Chapter 7 Chapter 5 Dawn
one
The chapter belonging to Heisenberg begins in July 1924.It was a month of good news for Heisenberg, when his thesis on the anomalous Zeeman effect was approved, resulting in his promotion to lecturer status and the eligibility to teach at any level in a German university.And Bohr - who clearly had a clear affection for this wonderful young man - also wrote to tell him that he had been awarded a prize of $1,000 by the International Educational Foundation (IEB), funded by the Rockefeller consortium. , which gave him the opportunity to go to Copenhagen and work with Bohr himself and his colleagues for a year.It was no coincidence that Heisenberg's original tutor in Göttingen, Bonn, was going to give lectures in the United States, so he agreed to go to Copenhagen, as long as he came back before the summer semester started in May next year.Judging from the later situation, Heisenberg's visit to Copenhagen undoubtedly has positive significance for the development of quantum mechanics.
Bohr's institute in Copenhagen already had a worldwide reputation at that time, and together with Göttingen and Munich, it became the "golden triangle" in the history of quantum mechanics.Scholars from all over the world came to visit and study. In the autumn of 1924, there were nearly 10 visiting scholars, 6 of whom were funded by IEB. It was crowded and had to be expanded.Heisenberg arrived in Copenhagen on September 17, 1924 after finishing his summer vacation trip. He and another Dr. King from the United States lived in the home of a professor who had just passed away, and were escorted by his widowed wife Take care of their diet and daily life.For Heisenberg, the place was more of a language school—his poor English and Danish improved by leaps and bounds during his stay.
Closer to home.As we mentioned earlier, at the turn of 1924 and 1925, physics was in a very difficult and confused situation.Tiny cracks have appeared in the delicate atomic structure of Borna, and whether the essence of radiation is particles or waves, the two sides are still fighting fiercely.Compton's experiment has forced even the most skeptical physicists to admit that particle nature is undeniable, but this is bound to overthrow the electromagnetic system, a behemoth that has been rooted in physics for more than a hundred years.And the foundation on which the latter rests--Maxwell's theory seems so unbreakable and unshakable.
We have also mentioned that shortly before Heisenberg came to Copenhagen, Bohr and his assistants Kramers and Slater published a theory called BKS in an attempt to solve the problem of waves and particles dilemma.According to the BKS theory, there are some "virtual vibrations" (virtual oscillators) near every stable atom. These mysterious virtual vibrations correspond to the classical vibrations one by one through the correspondence principle, so that after quantization It still retains all the advantages of the classical wave theory (in fact, it wants to further consider the particles as waves at different levels).However, this seemingly happy theory has unspeakable difficulties. In order to mediate the feud between waves and particles, it even abandons one of the cornerstones of physics: the laws of conservation of energy and conservation of momentum, thinking that they are nothing more than a statistical theory. below the average case.This price was too high, which was strongly opposed by Einstein, and Pauli quickly changed his attitude under his influence. He wrote to Heisenberg more than once to complain about "virtual vibration" and "virtual physics".
Some ideas of BKS are not meaningless.Kramer used the idea of virtual oscillators to study the phenomenon of dispersion and obtained positive results.Heisenberg became interested in this aspect when he was studying in Copenhagen, and jointly published a paper with Kramer in a physics journal. These ideas undoubtedly played an important role in the establishment of quantum mechanics.But the BKS theory finally died halfway. The experiment in April 1925 denied the statement that the conservation is only established in the statistical sense. The light quantum is indeed a real thing, not a virtual wave. The collapse of BKS marked the complete chaos of physics. The problem of particles and waves was so confusing and headache that Bohr said that it was really a kind of "torture".For Heisenberg who once believed in BKS, this is of course bad news, but like a basin of cold water, it can also make him sober up and seriously consider the way out for the future.
The days in Copenhagen were tense and meaningful.Heisenberg undoubtedly sensed an air of competition, and with his competitive nature redoubled his efforts.Of course, competition is one thing, and Copenhagen's free spirit and academic atmosphere are almost unmatched in Europe, and all of these are closely related to Niels Bohr, the "Godfather" of quantum theory.There is no doubt that everyone in Copenhagen is a genius, but they all better set off the greatness of Bohr himself.The genial Dane puts a good smile on everyone and leads people to talk about all kinds of issues.People surrounded him like stars and moons, and everyone was impressed by his knowledge and personality, and Heisenberg was no exception, and he would become one of Bohr's closest students and friends.Bohr often invited Heisenberg to his home (on the second floor of the Institute) to share the old wines in his home collection, or to go for a walk in the woods behind the Institute and discuss academic issues.Bohr was a very philosophical person, and his views on many physical issues were deeply philosophical, which shocked Heisenberg and greatly influenced his own way of thinking.In some ways, the edification in the "quantum atmosphere" in Copenhagen and the communication with Bohr may be more valuable than the actual research Heisenberg did during that time.
At that time, a trend of thought was popular in Copenhagen.I don't know who initiated this idea at the time, but it can be traced back to Mach in history.This trend of thought says that the research objects of physics should only be things that can be observed and practiced, and physics can only start from these things, rather than building on things that cannot be observed or are purely inferred.This point of view had a great influence on Heisenberg and Pauli, who also came to Copenhagen to visit soon after. can be detected experimentally.The most obvious example is the "orbit" of an electron and the "frequency" at which it orbits.We shall soon take a serious look at this question.
On April 27, 1925, Heisenberg returned to Göttingen after his visit to Copenhagen, and began to re-study the spectral lines of the hydrogen atom—should he be able to find out the basic principles of the quantum system from it?Heisenberg's plan is to still adopt the method of virtual oscillators, although BKS has fallen, but this has been proved to be an effective method in the dispersion theory.Heisenberg believed that this idea should be able to solve some problems that the Bohr system could not solve, such as the intensity of spectral lines.But when he carried out the calculation enthusiastically, his optimism quickly disappeared: in fact, if the electron radiation is expanded according to the algebraic method of virtual oscillators, the mathematical difficulties he encountered are almost insurmountable, This forced Heisenberg to abandon his original plan.Pauli was stumped on the same problem, the obstacles were so great that he could barely move forward, and the irascible physicist was so enraged that he was almost ready to give up physics. "There's something wrong with physics," he yelled. "Everything is too hard for me. I'd rather be a movie comedian than have never heard of physics!" (insert, bubble Leigh said he would rather be a comedian because he was one of Chaplin's fans)
In desperation, Heisenberg decided to change the method, temporarily ignoring the intensity of spectral lines, and starting from the movement of electrons in atoms, first established a basic movement model.Facts have proved that he is on the right path, and a new quantum mechanics will be established soon, but it is a form that people have never heard of, and they dare not even imagine before—Matrix.
Matrix is undoubtedly a word with a bit of mystery in itself, like an Enigma.Whether it is from its mathematical meaning or its meaning in the movie (even including the movie sequel), it is so confusing, difficult to grasp, and daunting.In fact, to this day, many people can hardly believe that our universe is built on these monsters.But whether you are reluctant or don't believe it, Matrix has become an indispensable concept in our lives.College students of science cannot escape the linear algebra class, engineers cannot do without MatLab software, and beautiful girls often miss Gino Reeves, there is no way.
Translated in the sense of mathematics, Matrix is translated as "matrix" in Chinese, which is essentially a two-dimensional table.For example, the following 2*2 matrix is actually a 2*2 square table:
Readers may already be confused. Everyone has long been accustomed to ordinary physical formulas represented by letters and symbols. What physical meaning can this weird form represent?What is even more incomprehensible is, can this kind of "table" be able to perform calculations like ordinary physical variables?How do you add, or multiply, two tables?Heisenberg must have gone mad.
However, I have already reminded everyone that we are about to enter an incredible and bizarre quantum world.In this world, everything looks so weird and unreasonable, even a little crazy.Our everyday experience here is completely useless and often even unreliable.The concepts and habits that have been used in the physical world for thousands of years have collapsed in the quantum world. Things that were once taken for granted must be ruthlessly abandoned and replaced by some strange principles that are closer to the truth.Yes, the world is built by these tables.They can not only add and multiply, but also have jaw-dropping calculation rules, which lead to some even more shocking conclusions.Moreover, all this is not imaginary, it is inferred from facts—and the only facts that can be observed and tested.The time has come for physics to change, Heisenberg said.
Let's set off to start this fantastic journey.
two
Physics, Heisenberg firmly thought, should have a firm foundation.It can only start from something that can be directly observed and tested by experiments. A physicist should always adhere to strict empiricism instead of imagining some images as the basis of the theory.The fault of Bohr's theory lies in this.
Let's look back at what Bohr's theory says.It says that the electrons in an atom move around certain orbits at a certain frequency and jump from one orbit to another from time to time.Each electron orbital represents a specific energy level, so when this transition occurs, the electron absorbs or emits energy in a quantized manner equal to the energy difference between the two orbitals.
Well, that sounds good, and the model does work in many cases.But Heisenberg began to ask himself.An electron's "orbit", what exactly is it?Are there any experiments that would allow us to see that electrons do indeed orbit a certain way?Are there any experiments that can reliably measure the actual distance of an orbital from the nucleus?It is true that the picture of the orbit is familiar to people and can be compared to the orbit of a planet, but unlike a planet, is there any way for people to actually see such an "orbit" of an electron and actually measure the "orbit" represented by an orbit? energy"?There is no way, the orbit of an electron and the frequency at which it revolves around the orbit cannot be actually observed, so how do people come up with these concepts and build an atomic model based on it?
Let's recall the relevant part of the previous history. The establishment of Bohr's model has the support of hydrogen atom spectrum.Each spectral line has a specific frequency, and by the quantum formula E1-E2 =
hν, which we know is the result of electrons transitioning between two energy levels.But, argues Heisenberg, you still haven't solved my doubts.There is no actual observation to prove what the "energy level" represented by a certain orbit is, and each spectral line only represents the "energy difference" between two "energy levels".Therefore, only "energy level difference" or "orbital difference" can be directly observed, but "energy level" and "orbital" are not.
To illustrate the problem, let's make an analogy.One of my childhood joys was to collect all kinds of tram tickets to pretend to be a conductor. At that time, tickets in Shanghai were usually very cheap, only a few cents at most.But the rule is this: No matter which station you get on the train from, the farther you sit, the more expensive the ticket is.For example, if I get on the bus from Xujiahui, it may cost only 3 cents to get to Huaihai Road, 5 cents to People’s Square, 7 cents to the Bund, and 10 cents if I go all the way to Hongkou Stadium .Of course, when I went back in the past two years, the bus has long been replaced by unmanned ticket sales and unified billing-the price is the same no matter how far it is, and the fare has long since changed from what it used to be.
Let us assume that there is a bus departing from station A, passing through BCD three stations to arrive at the terminal station E.The charge for this car follows the old tradition of our nostalgic era. Instead of paying 2 yuan for every boarding, it is billed separately according to the starting point and the ending point.We might as well set a charging standard: the distance between station A and station B is 1 yuan, and the distance between station B and C is 0.5 yuan. The distance between C and D is still 1 yuan, while D and E are far away, 2 yuan.In this way, the fare is easy to calculate. For example, if I get on the bus from station B to station E, then I should give 0.5+1+2=3.5 yuan as the fare.Conversely, if I get on the train from station D to station A, the reason is the same: 1+0.5+1=2.5 yuan.
