross it out three years later.

He accepted this matter.

It wasn't liking it, it was accepting it.

No matter how slow, the Soybean still needed to be watered.

Quantum Mechanics, continue gnawing away at it tomorrow.