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Join the ServerHeisenberg felt like he was going insane.
He locked himself in his dormitory and didn’t step out for a whole week.
His desk was piled high with parchment covered in calculations, each sheet a record of his battle with that bizarre “non-commutative multiplication.”
He tried all sorts of methods.
He attempted to define new rules for this new type of multiplication, but every attempt ended in logical self-contradiction.
He was like a traveler lost in a forest, where every path before him seemed to lead to a dead end.
He began to wonder if his entire approach was wrong from the start.
Perhaps Professor Lia’s words were just a directional guide, and the path he had chosen—using arrays to describe physical quantities—was a dead end.
A tide of frustration washed over him.
He even began to feel nostalgic for the old classical orbit model, which, though incorrect, was at least logically self-consistent.
In his despair, he decided to temporarily set aside his calculations and seek some external help.
He didn’t choose to consult the professors at the academy.
He knew that the problem he was facing was beyond anyone’s current scope of understanding.
He went to the Royal Library.
It was the ocean of knowledge for the entire kingdom, holding the crystallized wisdom of countless sages since the dawn of magic.
He hoped to find a clue, a thread, from the vast collection of texts.
Even a few words of inspiration would be enough.
For the next few weeks, Heisenberg practically lived in the library.
He went through everything from basic algebra to advanced geometry, to the rune theories describing the transformations of magical arrays, one book at a time.
Like a starving man, he greedily devoured any knowledge that might be related to “arrays” and “non-commutativity.”
However, his hopes were dashed time and time again.
All the mathematical theories he found were built upon the unshakable foundation of the commutative law of multiplication.
The problem he had encountered seemed to be an unprecedented monster in the world of mathematics.
Just as he was about to give up and head back, he discovered a row of dust-covered bookshelves in a remote corner of the library.
The books here were mostly obscure, even considered “useless” theoretical manuscripts.
Clinging to his last shred of hope, Heisenberg began to browse through this forgotten knowledge.
Most of the manuscripts were esoteric and difficult to understand, or were theoretical conjectures that had long been proven wrong.
Until he picked up a thin booklet.
On the cover of the booklet, the title was written in an archaic script—”Foundations of Linear Transformations.”
The author’s name was Arthur Cayley.
Heisenberg had some recollection of the name.
He seemed to have been a Ninth-Circle Archmage three hundred years ago who had made significant contributions to mathematics, but his theories were considered too abstract at the time and had not received much attention.
With the attitude of just taking a look, he opened the manuscript.
The first few pages dealt with basic concepts of vector spaces and coordinate transformations, which were not unfamiliar to Heisenberg.
But when he turned to the middle section, his breath caught in his throat.
A brand-new term appeared before his eyes—Matrix.
The manuscript stated that a matrix is a collection of complex or real numbers arranged in a rectangular array.
‘Isn’t this the two-dimensional array I’ve been using?’
Heisenberg’s heart began to race.
He eagerly read on.
The author of the manuscript, the Ninth-Circle Archmage Arthur, had defined in detail the rules for matrix addition and multiplication.
Addition was simple: just add the corresponding elements.
And the multiplication… it was precisely the operation he had been wrestling with.
The row of the first matrix multiplied by the column of the second matrix, their corresponding elements multiplied and then added, yielding the element in the corresponding position of the new matrix.
This definition might seem complex and odd to others.
But when Heisenberg saw the discussion on the properties of matrix multiplication in the manuscript, he froze completely.
On the parchment, a conclusion was written in clear script.
“It should be noted that the multiplication of matrices is generally not commutative. That is, in most cases, matrix A multiplied by matrix B is not equal to matrix B multiplied by matrix A.”
AB ≠ BA.
That sentence in particular.
Heisenberg felt his scalp tingle, a current shooting up his spine to the top of his head.
He had found it.
The monster he had been desperately hunting, the one abandoned by the entire mathematical world, had in fact been tamed by a sage three hundred years ago.
It wasn’t called an array; it was called a matrix.
Its bizarre non-commutative multiplication wasn’t a logical fallacy but a real, existing thing—another rigorous mathematical law.
This was exactly the mathematical tool he needed!
Heisenberg was ecstatic.
He hugged the thin manuscript tightly, as if it were a priceless treasure.
He felt not as if he were reading an old book from three hundred years ago, but as if he were having a conversation with a great prophet across time and space.
He immediately incorporated matrix operations into his theoretical framework.
All previous obstacles were instantly resolved.
The seemingly chaotic and disorderly array operations became clear and orderly under the laws of matrices.
He no longer viewed observable physical quantities as ordinary numbers, but represented each of them as a matrix.
Position was a matrix.
Momentum, another matrix.
Energy was also a matrix.
The entire microscopic world, in his eyes, transformed into a magnificent mathematical structure composed of countless matrices.
He found that describing physical quantities with matrices perfectly matched the patterns he had summarized from experimental data.
A brand-new mechanical framework began to gradually emerge in his mind.
He no longer needed the vague, classical image of an “orbit.”
An electron transitioning from one energy level to another was described in the new theory as the matrix representing the system’s state undergoing a certain mathematical transformation.
The intensity and frequency of the spectral lines could be precisely calculated from the elements of these matrices.
Everything became precise, quantifiable, and logically self-consistent.
After weeks of tireless calculation, working day and night, Heisenberg finally achieved a breakthrough.
Using the commutation relations of matrices, he successfully derived a core equation describing the relationship between fundamental physical quantities.
This equation was concise and profound, revealing that in the microscopic world, we can never know both the precise position and the precise momentum of a particle at the same time.
This was the profound physical meaning hidden behind that bizarre “non-commutativity.”
He first applied this new theory to the simplest model: the hydrogen atom.
Using matrix operations, he recalculated the spectral data of the hydrogen atom.
When the final result appeared on the parchment, Heisenberg trembled with excitement.
The positions of the spectral lines he calculated were astonishingly consistent with Balmer’s experimental results.
More importantly, his theory could even provide a reasonable, quantitative explanation for the intensity of the spectral lines.
This was something the old atomic model was completely incapable of doing.
At that moment, Heisenberg realized that he might have found the key to the essence of the microscopic world.
He had found a brand-new mathematical language to describe the underlying laws of the world.
He compiled all his research from this period into a complete paper.
He named it, “On the Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations.”
He didn’t submit the paper to the academy’s journal.
He chose “The Journal of Magical Theory.”
The moment he dropped the thick manuscript into the magical mailbox of the Royal Academy of Sciences, Heisenberg felt a sense of calm.
A physics revolution, more profound and more thorough than the particle-wave war, was about to begin in the capital city.
And he would be its vanguard.
The excitement doesn't stop here! If you enjoyed this, you’ll adore How to Avoid Regression. Start reading now!
Read : How to Avoid Regression
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