Now Bohr and Heisenberg were called to write a note about the fare and posted it in the car for reference.Bohr readily agreed. He said: This problem is very simple. The problem of fare is actually the problem of the distance between two stations. We only need to write down the location of each station, and passengers can understand it at a glance.So he assumed that the coordinate of station A is 0, and deduced that: the coordinate of station B is 1, the coordinate of station C is 1.5, the coordinate of station D is 2.5, and the coordinate of station E is 4.5.That's enough, Bohr said, the fare is the absolute value of the coordinates of the origin station minus the coordinates of the destination station, our "coordinates" can actually be regarded as a kind of "fare energy level", all situations are completely fine included in the form below:
This is a classic solution, where each station is assumed to have some absolute "fare energy level", just as each orbital of an electron in an atom is assumed to have some specific energy level.All fares, no matter from which station to which station, can be solved with this single variable, which is a one-dimensional traditional table, which can be expressed as an ordinary formula.This is also the traditional solution to all physics problems.
Now, Heisenberg spoke.No, Heisenberg argued that there is a fundamental error in this line of thinking, that is, as a passenger, he is completely unable to realize, and it is impossible to observe what the "absolute coordinates" of a certain station are.For example, if I take a bus from station C to station D, no matter what, I cannot observe the conclusion that "the coordinate of station C is 1.5" or "the coordinate of station D is 2.5".As me, a passenger, the only thing I can observe and understand is that "it costs 1 yuan to get from station C to station D", which is the most conclusive and solid thing.Our fare rules can only be based on such facts, not on unobservable so-called "coordinates" or "energy levels".
How, then, can we build our fare rules from only these observable facts?Heisenberg said that the traditional one-dimensional table is no longer applicable, we need a new type of table, like the following:
Here, the vertical one is the starting station, and the horizontal one is the terminal station.Now every number in this table is actually observable and testable.For example, the 1.5 in the third column of the first row, its abscissa is A, indicating that it departs from station A.Its ordinate is C, indicating to get off at C station.Then, as long as a passenger actually sits from station A to station C, he can verify that the number is correct: this journey does require a fare of 1.5 yuan.
Well, some readers may be impatient, they are indeed two different types of things, but is the significance of this difference so great?After all, don't they express the same charging rule?But things are much more complicated than we imagined. For example, the reason why Bohr's table is so concise is actually the assumption that "from A to B" and "from B to A" require the same amount of money of.This may not be the case. It costs 1 yuan to go from A to B, but it is likely to cost 1.5 yuan from B to A.In this way, Bohr's traditional method will be a big headache, but Heisenberg's table is concise and clear: just modify the number of B to be the abscissa and A to be the ordinate, but the table is no longer symmetrical according to the diagonal That's all.
More importantly, Heisenberg argued that all the laws of physics should also be rewritten according to this form.We already have the classical kinetic equations, now we have to rewrite them all in a quantum way into some kind of tabular equations.Many traditional physical variables are now treated as independent matrices.
In classical mechanics, a periodic vibration can be decomposed into a series of simple harmonic vibration superposition by mathematical method, this method is called Fourier expansion.Imagine our ears, which can sensitively distinguish various sounds, even if these sounds sound at the same time, it doesn't matter if they are mixed together. An audiophile can even distinguish the subtle rustle of a player flipping the score in CD music.The human ear is amazing, of course, but in essence, mathematicians can do all this, too, by breaking down a mixed sound wave into a series of simple harmonics through Fourier analysis.You may want to sigh that the human ear can complete such complex mathematical analysis in an instant, but this is actually a natural evolution.For example, when a goalkeeper hugs a flying football, mathematically speaking, it is equivalent to analyzing a lot of differential equations of gravity and aerodynamics and finding out the trajectory of the ball. From the point of view, it is also equivalent to the calculation of countless risk probabilities and future profits.But this is only due to the forces of evolution that tend to make organisms have such abilities, and this ability is conducive to natural selection, not because of any special mathematical ability.
Back to the topic, in the old atomic model of Bohr and Sommerfeld, we already have the electron motion equation and quantization conditions.This motion can also be transformed into a series of superposition of simple harmonic motion by means of Fourier analysis.Each term in this expansion represents a specific frequency.Now, Heisenberg is going to perform surgery on this old equation, completely transforming it into the latest matrix version.But here comes the difficulty. We now have a variable p, which represents the momentum of the electron, and a variable q, which represents the position of the electron.Originally, these two variables should be multiplied in the old equation, but now Heisenberg has turned p and q into matrices, so how should p and q be multiplied now?
Good question: how do you multiply two "tables" together?
Or we might as well ask ourselves this question first: What does it mean to multiply two tables together?
For ease of understanding, let's go back to our bus fare analogy.Now suppose we have two fare tables formulated by Heisenberg: Matrix I and Matrix II, which respectively represent the fare situation of bus line I and bus line II in a certain place.For simplicity, let's assume that each line has only two stations, A and B.The two tables are as follows:
Well, let's review what these two tables represent.According to Heisenberg's rule, the abscissa of the number represents the starting station, and the ordinate represents the terminal station.Then the 1 in the first row and first column of matrix I means that if you take bus line I, start from point A, and get off at point A, the fare is 1 yuan (ah? Why do you have to stay where you are? What about paying 1 yuan? This... On the one hand, it’s just a metaphor, and you can think of 1 yuan as a starting fee. Besides, in most cities’ subways, if you go in and get out immediately, you really need to use the electronic card A little money will be deducted).Similarly, the 2 in the first row and second column of the matrix I means that you need 2 yuan to travel from A to B by line I.However, if you return from B to A, then you need to look at the number whose abscissa is B and ordinate is A, that is, the 3 in the first column of the second row.The same is true for Matrix II.
Alright, now let's do an elementary school level math exercise: multiplication.It's just that this time it's not ordinary numbers, but two tables: I and II. What is I×II equal to?
Let's write out the exercises in full.Now, boys and girls, what is the answer to this question?
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After-dinner gossip: Physics for boys
In 1925, when Heisenberg made his breakthrough contribution, he was just 24 years old.Despite his astonishing genius in physics, Heisenberg is undoubtedly just a childish child in other respects.He enthusiastically followed the youth group to travel around. During his stay in Copenhagen, he took time to go skiing in Bavaria, but he broke his knee and lay down for several weeks.When swimming in the valley and fields, he was so happy that he even said, "I don't even want to think about physics for a second."
The development of quantum theory is almost the world of young people.Einstein was only 26 years old when he proposed the light quantum hypothesis in 1905.Bohr was 28 years old when he proposed his atomic structure in 1913.When de Broglie proposed Xiangbo in 1923, he was 31 years old.And in 1925, when quantum mechanics made a breakthrough in the hands of Heisenberg, the main figures who later shone in history were almost as young as Heisenberg: Pauli was 25 years old, Dirac was 23 years old, and Uganda was 23 years old. Lunbeck is 25, Gudschmidt is 23 and Jordang is 23.Compared with them, the 36-year-old Schrödinger and the 43-year-old Bonn are simply grandpas.Quantum mechanics is jokingly called "boy physics", and Bonn's theoretical class in Göttingen is also called "Bonn Kindergarten".
However, this only shows the vigor and vigor of quantum theory.In that mythical era, it symbolized the fearless progress of science and created an unprecedented era. The legendary term "boy physics" will also be engraved with eternal light in the history of physics.
three
Last time we assigned a practice question, now let's find out its answer together.
If you remember our public bus analogy, the matrix I on the left side of the multiplication sign represents the fare table of our bus line I, and the matrix II on the right side of the multiplication sign represents the fare table of line II. I is a 2×2 form, and II is also a 2×2 form. We have reason to believe that their product should be in a similar form, which is also a 2×2 form.
But what exactly is the answer?How do we find the four unknowns abcd?More importantly, what is the significance of I×II?
Heisenberg said, I×II, which means you took bus line I first and then changed to line II.What is the a in the answer? a is in the first row and first column, and it must also represent a certain charging situation from A to A's underground car.Heisenberg said, a, in fact, it means that you start from point A on line I, transfer to line II during the period, and finally return to underground station A.Because it is a multiplication, it means the product of "Toll for Line I" and "Toll for Line II".However, the situation is not that simple, because we may have more than one route, and a actually represents the "sum" of all charging situations.
If this is difficult to understand, then we simply make the topic out.The a in the answer, as we have already explained, means that I take line I to start from point A, then transfer to line II, and return to the sum of charges for subway A.So how do we do this concretely?There are two ways: first, we can take line I from point A to point B, then transfer to line II at point B, and then return from point B to point A.In addition, there is another way, that is, we get on line I at ground A, and then get off at the original place.Then get on Line II at A, and get off at the same place.While this may sound unwise, it is certainly a way.Then, a in our answer is actually the sum of the charges of these two methods.
Now let's see how much the specific number should be: the first method, we first take line I from point A to point B, how much should the fare be?We still remember Heisenberg's fare rules, so look at the number whose abscissa I is A and the ordinate is B, that is, the 2, 2 yuan in the first row and second column.Well, then we transferred from point B to line II and returned to point A. The fare here corresponds to the 4 in the first column of the second row of matrix II.So the "charge product" of the first method is 2*4=8.However, we mentioned that there is another possibility, that is, we get on line I at point A and then get off, and then get on line II and get off again, which is also in line with the condition that we start from point A and end at point A. .This corresponds to the product of the two numbers in the first row and first column of the two matrices, 1×1=1.Then, our final answer, a, is equal to the superposition of these two possibilities, that is, a=2×4+1×1=9.Because there is no third possibility.
In the same way, we come to seek b. b represents the sum of charges for taking Line I first and then transferring to Line II, departing from Point A and finally arriving at Point B.There are also two ways to do this: first get off on line I on ground A, and then take line II from ground A to ground B.The charges are 1 block (first row, first column of matrix I) and 3 blocks (first row, second column of matrix II), so 1×3=3.Another way is to take line I to go from A to B, charge 2 yuan (the first row and second column of matrix I), and then transfer to line II at the same place at B, and charge 1 yuan (the second column of matrix II Two rows and second column), so 2×1=1.So the final answer: b=1×3+2×1=5.
You can try to find c and d by yourself without peeking at the answer.In the end it should be like this: c=3×1+1×4=7, d=3×3+1×1=10.so:
Sorry to make you all so miserable, but we do learn new things.If this kind of multiplication sounds foreign to you, we'll soon have an even bigger surprise, but first we need to get acquainted with this new algorithm.The sage said, learn the new by reviewing the past, we don't have to be complacent about what we have learned, but let's consolidate our foundation. Let's check the above question again.Oh, don't faint, don't faint, it's not that tedious, we can reverse the order of multiplication, and now check out II×I:
I know everyone is moaning, but I still insist that reviewing homework is beneficial and harmless.Let's take a look at what a is. Now we take Line II first and then transfer to Line I, so we can get on Line II from A and then get off.Get on Line I again, and then get off again.The corresponding is 1×1.In addition, we can take line II to point B, and transfer to line I at point B to return to point A, so it is 3×3=9.So a=1×1+3×3=10.
Hey sleepy folks, wake up, we have a problem.In our calculation, a=10, but I still remember that our answer said a=9.Everyone, turn back a few pages in your notebook to see if I remember correctly?Well, although everyone did not take notes, I still remember correctly, our a=2×4+1×1=9 just now.It seems that I made a mistake, let's do the calculation again, this time we have to cheer up: a stands for A getting on the bus and A getting off the bus.So the possible situation is: I take Line II and get on A and get off at A (the first row and first column of matrix II), 1 block.Then turn to Line I and get on A and get off at A (the first row and first column of matrix I), which is also 1 block. 1×1=1.Another possibility is that I take Line II and get off at A and B (the first row and second column of matrix II), 3 yuan.Then turn to Line I at B and return to A from B (second row, first column of matrix II), 3 blocks. 3×3=9.So a=1+9=10.
Well, strange, yes.So is it wrong before?Let's do the calculation again, and it seems to be right, a=1+8=9 in front.So, so... who was wrong?Haha, Heisenberg was wrong, he was ashamed this time, what kind of table multiplication he invented, it led to such a ridiculous result: I×II
≠II×I.
Let's calculate the result in its entirety:
Indeed, I×II ≠
II x I.This is really a pity, we originally thought that this tabular calculation was at least a little creative, but now it seems that we have wasted a lot of time, so we have to say sorry.But wait, Heisenberg has something to say, don't mourn our dead brain cells, their death may not be completely meaningless.
Everyone calm down, everyone calm down, Heisenberg said, shaking his beautiful hair, we must learn to face reality.We have already said that physics must start from the only data that can be practiced, rather than relying on imagination and common sense habits.We must learn to rely on mathematics instead of everyday language, because only mathematics has the only meaning and can tell us the only truth.We must realize this: we have to accept what mathematics says.If math says I×II
≠
II×I, then we have to think so, even if the world ridicules us in a more mocking tone, we cannot change this position.What's more, if you carefully examine the meaning of this, it is not too ridiculous: take Line I first, then transfer to Line II, which may lead to different results than taking Line II first, and then transfer to Line I Yes, what's the problem?
Well, someone said sarcastically, so is Newton's second law F=ma or F=am?
Heisenberg said coldly that Newtonian mechanics is a classical system, and what we are discussing is a quantum system.Never make too much fuss about any weird properties of the quantum world, it will drive you crazy.The rules of quantum do not necessarily have to be bound by the exchange rate of multiplication.
He could not do more verbal disputes, and in the summer of 1925, he was infected by a fever and had to leave Göttingen to Helgoland, a small island in the North Sea, to recuperate.But his brain did not stagnate. On a small island far away from the hustle and bustle, Heisenberg firmly followed this peculiar tabular path to explore the future of physics.Moreover, he succeeded very quickly: in fact, as long as the rules of the matrix are applied to the classical dynamical formulas, the old quantum conditions of Bohr and Sommerfeld are transformed into new structures made of solid matrix bricks. Based on the equation, Heisenberg can naturally deduce the quantized atomic energy level and radiation frequency.Moreover, all of these can be solved logically from the equation itself, and there is no need to impose an unnatural quantum condition like Bohr's old model.Heisenberg's table does work!Mathematics explains everything, our imagination is not reliable.
Although what this weird matrix multiplication that does not obey the commutation rate really means is still a mystery to Heisenberg and everyone at the time, the basic form of quantum mechanics has been obtained. Breakthrough progress.From then on, quantum theory will move forward with a majestic attitude, each step is so majestic and magnificent, stirring up monstrous waves and beautiful waves.The next three years will be fantastic and unimaginable in the history of physics. The golden age of theoretical physics will finally radiate its most dazzling brilliance and make the entire 20th century sacred.
Heisenberg later recalled in a letter to his friend Van der Voorden that when he was on the small stone island, one night he suddenly thought that the total energy of the system should be a constant.So he tried to use his rules to solve this equation to find the oscillator energy.It was not easy to solve it. He did it all night, but the result was in good agreement with the experiment.So he climbed a cliff to watch the sunrise, and felt very lucky at the same time.
Yes, the dawn has appeared, the sun is rising slowly from the sea level, the sea surface and the clouds in the sky are dyed red by thousands of rays, and there is a fantastic glow flowing between the sky and the earth.On the top of the high rocky cliff, Heisenberg faced the spectacular sunrise scene, and the blue sea was rising under his feet, extending to the endless distance.Yes, he knows, this is the moment, he has made the most important breakthrough in his life, and the dawn of physics has finally arrived.
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After-dinner gossip: The Matrix
We have seen that Heisenberg invented this peculiar table, I×II ≠ II×I, and even he himself was not sure what kind of monster it was.当他结束养病，回到哥廷根后，就把论文草稿送给老师波恩，让他评论评论。波恩看到这种表格运算大吃一惊，原来这不是什么新鲜东西，正是线性代数里学到的“矩阵”！回溯历史，这种工具早在1858年就已经由一位剑桥的数学家Arthur
Cayley所发明，不过当时不叫“矩阵”而叫做“行列式”（determinant，这个字后来变成了另外一个意思，虽然还是和矩阵关系很紧密）。发明矩阵最初的目的，是简洁地来求解某些微分方程组（事实上直到今天，大学线性代数课还是主要解决这个问题）。但海森堡对此毫不知情，他实际上不知不觉地“重新发明”了矩阵的概念。波恩和他那精通矩阵运算的助教约尔当随即在严格的数学基础上发展了海森堡的理论，进一步完善了量子力学，我们很快就要谈到。
数学在某种意义上来说总是领先的。Cayley创立矩阵的时候，自然想不到它后来会在量子论的发展中起到关键作用。同样，黎曼创立黎曼几何的时候，又怎会料到他已经给爱因斯坦和他伟大的相对论提供了最好的工具。
乔治?盖莫夫在那本受欢迎的老科普书《从一到无穷大》（One, Two, Three…Infinity）里说，目前数学还有一个大分支没有派上用场（除了智力体操的用处之外），那就是数论。古老的数论领域里已经有许多难题被解开，比如四色问题，费马大定理。也有比如著名的哥德巴赫猜想，至今悬而未决。天知道，这些理论和思路是不是在将来会给某个物理或者化学理论开道，打造出一片全新的天地来。
Four
从赫尔格兰回来后，海森堡找到波恩，请求允许他离开哥廷根一阵，去剑桥讲课。同时，他也把自己的论文给了波恩过目，问他有没有发表的价值。波恩显然被海森堡的想法给迷住了，正如他后来回忆的那样：“我对此着了迷……海森堡的思想给我留下了深刻的印象，对于我们一直追求的那个体系来说，这是一次伟大的突破。”
于是当海森堡去到英国讲学的时候，波恩就把他的这篇论文寄给了《物理学杂志》（Zeitschrift fur Physik）,并于7月29日发表。这无疑标志着新生的量子力学在公众面前的首次亮相。
但海森堡古怪的表格乘法无疑也让波恩困扰，他在7月15日写给爱因斯坦的信中说：“海森堡新的工作看起来有点神秘莫测，不过无疑是很深刻的，而且是正确的。”但是，有一天，波恩突然灵光一闪：他终于想起来这是什么了。海森堡的表格，正是他从前所听说过的那个“矩阵”！
但是对于当时的欧洲物理学家来说，矩阵几乎是一个完全陌生的名字。甚至连海森堡自己，也不见得对它的性质有着完全的了解。波恩决定为海森堡的理论打一个坚实的数学基础，他找到泡利，希望与之合作，可是泡利对此持有强烈的怀疑态度，他以他标志性的尖刻语气对波恩说：“是的，我就知道你喜欢那种冗长和复杂的形式主义，但你那无用的数学只会损害海森堡的物理思想。”波恩在泡利那里碰了一鼻子灰，不得不转向他那熟悉矩阵运算的年轻助教约尔当（Pascual
Jordan,再过一个礼拜，就是他101年诞辰），两人于是欣然合作，很快写出了著名的论文《论量子力学》（Zur
Quantenmechanik），发表在《物理学杂志》上。在这篇论文中，两人用了很大的篇幅来阐明矩阵运算的基本规则，并把经典力学的哈密顿变换统统改造成为矩阵的形式。传统的动量p和位置q这两个物理变量，现在成为了两个含有无限数据的庞大表格，而且，正如我们已经看到的那样，它们并不遵守传统的乘法交换率，p×q
≠ q×p。
波恩和约尔当甚至把p×q和q×p之间的差值也算了出来，结果是这样的：
pq – qp = (h/2πi) I
h是我们已经熟悉的普朗克常数，i是虚数的单位，代表-1的平方根，而I叫做单位矩阵，相当于矩阵运算中的1。波恩和约尔当奠定了一种新的力学——矩阵力学的基础。在这种新力学体系的魔法下，普朗克常数和量子化从我们的基本力学方程中自然而然地跳了出来，成为自然界的内在禀性。如果认真地对这种力学形式做一下探讨，人们会惊奇地发现，牛顿体系里的种种结论，比如能量守恒，从新理论中也可以得到。这就是说，新力学其实是牛顿理论的一个扩展，老的经典力学其实被“包含”在我们的新力学中，成为一种特殊情况下的表现形式。
这种新的力学很快就得到进一步完善。从剑桥返回哥廷根后，海森堡本人也加入了这个伟大的开创性工作中。11月26日，《论量子力学II》在《物理学杂志》上发表，作者是波恩，海森堡和约尔当。这篇论文把原来只讨论一个自由度的体系扩展到任意个自由度，从而彻底建立了新力学的主体。现在，他们可以自豪地宣称，长期以来人们所苦苦追寻的那个目标终于达到了，多年以来如此困扰着物理学家的原子光谱问题，现在终于可以在新力学内部完美地解决。《论量子力学II》这篇文章，被海森堡本人亲切地称呼为“三人论文”（Dreimannerarbeit）的，也终于注定要在物理史上流芳百世。
新体系显然在理论上获得了巨大的成功。泡利很快就改变了他的态度，在写给克罗尼格（Ralph Laer Kronig）的信里，他说：“海森堡的力学让我有了新的热情和希望。”随后他很快就给出了极其有说服力的证明，展示新理论的结果和氢分子的光谱符合得非常完美，从量子规则中，巴尔末公式可以被自然而然地推导出来。非常好笑的是，虽然他不久前还对波恩咆哮说“冗长和复杂的形式主义”，但他自己的证明无疑动用了最最复杂的数学。
不过，对于当时其他的物理学家来说，海森堡的新体系无疑是一个怪物。矩阵这种冷冰冰的东西实在太不讲情面，不给人以任何想象的空间。人们一再追问，这里面的物理意义是什么？矩阵究竟是个什么东西？海森堡却始终护定他那让人沮丧的立场：所谓“意义”是不存在的，如果有的话，那数学就是一切“意义”所在。物理学是什么？就是从实验观测量出发，并以庞大复杂的数学关系将它们联系起来的一门科学，如果说有什么图像能够让人们容易理解和记忆的话，那也是靠不住的。但是，不管怎么样，毕竟矩阵力学对于大部分人来说都太陌生太遥远了，而隐藏在它背后的深刻含义，当时还远远没有被发掘出来。特别是，p×q ≠ q×p，这究竟代表了什么，令人头痛不已。
一年后，当薛定谔以人们所喜闻乐见的传统方式发布他的波动方程后，几乎全世界的物理学家都松了一口气：他们终于解脱了，不必再费劲地学习海森堡那异常复杂和繁难的矩阵力学。当然，人人都必须承认，矩阵力学本身的伟大含义是不容怀疑的。
但是，如果说在1925年，欧洲大部分物理学家都还对海森堡，波恩和约尔当的力学一知半解的话，那我们也不得不说，其中有一个非常显著的例外，他就是保罗?狄拉克。在量子力学大发展的年代，哥本哈根，哥廷根以及慕尼黑三地抢尽了风头，狄拉克的崛起总算也为老牌的剑桥挽回了一点颜面。
保罗?埃德里安?莫里斯?狄拉克（Paul Adrien Maurice Dirac）于1902年8月8日出生于英国布里斯托尔港。他的父亲是瑞士人，当时是一位法语教师，狄拉克是家里的第二个孩子。许多大物理学家的童年教育都是多姿多彩的，比如玻尔，海森堡，还有薛定谔。但狄拉克的童年显然要悲惨许多，他父亲是一位非常严肃而刻板的人，给保罗制定了众多的严格规矩。比如他规定保罗只能和他讲法语（他认为这样才能学好这种语言），于是当保罗无法表达自己的时候，只好选择沉默。在小狄拉克的童年里，音乐、文学、艺术显然都和他无缘，社交活动也几乎没有。这一切把狄拉克塑造成了一个沉默寡言，喜好孤独，淡泊名利，在许多人眼里显得geeky的人。有一个流传很广的关于狄拉克的笑话是这样说的：有一次狄拉克在某大学演讲，讲完后一个观众起来说：“狄拉克教授，我不明白你那个公式是如何推导出来的。”狄拉克看着他久久地不说话，主持人不得不提醒他，他还没有回答问题。
“回答什么问题？”狄拉克奇怪地说，“他刚刚说的是一个陈述句，不是一个疑问句。”
1921年，狄拉克从布里斯托尔大学电机工程系毕业，恰逢经济大萧条，结果没法找到工作。事实上，很难说他是否会成为一个出色的工程师，狄拉克显然长于理论而拙于实验。不过幸运的是，布里斯托尔大学数学系又给了他一个免费进修数学的机会，2年后，狄拉克转到剑桥，开始了人生的新篇章。
我们在上面说到，1925年秋天，当海森堡在赫尔格兰岛作出了他的突破后，他获得波恩的批准来到剑桥讲学。当时海森堡对自己的发现心中还没有底，所以没有在公开场合提到自己这方面的工作，不过7月28号，他参加了所谓“卡皮察俱乐部”的一次活动。卡皮察（PLKapitsa）是一位年轻的苏联学生，当时在剑桥跟随卢瑟福工作。他感到英国的学术活动太刻板，便自己组织了一个俱乐部，在晚上聚会，报告和讨论有关物理学的最新进展。我们在前面讨论卢瑟福的时候提到过卡皮察的名字，他后来也获得了诺贝尔奖。
狄拉克也是卡皮察俱乐部的成员之一，他当时不在剑桥，所以没有参加这个聚会。不过他的导师福勒（William Alfred Fowler）参加了，而且大概在和海森堡的课后讨论中，得知他已经发明了一种全新的理论来解释原子光谱问题。后来海森堡把他的证明寄给了福勒，而福勒给了狄拉克一个复印本。这一开始没有引起狄拉克的重视，不过大概一个礼拜后，他重新审视海森堡的论文，这下他把握住了其中的精髓：别的都是细枝末节，只有一件事是重要的，那就是我们那奇怪的矩阵乘法规则：p×q
≠ q×p。
***********
饭后闲话：约尔当
恩斯特?帕斯库尔?约尔当（Ernst Pascual Jordan）出生于汉诺威。在我们的史话里已经提到，他是物理史上两篇重要的论文《论量子力学》I和II的作者之一，可以说也是量子力学的主要创立者。但是，他的名声显然及不上波恩或者海森堡。
这里面的原因显然也是多方面的，1925年，约尔当才22岁，无论从资格还是名声来说，都远远及不上元老级的波恩和少年成名的海森堡。当时和他一起做出贡献的那些人，后来都变得如此著名：波恩，海森堡，泡利，他们的光辉耀眼，把约尔当完全给盖住了。
从约尔当本人来说，他是一个害羞和内向的人，说话有口吃的毛病，总是结结巴巴的，所以他很少授课或发表演讲。更严重的是，约尔当在二战期间站到了希特勒的一边，成为一个纳粹的同情者，被指责曾经告密。这大大损害了他的声名。
约尔当是一个作出了许多伟大成就的科学家。除了创立了基本的矩阵力学形式，为量子论打下基础之外，他同样在量子场论，电子自旋，量子电动力学中作出了巨大的贡献。他是最先证明海森堡和薛定谔体系同等性的人之一，他发明了约尔当代数，后来又广泛涉足生物学、心理学和运动学。他曾被提名为诺贝尔奖得主，却没有成功。约尔当后来显然也对自己的成就被低估有些恼火，1964年，他声称《论量子力学》一文其实几乎都是他一个人的贡献——波恩那时候病了。这引起了广泛的争议，不过许多人显然同意，约尔当的贡献应当得到更多的承认。
Fives
p×q ≠ q×p。如果说狄拉克比别人天才在什么地方，那就是他可以一眼就看出这才是海森堡体系的精髓。那个时候，波恩和约尔当还在苦苦地钻研讨厌的矩阵，为了建立起新的物理大厦而努力地搬运着这种庞大而又沉重的表格式方砖，而他们的文章尚未发表。但狄拉克是不想做这种苦力的，他轻易地透过海森堡的表格，把握住了这种代数的实质。不遵守交换率，这让我想起了什么？狄拉克的脑海里闪过一个名词，他以前在上某一门动力学课的时候，似乎听说过一种运算，同样不符合乘法交换率。但他还不是十分确定，他甚至连那种运算的定义都给忘了。那天是星期天，所有的图书馆都关门了，这让狄拉克急得像热锅上的蚂蚁。第二天一早，图书馆刚刚开门，他就冲了进去，果然，那正是他所要的东西：它的名字叫做“泊松括号”。
我们还在第一章讨论光和菲涅尔的时候，就谈到过泊松，还有著名的泊松光斑。泊松括号也是这位法国科学家的杰出贡献，不过我们在这里没有必要深入它的数学意义。总之，狄拉克发现，我们不必花九牛二虎之力去搬弄一个晦涩的矩阵，以此来显示和经典体系的决裂。我们完全可以从经典的泊松括号出发，建立一种新的代数。这种代数同样不符合乘法交换率，狄拉克把它称作“q数”（q表示“奇异”或者“量子”）。我们的动量、位置、能量、时间等等概念，现在都要改造成这种q数。而原来那些老体系里的符合交换率的变量，狄拉克把它们称作“c数”（c代表“普通”）。
“看。”狄拉克说，“海森堡的最后方程当然是对的，但我们不用他那种大惊小怪，牵强附会的方式，也能够得出同样的结果。用我的方式，同样能得出xy-yx的差值，只不过把那个让人看了生厌的矩阵换成我们的经典泊松括号[x,y]罢了。然后把它用于经典力学的哈密顿函数，我们可以顺理成章地导出能量守恒条件和玻尔的频率条件。重要的是，这清楚地表明了，我们的新力学和经典力学是一脉相承的，是旧体系的一个扩展。c数和q数，可以以清楚的方式建立起联系来。”
狄拉克把论文寄给海森堡，海森堡热情地赞扬了他的成就，不过带给狄拉克一个糟糕的消息：他的结果已经在德国由波恩和约尔当作出了，是通过矩阵的方式得到的。想来狄拉克一定为此感到很郁闷，因为显然他的法子更简洁明晰。随后狄拉克又出色地证明了新力学和氢分子实验数据的吻合，他又一次郁闷了——泡利比他快了一点点，五天而已。哥廷根的这帮家伙，海森堡，波恩，约尔当，泡利，他们是大军团联合作战，而狄拉克在剑桥则是孤军奋斗，因为在英国懂得量子力学的人简直屈指可数。但是，虽然狄拉克慢了那么一点，但每一次他的理论都显得更为简洁、优美、深刻。而且，上天很快会给他新的机会，让他的名字在历史上取得不逊于海森堡、波恩等人的地位。
现在，在旧的经典体系的废墟上，矗立起了一种新的力学，由海森堡为它奠基，波恩，约尔当用矩阵那实心的砖块为它建造了坚固的主体，而狄拉克的优美的q数为它做了最好的装饰。现在，唯一缺少的就是一个成功的广告和落成典礼，把那些还在旧废墟上唉声叹气的人们都吸引到新大厦里来定居。这个庆典在海森堡取得突破后3个月便召开了，它的主题叫做“电子自旋”。
我们还记得那让人头痛的“反常塞曼效应”，这种复杂现象要求引进1/2的量子数。为此，泡利在1925年初提出了他那著名的“不相容原理”的假设，我们前面已经讨论过，这个规定是说，在原子大厦里，每一间房间都有一个4位数的门牌号码，而每间房只能入住一个电子。所以任何两个电子也不能共享同一组号码。
这个“4位数的号码”，其每一位都代表了电子的一个量子数。当时人们已经知道电子有3个量子数，这第四个是什么，便成了众说纷纭的谜题。不相容原理提出后不久，当时在哥本哈根访问的克罗尼格（Ralph Kronig）想到了一种可能：就是把这第四个自由度看成电子绕着自己的轴旋转。他找到海森堡和泡利，提出了这一思路，结果遭到两个德国年轻人的一致反对。因为这样就又回到了一种图像化的电子概念那里，把电子想象成一个实实在在的小球，而违背了我们从观察和数学出发的本意了。如果电子真是这样一个带电小球的话，在麦克斯韦体系里是不稳定的，再说也违反相对论——它的表面旋转速度要高于光速。
到了1925年秋天，自旋的假设又在荷兰莱顿大学的两个学生，乌仑贝克（George Eugene Uhlenbeck）和古德施密特（Somul Abraham Goudsmit）那里死灰复燃了。当然，两人不知道克罗尼格曾经有过这样的意见，他们是在研究光谱的时候独立产生这一想法的。于是两人找到导师埃仑费斯特（Paul Ehrenfest）征求意见。埃仑费斯特也不是很确定，他建议两人先写一个小文章发表。于是两人当真写了一个短文交给埃仑费斯特，然后又去求教于老资格的洛仑兹。洛仑兹帮他们算了算，结果在这个模型里电子表面的速度达到了光速的10倍。两人大吃一惊，风急火燎地赶回大学要求撤销那篇短文，结果还是晚了，埃仑费斯特早就给Nature杂志寄了出去。据说，两人当时懊恼得都快哭了，埃仑费斯特只好安慰他们说：“你们还年轻，做点蠢事也没关系。”
还好，事情并没有想象的那么糟糕。玻尔首先对此表示赞同，海森堡用新的理论去算了算结果后，也转变了反对的态度。到了1926年，海森堡已经在说：“如果没有古德施密特，我们真不知该如何处理塞曼效应。”一些技术上的问题也很快被解决了，比如有一个系数2，一直和理论所抵触，结果在玻尔研究所访问的美国物理学家托马斯发现原来人们都犯了一个计算错误，而自旋模型是正确的。很快海森堡和约尔当用矩阵力学处理了自旋，结果大获全胜，很快没有人怀疑自旋的正确性了。
哦，不过有一个例外，就是泡利，他一直对自旋深恶痛绝。在他看来，原本电子已经在数学当中被表达得很充分了——现在可好，什么形状、轨道、大小、旋转……种种经验性的概念又幽灵般地回来了。原子系统比任何时候都像个太阳系，本来只有公转，现在连自转都有了。他始终按照自己的路子走，决不向任何力学模型低头。事实上，在某种意义上泡利是对的，电子的自旋并不能想象成传统行星的那种自转，它具有1/2的量子数，也就是说，它要转两圈才露出同一个面孔，这里面的意义只能由数学来把握。后来泡利真的从特定的矩阵出发，推出了这一性质，而一切又被伟大的狄拉克于1928年统统包含于他那相对论化了的量子体系中，成为电子内禀的自然属性。
但是，无论如何，1926年海森堡和约尔当的成功不仅是电子自旋模型的胜利，更是新生的矩阵力学的胜利。不久海森堡又天才般地指出了解决有着两个电子的原子——氦原子的道路，使得新体系的威力再次超越了玻尔的老系统，把它的疆域扩大到以前未知的领域中。已经在迷雾和荆棘中彷徨了好几年的物理学家们这次终于可以扬眉吐气，把长久郁积的坏心情一扫而空，好好地呼吸一下那新鲜的空气。
但是，人们还没有来得及歇一歇脚，欣赏一下周围的风景，为目前的成就自豪一下，我们的快艇便又要前进了。物理学正处在激流之中，它飞流直下，一泻千里，带给人晕眩的速度和刺激。自牛顿起250年来，科学从没有在哪个时期可以像如今这般翻天覆地，健步如飞。量子的力量现在已经完全苏醒了，在接下来的3年间，它将改变物理学的一切，在人类的智慧中刻下最深的烙印，并影响整个20世纪的面貌。
当乌仑贝克和古德施密特提出自旋的时候，玻尔正在去往莱登（Leiden）的路上。当他的火车到达汉堡的时候，他发现泡利和斯特恩（Stern）站在站台上，只是想问问他关于自旋的看法，玻尔不大相信，但称这很有趣。到达莱登以后，他又碰到了爱因斯坦和埃仑费斯特，爱因斯坦详细地分析了这个理论，于是玻尔改变了看法。在回去的路上，玻尔先经过哥廷根，海森堡和约尔当站在站台上。同样的问题：怎么看待自旋？最后，当玻尔的火车抵达柏林，泡利又站在了站台上——他从汉堡一路赶到柏林，想听听玻尔一路上有了什么看法的变化。
人们后来回忆起那个年代，简直像是在讲述一个童话。物理学家们一个个都被洪流冲击得站不住脚：节奏快得几乎不给人喘息的机会，爆炸性的概念一再地被提出，每一个都足以改变整个科学的面貌。但是，每一个人都感到深深的骄傲和自豪，在理论物理的黄金年代，能够扮演历史舞台上的那一个角色。人们常说，时势造英雄，在量子物理的大发展时代，英雄们的确留下了最最伟大的业绩，永远让后人心神向往。
回到我们的史话中来。现在，花开两朵，各表一支。我们去看看量子论是如何沿着另一条完全不同的思路，取得同样伟大的突破的。
Readers may already be confused. Everyone has long been accustomed to ordinary physical formulas represented by letters and symbols. What physical meaning can this weird form represent?What is even more incomprehensible is, can this kind of "table" be able to perform calculations like ordinary physical variables?How do you add, or multiply, two tables?Heisenberg must have gone mad.
However, I have already reminded everyone that we are about to enter an incredible and bizarre quantum world.In this world, everything looks so weird and unreasonable, even a little crazy.Our everyday experience here is completely useless and often even unreliable.The concepts and habits that have been used in the physical world for thousands of years have collapsed in the quantum world. Things that were once taken for granted must be ruthlessly abandoned and replaced by some strange principles that are closer to the truth.Yes, the world is built by these tables.They can not only add and multiply, but also have jaw-dropping calculation rules, which lead to some even more shocking conclusions.Moreover, all this is not imaginary, it is inferred from facts—and the only facts that can be observed and tested.The time has come for physics to change, Heisenberg said.
Let's set off to start this fantastic journey.
two
Physics, Heisenberg firmly thought, should have a firm foundation.It can only start from something that can be directly observed and tested by experiments. A physicist should always adhere to strict empiricism instead of imagining some images as the basis of the theory.The fault of Bohr's theory lies in this.
Let's look back at what Bohr's theory says.It says that the electrons in an atom move around certain orbits at a certain frequency and jump from one orbit to another from time to time.Each electron orbital represents a specific energy level, so when this transition occurs, the electron absorbs or emits energy in a quantized manner equal to the energy difference between the two orbitals.
Well, that sounds good, and the model does work in many cases.But Heisenberg began to ask himself.An electron's "orbit", what exactly is it?Are there any experiments that would allow us to see that electrons do indeed orbit a certain way?Are there any experiments that can reliably measure the actual distance of an orbital from the nucleus?It is true that the picture of the orbit is familiar to people and can be compared to the orbit of a planet, but unlike a planet, is there any way for people to actually see such an "orbit" of an electron and actually measure the "orbit" represented by an orbit? energy"?There is no way, the orbit of an electron and the frequency at which it revolves around the orbit cannot be actually observed, so how do people come up with these concepts and build an atomic model based on it?
Let's recall the relevant part of the previous history. The establishment of Bohr's model has the support of hydrogen atom spectrum.Each spectral line has a specific frequency, and by the quantum formula E1-E2 =
hν, which we know is the result of electrons transitioning between two energy levels.But, argues Heisenberg, you still haven't solved my doubts.There is no actual observation to prove what the "energy level" represented by a certain orbit is, and each spectral line only represents the "energy difference" between two "energy levels".Therefore, only "energy level difference" or "orbital difference" can be directly observed, but "energy level" and "orbital" are not.
To illustrate the problem, let's make an analogy.One of my childhood joys was to collect all kinds of tram tickets to pretend to be a conductor. At that time, tickets in Shanghai were usually very cheap, only a few cents at most.But the rule is this: No matter which station you get on the train from, the farther you sit, the more expensive the ticket is.For example, if I get on the bus from Xujiahui, it may cost only 3 cents to get to Huaihai Road, 5 cents to People’s Square, 7 cents to the Bund, and 10 cents if I go all the way to Hongkou Stadium .Of course, when I went back in the past two years, the bus has long been replaced by unmanned ticket sales and unified billing-the price is the same no matter how far it is, and the fare has long since changed from what it used to be.
Let us assume that there is a bus departing from station A, passing through BCD three stations to arrive at the terminal station E.The charge for this car follows the old tradition of our nostalgic era. Instead of paying 2 yuan for every boarding, it is billed separately according to the starting point and the ending point.We might as well set a charging standard: the distance between station A and station B is 1 yuan, and the distance between station B and C is 0.5 yuan. The distance between C and D is still 1 yuan, while D and E are far away, 2 yuan.In this way, the fare is easy to calculate. For example, if I get on the bus from station B to station E, then I should give 0.5+1+2=3.5 yuan as the fare.Conversely, if I get on the train from station D to station A, the reason is the same: 1+0.5+1=2.5 yuan.
Now Bohr and Heisenberg were called to write a note about the fare and posted it in the car for reference.Bohr readily agreed. He said: This problem is very simple. The problem of fare is actually the problem of the distance between two stations. We only need to write down the location of each station, and passengers can understand it at a glance.So he assumed that the coordinate of station A is 0, and deduced that: the coordinate of station B is 1, the coordinate of station C is 1.5, the coordinate of station D is 2.5, and the coordinate of station E is 4.5.That's enough, Bohr said, the fare is the absolute value of the coordinates of the origin station minus the coordinates of the destination station, our "coordinates" can actually be regarded as a kind of "fare energy level", all situations are completely fine included in the form below:
This is a classic solution, where each station is assumed to have some absolute "fare energy level", just as each orbital of an electron in an atom is assumed to have some specific energy level.All fares, no matter from which station to which station, can be solved with this single variable, which is a one-dimensional traditional table, which can be expressed as an ordinary formula.This is also the traditional solution to all physics problems.
Now, Heisenberg spoke.No, Heisenberg argued that there is a fundamental error in this line of thinking, that is, as a passenger, he is completely unable to realize, and it is impossible to observe what the "absolute coordinates" of a certain station are.For example, if I take a bus from station C to station D, no matter what, I cannot observe the conclusion that "the coordinate of station C is 1.5" or "the coordinate of station D is 2.5".As me, a passenger, the only thing I can observe and understand is that "it costs 1 yuan to get from station C to station D", which is the most conclusive and solid thing.Our fare rules can only be based on such facts, not on unobservable so-called "coordinates" or "energy levels".
How, then, can we build our fare rules from only these observable facts?Heisenberg said that the traditional one-dimensional table is no longer applicable, we need a new type of table, like the following:
Here, the vertical one is the starting station, and the horizontal one is the terminal station.Now every number in this table is actually observable and testable.For example, the 1.5 in the third column of the first row, its abscissa is A, indicating that it departs from station A.Its ordinate is C, indicating to get off at C station.Then, as long as a passenger actually sits from station A to station C, he can verify that the number is correct: this journey does require a fare of 1.5 yuan.
Well, some readers may be impatient, they are indeed two different types of things, but is the significance of this difference so great?After all, don't they express the same charging rule?But things are much more complicated than we imagined. For example, the reason why Bohr's table is so concise is actually the assumption that "from A to B" and "from B to A" require the same amount of money of.This may not be the case. It costs 1 yuan to go from A to B, but it is likely to cost 1.5 yuan from B to A.In this way, Bohr's traditional method will be a big headache, but Heisenberg's table is concise and clear: just modify the number of B to be the abscissa and A to be the ordinate, but the table is no longer symmetrical according to the diagonal That's all.
More importantly, Heisenberg argued that all the laws of physics should also be rewritten according to this form.We already have the classical kinetic equations, now we have to rewrite them all in a quantum way into some kind of tabular equations.Many traditional physical variables are now treated as independent matrices.
In classical mechanics, a periodic vibration can be decomposed into a series of simple harmonic vibration superposition by mathematical method, this method is called Fourier expansion.Imagine our ears, which can sensitively distinguish various sounds, even if these sounds sound at the same time, it doesn't matter if they are mixed together. An audiophile can even distinguish the subtle rustle of a player flipping the score in CD music.The human ear is amazing, of course, but in essence, mathematicians can do all this, too, by breaking down a mixed sound wave into a series of simple harmonics through Fourier analysis.You may want to sigh that the human ear can complete such complex mathematical analysis in an instant, but this is actually a natural evolution.For example, when a goalkeeper hugs a flying football, mathematically speaking, it is equivalent to analyzing a lot of differential equations of gravity and aerodynamics and finding out the trajectory of the ball. From the point of view, it is also equivalent to the calculation of countless risk probabilities and future profits.But this is only due to the forces of evolution that tend to make organisms have such abilities, and this ability is conducive to natural selection, not because of any special mathematical ability.
Back to the topic, in the old atomic model of Bohr and Sommerfeld, we already have the electron motion equation and quantization conditions.This motion can also be transformed into a series of superposition of simple harmonic motion by means of Fourier analysis.Each term in this expansion represents a specific frequency.Now, Heisenberg is going to perform surgery on this old equation, completely transforming it into the latest matrix version.But here comes the difficulty. We now have a variable p, which represents the momentum of the electron, and a variable q, which represents the position of the electron.Originally, these two variables should be multiplied in the old equation, but now Heisenberg has turned p and q into matrices, so how should p and q be multiplied now?
Good question: how do you multiply two "tables" together?
Or we might as well ask ourselves this question first: What does it mean to multiply two tables together?
For ease of understanding, let's go back to our bus fare analogy.Now suppose we have two fare tables formulated by Heisenberg: Matrix I and Matrix II, which respectively represent the fare situation of bus line I and bus line II in a certain place.For simplicity, let's assume that each line has only two stations, A and B.The two tables are as follows:
Well, let's review what these two tables represent.According to Heisenberg's rule, the abscissa of the number represents the starting station, and the ordinate represents the terminal station.Then the 1 in the first row and first column of matrix I means that if you take bus line I, start from point A, and get off at point A, the fare is 1 yuan (ah? Why do you have to stay where you are? What about paying 1 yuan? This... On the one hand, it’s just a metaphor, and you can think of 1 yuan as a starting fee. Besides, in most cities’ subways, if you go in and get out immediately, you really need to use the electronic card A little money will be deducted).Similarly, the 2 in the first row and second column of the matrix I means that you need 2 yuan to travel from A to B by line I.However, if you return from B to A, then you need to look at the number whose abscissa is B and ordinate is A, that is, the 3 in the first column of the second row.The same is true for Matrix II.
Alright, now let's do an elementary school level math exercise: multiplication.It's just that this time it's not ordinary numbers, but two tables: I and II. What is I×II equal to?
Let's write out the exercises in full.Now, boys and girls, what is the answer to this question?
***********
After-dinner gossip: Physics for boys
In 1925, when Heisenberg made his breakthrough contribution, he was just 24 years old.Despite his astonishing genius in physics, Heisenberg is undoubtedly just a childish child in other respects.He enthusiastically followed the youth group to travel around. During his stay in Copenhagen, he took time to go skiing in Bavaria, but he broke his knee and lay down for several weeks.When swimming in the valley and fields, he was so happy that he even said, "I don't even want to think about physics for a second."
The development of quantum theory is almost the world of young people.Einstein was only 26 years old when he proposed the light quantum hypothesis in 1905.Bohr was 28 years old when he proposed his atomic structure in 1913.When de Broglie proposed Xiangbo in 1923, he was 31 years old.And in 1925, when quantum mechanics made a breakthrough in the hands of Heisenberg, the main figures who later shone in history were almost as young as Heisenberg: Pauli was 25 years old, Dirac was 23 years old, and Uganda was 23 years old. Lunbeck is 25, Gudschmidt is 23 and Jordang is 23.Compared with them, the 36-year-old Schrödinger and the 43-year-old Bonn are simply grandpas.Quantum mechanics is jokingly called "boy physics", and Bonn's theoretical class in Göttingen is also called "Bonn Kindergarten".
However, this only shows the vigor and vigor of quantum theory.In that mythical era, it symbolized the fearless progress of science and created an unprecedented era. The legendary term "boy physics" will also be engraved with eternal light in the history of physics.
three
Last time we assigned a practice question, now let's find out its answer together.
If you remember our public bus analogy, the matrix I on the left side of the multiplication sign represents the fare table of our bus line I, and the matrix II on the right side of the multiplication sign represents the fare table of line II. I is a 2×2 form, and II is also a 2×2 form. We have reason to believe that their product should be in a similar form, which is also a 2×2 form.
But what exactly is the answer?How do we find the four unknowns abcd?More importantly, what is the significance of I×II?
Heisenberg said, I×II, which means you took bus line I first and then changed to line II.What is the a in the answer? a is in the first row and first column, and it must also represent a certain charging situation from A to A's underground car.Heisenberg said, a, in fact, it means that you start from point A on line I, transfer to line II during the period, and finally return to underground station A.Because it is a multiplication, it means the product of "Toll for Line I" and "Toll for Line II".However, the situation is not that simple, because we may have more than one route, and a actually represents the "sum" of all charging situations.
If this is difficult to understand, then we simply make the topic out.The a in the answer, as we have already explained, means that I take line I to start from point A, then transfer to line II, and return to the sum of charges for subway A.So how do we do this concretely?There are two ways: first, we can take line I from point A to point B, then transfer to line II at point B, and then return from point B to point A.In addition, there is another way, that is, we get on line I at ground A, and then get off at the original place.Then get on Line II at A, and get off at the same place.While this may sound unwise, it is certainly a way.Then, a in our answer is actually the sum of the charges of these two methods.
Now let's see how much the specific number should be: the first method, we first take line I from point A to point B, how much should the fare be?We still remember Heisenberg's fare rules, so look at the number whose abscissa I is A and the ordinate is B, that is, the 2, 2 yuan in the first row and second column.Well, then we transferred from point B to line II and returned to point A. The fare here corresponds to the 4 in the first column of the second row of matrix II.So the "charge product" of the first method is 2*4=8.However, we mentioned that there is another possibility, that is, we get on line I at point A and then get off, and then get on line II and get off again, which is also in line with the condition that we start from point A and end at point A. .This corresponds to the product of the two numbers in the first row and first column of the two matrices, 1×1=1.Then, our final answer, a, is equal to the superposition of these two possibilities, that is, a=2×4+1×1=9.Because there is no third possibility.
In the same way, we come to seek b. b represents the sum of charges for taking Line I first and then transferring to Line II, departing from Point A and finally arriving at Point B.There are also two ways to do this: first get off on line I on ground A, and then take line II from ground A to ground B.The charges are 1 block (first row, first column of matrix I) and 3 blocks (first row, second column of matrix II), so 1×3=3.Another way is to take line I to go from A to B, charge 2 yuan (the first row and second column of matrix I), and then transfer to line II at the same place at B, and charge 1 yuan (the second column of matrix II Two rows and second column), so 2×1=1.So the final answer: b=1×3+2×1=5.
You can try to find c and d by yourself without peeking at the answer.In the end it should be like this: c=3×1+1×4=7, d=3×3+1×1=10.so:
Sorry to make you all so miserable, but we do learn new things.If this kind of multiplication sounds foreign to you, we'll soon have an even bigger surprise, but first we need to get acquainted with this new algorithm.The sage said, learn the new by reviewing the past, we don't have to be complacent about what we have learned, but let's consolidate our foundation. Let's check the above question again.Oh, don't faint, don't faint, it's not that tedious, we can reverse the order of multiplication, and now check out II×I:
I know everyone is moaning, but I still insist that reviewing homework is beneficial and harmless.Let's take a look at what a is. Now we take Line II first and then transfer to Line I, so we can get on Line II from A and then get off.Get on Line I again, and then get off again.The corresponding is 1×1.In addition, we can take line II to point B, and transfer to line I at point B to return to point A, so it is 3×3=9.So a=1×1+3×3=10.
Hey sleepy folks, wake up, we have a problem.In our calculation, a=10, but I still remember that our answer said a=9.Everyone, turn back a few pages in your notebook to see if I remember correctly?Well, although everyone did not take notes, I still remember correctly, our a=2×4+1×1=9 just now.It seems that I made a mistake, let's do the calculation again, this time we have to cheer up: a stands for A getting on the bus and A getting off the bus.So the possible situation is: I take Line II and get on A and get off at A (the first row and first column of matrix II), 1 block.Then turn to Line I and get on A and get off at A (the first row and first column of matrix I), which is also 1 block. 1×1=1.Another possibility is that I take Line II and get off at A and B (the first row and second column of matrix II), 3 yuan.Then turn to Line I at B and return to A from B (second row, first column of matrix II), 3 blocks. 3×3=9.So a=1+9=10.
Well, strange, yes.So is it wrong before?Let's do the calculation again, and it seems to be right, a=1+8=9 in front.So, so... who was wrong?Haha, Heisenberg was wrong, he was ashamed this time, what kind of table multiplication he invented, it led to such a ridiculous result: I×II
≠II×I.
Let's calculate the result in its entirety:
Indeed, I×II ≠
II x I.This is really a pity, we originally thought that this tabular calculation was at least a little creative, but now it seems that we have wasted a lot of time, so we have to say sorry.But wait, Heisenberg has something to say, don't mourn our dead brain cells, their death may not be completely meaningless.
Everyone calm down, everyone calm down, Heisenberg said, shaking his beautiful hair, we must learn to face reality.We have already said that physics must start from the only data that can be practiced, rather than relying on imagination and common sense habits.We must learn to rely on mathematics instead of everyday language, because only mathematics has the only meaning and can tell us the only truth.We must realize this: we have to accept what mathematics says.If math says I×II
≠
II×I, then we have to think so, even if the world ridicules us in a more mocking tone, we cannot change this position.What's more, if you carefully examine the meaning of this, it is not too ridiculous: take Line I first, then transfer to Line II, which may lead to different results than taking Line II first, and then transfer to Line I Yes, what's the problem?
Well, someone said sarcastically, so is Newton's second law F=ma or F=am?
Heisenberg said coldly that Newtonian mechanics is a classical system, and what we are discussing is a quantum system.Never make too much fuss about any weird properties of the quantum world, it will drive you crazy.The rules of quantum do not necessarily have to be bound by the exchange rate of multiplication.
He could not do more verbal disputes, and in the summer of 1925, he was infected by a fever and had to leave Göttingen to Helgoland, a small island in the North Sea, to recuperate.But his brain did not stagnate. On a small island far away from the hustle and bustle, Heisenberg firmly followed this peculiar tabular path to explore the future of physics.Moreover, he succeeded very quickly: in fact, as long as the rules of the matrix are applied to the classical dynamical formulas, the old quantum conditions of Bohr and Sommerfeld are transformed into new structures made of solid matrix bricks. Based on the equation, Heisenberg can naturally deduce the quantized atomic energy level and radiation frequency.Moreover, all of these can be solved logically from the equation itself, and there is no need to impose an unnatural quantum condition like Bohr's old model.Heisenberg's table does work!Mathematics explains everything, our imagination is not reliable.
Although what this weird matrix multiplication that does not obey the commutation rate really means is still a mystery to Heisenberg and everyone at the time, the basic form of quantum mechanics has been obtained. Breakthrough progress.From then on, quantum theory will move forward with a majestic attitude, each step is so majestic and magnificent, stirring up monstrous waves and beautiful waves.The next three years will be fantastic and unimaginable in the history of physics. The golden age of theoretical physics will finally radiate its most dazzling brilliance and make the entire 20th century sacred.
Heisenberg later recalled in a letter to his friend Van der Voorden that when he was on the small stone island, one night he suddenly thought that the total energy of the system should be a constant.So he tried to use his rules to solve this equation to find the oscillator energy.It was not easy to solve it. He did it all night, but the result was in good agreement with the experiment.So he climbed a cliff to watch the sunrise, and felt very lucky at the same time.
Yes, the dawn has appeared, the sun is rising slowly from the sea level, the sea surface and the clouds in the sky are dyed red by thousands of rays, and there is a fantastic glow flowing between the sky and the earth.On the top of the high rocky cliff, Heisenberg faced the spectacular sunrise scene, and the blue sea was rising under his feet, extending to the endless distance.Yes, he knows, this is the moment, he has made the most important breakthrough in his life, and the dawn of physics has finally arrived.
***********
After-dinner gossip: The Matrix
We have seen that Heisenberg invented this peculiar table, I×II ≠ II×I, and even he himself was not sure what kind of monster it was.当他结束养病，回到哥廷根后，就把论文草稿送给老师波恩，让他评论评论。波恩看到这种表格运算大吃一惊，原来这不是什么新鲜东西，正是线性代数里学到的“矩阵”！回溯历史，这种工具早在1858年就已经由一位剑桥的数学家Arthur
Cayley所发明，不过当时不叫“矩阵”而叫做“行列式”（determinant，这个字后来变成了另外一个意思，虽然还是和矩阵关系很紧密）。发明矩阵最初的目的，是简洁地来求解某些微分方程组（事实上直到今天，大学线性代数课还是主要解决这个问题）。但海森堡对此毫不知情，他实际上不知不觉地“重新发明”了矩阵的概念。波恩和他那精通矩阵运算的助教约尔当随即在严格的数学基础上发展了海森堡的理论，进一步完善了量子力学，我们很快就要谈到。
数学在某种意义上来说总是领先的。Cayley创立矩阵的时候，自然想不到它后来会在量子论的发展中起到关键作用。同样，黎曼创立黎曼几何的时候，又怎会料到他已经给爱因斯坦和他伟大的相对论提供了最好的工具。
乔治?盖莫夫在那本受欢迎的老科普书《从一到无穷大》（One, Two, Three…Infinity）里说，目前数学还有一个大分支没有派上用场（除了智力体操的用处之外），那就是数论。古老的数论领域里已经有许多难题被解开，比如四色问题，费马大定理。也有比如著名的哥德巴赫猜想，至今悬而未决。天知道，这些理论和思路是不是在将来会给某个物理或者化学理论开道，打造出一片全新的天地来。
Four
从赫尔格兰回来后，海森堡找到波恩，请求允许他离开哥廷根一阵，去剑桥讲课。同时，他也把自己的论文给了波恩过目，问他有没有发表的价值。波恩显然被海森堡的想法给迷住了，正如他后来回忆的那样：“我对此着了迷……海森堡的思想给我留下了深刻的印象，对于我们一直追求的那个体系来说，这是一次伟大的突破。”
于是当海森堡去到英国讲学的时候，波恩就把他的这篇论文寄给了《物理学杂志》（Zeitschrift fur Physik）,并于7月29日发表。这无疑标志着新生的量子力学在公众面前的首次亮相。
但海森堡古怪的表格乘法无疑也让波恩困扰，他在7月15日写给爱因斯坦的信中说：“海森堡新的工作看起来有点神秘莫测，不过无疑是很深刻的，而且是正确的。”但是，有一天，波恩突然灵光一闪：他终于想起来这是什么了。海森堡的表格，正是他从前所听说过的那个“矩阵”！
但是对于当时的欧洲物理学家来说，矩阵几乎是一个完全陌生的名字。甚至连海森堡自己，也不见得对它的性质有着完全的了解。波恩决定为海森堡的理论打一个坚实的数学基础，他找到泡利，希望与之合作，可是泡利对此持有强烈的怀疑态度，他以他标志性的尖刻语气对波恩说：“是的，我就知道你喜欢那种冗长和复杂的形式主义，但你那无用的数学只会损害海森堡的物理思想。”波恩在泡利那里碰了一鼻子灰，不得不转向他那熟悉矩阵运算的年轻助教约尔当（Pascual
Jordan,再过一个礼拜，就是他101年诞辰），两人于是欣然合作，很快写出了著名的论文《论量子力学》（Zur
Quantenmechanik），发表在《物理学杂志》上。在这篇论文中，两人用了很大的篇幅来阐明矩阵运算的基本规则，并把经典力学的哈密顿变换统统改造成为矩阵的形式。传统的动量p和位置q这两个物理变量，现在成为了两个含有无限数据的庞大表格，而且，正如我们已经看到的那样，它们并不遵守传统的乘法交换率，p×q
≠ q×p。
波恩和约尔当甚至把p×q和q×p之间的差值也算了出来，结果是这样的：
pq – qp = (h/2πi) I
h是我们已经熟悉的普朗克常数，i是虚数的单位，代表-1的平方根，而I叫做单位矩阵，相当于矩阵运算中的1。波恩和约尔当奠定了一种新的力学——矩阵力学的基础。在这种新力学体系的魔法下，普朗克常数和量子化从我们的基本力学方程中自然而然地跳了出来，成为自然界的内在禀性。如果认真地对这种力学形式做一下探讨，人们会惊奇地发现，牛顿体系里的种种结论，比如能量守恒，从新理论中也可以得到。这就是说，新力学其实是牛顿理论的一个扩展，老的经典力学其实被“包含”在我们的新力学中，成为一种特殊情况下的表现形式。
这种新的力学很快就得到进一步完善。从剑桥返回哥廷根后，海森堡本人也加入了这个伟大的开创性工作中。11月26日，《论量子力学II》在《物理学杂志》上发表，作者是波恩，海森堡和约尔当。这篇论文把原来只讨论一个自由度的体系扩展到任意个自由度，从而彻底建立了新力学的主体。现在，他们可以自豪地宣称，长期以来人们所苦苦追寻的那个目标终于达到了，多年以来如此困扰着物理学家的原子光谱问题，现在终于可以在新力学内部完美地解决。《论量子力学II》这篇文章，被海森堡本人亲切地称呼为“三人论文”（Dreimannerarbeit）的，也终于注定要在物理史上流芳百世。
新体系显然在理论上获得了巨大的成功。泡利很快就改变了他的态度，在写给克罗尼格（Ralph Laer Kronig）的信里，他说：“海森堡的力学让我有了新的热情和希望。”随后他很快就给出了极其有说服力的证明，展示新理论的结果和氢分子的光谱符合得非常完美，从量子规则中，巴尔末公式可以被自然而然地推导出来。非常好笑的是，虽然他不久前还对波恩咆哮说“冗长和复杂的形式主义”，但他自己的证明无疑动用了最最复杂的数学。
不过，对于当时其他的物理学家来说，海森堡的新体系无疑是一个怪物。矩阵这种冷冰冰的东西实在太不讲情面，不给人以任何想象的空间。人们一再追问，这里面的物理意义是什么？矩阵究竟是个什么东西？海森堡却始终护定他那让人沮丧的立场：所谓“意义”是不存在的，如果有的话，那数学就是一切“意义”所在。物理学是什么？就是从实验观测量出发，并以庞大复杂的数学关系将它们联系起来的一门科学，如果说有什么图像能够让人们容易理解和记忆的话，那也是靠不住的。但是，不管怎么样，毕竟矩阵力学对于大部分人来说都太陌生太遥远了，而隐藏在它背后的深刻含义，当时还远远没有被发掘出来。特别是，p×q ≠ q×p，这究竟代表了什么，令人头痛不已。
一年后，当薛定谔以人们所喜闻乐见的传统方式发布他的波动方程后，几乎全世界的物理学家都松了一口气：他们终于解脱了，不必再费劲地学习海森堡那异常复杂和繁难的矩阵力学。当然，人人都必须承认，矩阵力学本身的伟大含义是不容怀疑的。
但是，如果说在1925年，欧洲大部分物理学家都还对海森堡，波恩和约尔当的力学一知半解的话，那我们也不得不说，其中有一个非常显著的例外，他就是保罗?狄拉克。在量子力学大发展的年代，哥本哈根，哥廷根以及慕尼黑三地抢尽了风头，狄拉克的崛起总算也为老牌的剑桥挽回了一点颜面。
保罗?埃德里安?莫里斯?狄拉克（Paul Adrien Maurice Dirac）于1902年8月8日出生于英国布里斯托尔港。他的父亲是瑞士人，当时是一位法语教师，狄拉克是家里的第二个孩子。许多大物理学家的童年教育都是多姿多彩的，比如玻尔，海森堡，还有薛定谔。但狄拉克的童年显然要悲惨许多，他父亲是一位非常严肃而刻板的人，给保罗制定了众多的严格规矩。比如他规定保罗只能和他讲法语（他认为这样才能学好这种语言），于是当保罗无法表达自己的时候，只好选择沉默。在小狄拉克的童年里，音乐、文学、艺术显然都和他无缘，社交活动也几乎没有。这一切把狄拉克塑造成了一个沉默寡言，喜好孤独，淡泊名利，在许多人眼里显得geeky的人。有一个流传很广的关于狄拉克的笑话是这样说的：有一次狄拉克在某大学演讲，讲完后一个观众起来说：“狄拉克教授，我不明白你那个公式是如何推导出来的。”狄拉克看着他久久地不说话，主持人不得不提醒他，他还没有回答问题。
“回答什么问题？”狄拉克奇怪地说，“他刚刚说的是一个陈述句，不是一个疑问句。”
1921年，狄拉克从布里斯托尔大学电机工程系毕业，恰逢经济大萧条，结果没法找到工作。事实上，很难说他是否会成为一个出色的工程师，狄拉克显然长于理论而拙于实验。不过幸运的是，布里斯托尔大学数学系又给了他一个免费进修数学的机会，2年后，狄拉克转到剑桥，开始了人生的新篇章。
我们在上面说到，1925年秋天，当海森堡在赫尔格兰岛作出了他的突破后，他获得波恩的批准来到剑桥讲学。当时海森堡对自己的发现心中还没有底，所以没有在公开场合提到自己这方面的工作，不过7月28号，他参加了所谓“卡皮察俱乐部”的一次活动。卡皮察（PLKapitsa）是一位年轻的苏联学生，当时在剑桥跟随卢瑟福工作。他感到英国的学术活动太刻板，便自己组织了一个俱乐部，在晚上聚会，报告和讨论有关物理学的最新进展。我们在前面讨论卢瑟福的时候提到过卡皮察的名字，他后来也获得了诺贝尔奖。
狄拉克也是卡皮察俱乐部的成员之一，他当时不在剑桥，所以没有参加这个聚会。不过他的导师福勒（William Alfred Fowler）参加了，而且大概在和海森堡的课后讨论中，得知他已经发明了一种全新的理论来解释原子光谱问题。后来海森堡把他的证明寄给了福勒，而福勒给了狄拉克一个复印本。这一开始没有引起狄拉克的重视，不过大概一个礼拜后，他重新审视海森堡的论文，这下他把握住了其中的精髓：别的都是细枝末节，只有一件事是重要的，那就是我们那奇怪的矩阵乘法规则：p×q
≠ q×p。
***********
饭后闲话：约尔当
恩斯特?帕斯库尔?约尔当（Ernst Pascual Jordan）出生于汉诺威。在我们的史话里已经提到，他是物理史上两篇重要的论文《论量子力学》I和II的作者之一，可以说也是量子力学的主要创立者。但是，他的名声显然及不上波恩或者海森堡。
这里面的原因显然也是多方面的，1925年，约尔当才22岁，无论从资格还是名声来说，都远远及不上元老级的波恩和少年成名的海森堡。当时和他一起做出贡献的那些人，后来都变得如此著名：波恩，海森堡，泡利，他们的光辉耀眼，把约尔当完全给盖住了。
从约尔当本人来说，他是一个害羞和内向的人，说话有口吃的毛病，总是结结巴巴的，所以他很少授课或发表演讲。更严重的是，约尔当在二战期间站到了希特勒的一边，成为一个纳粹的同情者，被指责曾经告密。这大大损害了他的声名。
约尔当是一个作出了许多伟大成就的科学家。除了创立了基本的矩阵力学形式，为量子论打下基础之外，他同样在量子场论，电子自旋，量子电动力学中作出了巨大的贡献。他是最先证明海森堡和薛定谔体系同等性的人之一，他发明了约尔当代数，后来又广泛涉足生物学、心理学和运动学。他曾被提名为诺贝尔奖得主，却没有成功。约尔当后来显然也对自己的成就被低估有些恼火，1964年，他声称《论量子力学》一文其实几乎都是他一个人的贡献——波恩那时候病了。这引起了广泛的争议，不过许多人显然同意，约尔当的贡献应当得到更多的承认。
Fives
p×q ≠ q×p。如果说狄拉克比别人天才在什么地方，那就是他可以一眼就看出这才是海森堡体系的精髓。那个时候，波恩和约尔当还在苦苦地钻研讨厌的矩阵，为了建立起新的物理大厦而努力地搬运着这种庞大而又沉重的表格式方砖，而他们的文章尚未发表。但狄拉克是不想做这种苦力的，他轻易地透过海森堡的表格，把握住了这种代数的实质。不遵守交换率，这让我想起了什么？狄拉克的脑海里闪过一个名词，他以前在上某一门动力学课的时候，似乎听说过一种运算，同样不符合乘法交换率。但他还不是十分确定，他甚至连那种运算的定义都给忘了。那天是星期天，所有的图书馆都关门了，这让狄拉克急得像热锅上的蚂蚁。第二天一早，图书馆刚刚开门，他就冲了进去，果然，那正是他所要的东西：它的名字叫做“泊松括号”。
我们还在第一章讨论光和菲涅尔的时候，就谈到过泊松，还有著名的泊松光斑。泊松括号也是这位法国科学家的杰出贡献，不过我们在这里没有必要深入它的数学意义。总之，狄拉克发现，我们不必花九牛二虎之力去搬弄一个晦涩的矩阵，以此来显示和经典体系的决裂。我们完全可以从经典的泊松括号出发，建立一种新的代数。这种代数同样不符合乘法交换率，狄拉克把它称作“q数”（q表示“奇异”或者“量子”）。我们的动量、位置、能量、时间等等概念，现在都要改造成这种q数。而原来那些老体系里的符合交换率的变量，狄拉克把它们称作“c数”（c代表“普通”）。
“看。”狄拉克说，“海森堡的最后方程当然是对的，但我们不用他那种大惊小怪，牵强附会的方式，也能够得出同样的结果。用我的方式，同样能得出xy-yx的差值，只不过把那个让人看了生厌的矩阵换成我们的经典泊松括号[x,y]罢了。然后把它用于经典力学的哈密顿函数，我们可以顺理成章地导出能量守恒条件和玻尔的频率条件。重要的是，这清楚地表明了，我们的新力学和经典力学是一脉相承的，是旧体系的一个扩展。c数和q数，可以以清楚的方式建立起联系来。”
狄拉克把论文寄给海森堡，海森堡热情地赞扬了他的成就，不过带给狄拉克一个糟糕的消息：他的结果已经在德国由波恩和约尔当作出了，是通过矩阵的方式得到的。想来狄拉克一定为此感到很郁闷，因为显然他的法子更简洁明晰。随后狄拉克又出色地证明了新力学和氢分子实验数据的吻合，他又一次郁闷了——泡利比他快了一点点，五天而已。哥廷根的这帮家伙，海森堡，波恩，约尔当，泡利，他们是大军团联合作战，而狄拉克在剑桥则是孤军奋斗，因为在英国懂得量子力学的人简直屈指可数。但是，虽然狄拉克慢了那么一点，但每一次他的理论都显得更为简洁、优美、深刻。而且，上天很快会给他新的机会，让他的名字在历史上取得不逊于海森堡、波恩等人的地位。
现在，在旧的经典体系的废墟上，矗立起了一种新的力学，由海森堡为它奠基，波恩，约尔当用矩阵那实心的砖块为它建造了坚固的主体，而狄拉克的优美的q数为它做了最好的装饰。现在，唯一缺少的就是一个成功的广告和落成典礼，把那些还在旧废墟上唉声叹气的人们都吸引到新大厦里来定居。这个庆典在海森堡取得突破后3个月便召开了，它的主题叫做“电子自旋”。
我们还记得那让人头痛的“反常塞曼效应”，这种复杂现象要求引进1/2的量子数。为此，泡利在1925年初提出了他那著名的“不相容原理”的假设，我们前面已经讨论过，这个规定是说，在原子大厦里，每一间房间都有一个4位数的门牌号码，而每间房只能入住一个电子。所以任何两个电子也不能共享同一组号码。
这个“4位数的号码”，其每一位都代表了电子的一个量子数。当时人们已经知道电子有3个量子数，这第四个是什么，便成了众说纷纭的谜题。不相容原理提出后不久，当时在哥本哈根访问的克罗尼格（Ralph Kronig）想到了一种可能：就是把这第四个自由度看成电子绕着自己的轴旋转。他找到海森堡和泡利，提出了这一思路，结果遭到两个德国年轻人的一致反对。因为这样就又回到了一种图像化的电子概念那里，把电子想象成一个实实在在的小球，而违背了我们从观察和数学出发的本意了。如果电子真是这样一个带电小球的话，在麦克斯韦体系里是不稳定的，再说也违反相对论——它的表面旋转速度要高于光速。
到了1925年秋天，自旋的假设又在荷兰莱顿大学的两个学生，乌仑贝克（George Eugene Uhlenbeck）和古德施密特（Somul Abraham Goudsmit）那里死灰复燃了。当然，两人不知道克罗尼格曾经有过这样的意见，他们是在研究光谱的时候独立产生这一想法的。于是两人找到导师埃仑费斯特（Paul Ehrenfest）征求意见。埃仑费斯特也不是很确定，他建议两人先写一个小文章发表。于是两人当真写了一个短文交给埃仑费斯特，然后又去求教于老资格的洛仑兹。洛仑兹帮他们算了算，结果在这个模型里电子表面的速度达到了光速的10倍。两人大吃一惊，风急火燎地赶回大学要求撤销那篇短文，结果还是晚了，埃仑费斯特早就给Nature杂志寄了出去。据说，两人当时懊恼得都快哭了，埃仑费斯特只好安慰他们说：“你们还年轻，做点蠢事也没关系。”
还好，事情并没有想象的那么糟糕。玻尔首先对此表示赞同，海森堡用新的理论去算了算结果后，也转变了反对的态度。到了1926年，海森堡已经在说：“如果没有古德施密特，我们真不知该如何处理塞曼效应。”一些技术上的问题也很快被解决了，比如有一个系数2，一直和理论所抵触，结果在玻尔研究所访问的美国物理学家托马斯发现原来人们都犯了一个计算错误，而自旋模型是正确的。很快海森堡和约尔当用矩阵力学处理了自旋，结果大获全胜，很快没有人怀疑自旋的正确性了。
哦，不过有一个例外，就是泡利，他一直对自旋深恶痛绝。在他看来，原本电子已经在数学当中被表达得很充分了——现在可好，什么形状、轨道、大小、旋转……种种经验性的概念又幽灵般地回来了。原子系统比任何时候都像个太阳系，本来只有公转，现在连自转都有了。他始终按照自己的路子走，决不向任何力学模型低头。事实上，在某种意义上泡利是对的，电子的自旋并不能想象成传统行星的那种自转，它具有1/2的量子数，也就是说，它要转两圈才露出同一个面孔，这里面的意义只能由数学来把握。后来泡利真的从特定的矩阵出发，推出了这一性质，而一切又被伟大的狄拉克于1928年统统包含于他那相对论化了的量子体系中，成为电子内禀的自然属性。
但是，无论如何，1926年海森堡和约尔当的成功不仅是电子自旋模型的胜利，更是新生的矩阵力学的胜利。不久海森堡又天才般地指出了解决有着两个电子的原子——氦原子的道路，使得新体系的威力再次超越了玻尔的老系统，把它的疆域扩大到以前未知的领域中。已经在迷雾和荆棘中彷徨了好几年的物理学家们这次终于可以扬眉吐气，把长久郁积的坏心情一扫而空，好好地呼吸一下那新鲜的空气。
但是，人们还没有来得及歇一歇脚，欣赏一下周围的风景，为目前的成就自豪一下，我们的快艇便又要前进了。物理学正处在激流之中，它飞流直下，一泻千里，带给人晕眩的速度和刺激。自牛顿起250年来，科学从没有在哪个时期可以像如今这般翻天覆地，健步如飞。量子的力量现在已经完全苏醒了，在接下来的3年间，它将改变物理学的一切，在人类的智慧中刻下最深的烙印，并影响整个20世纪的面貌。
当乌仑贝克和古德施密特提出自旋的时候，玻尔正在去往莱登（Leiden）的路上。当他的火车到达汉堡的时候，他发现泡利和斯特恩（Stern）站在站台上，只是想问问他关于自旋的看法，玻尔不大相信，但称这很有趣。到达莱登以后，他又碰到了爱因斯坦和埃仑费斯特，爱因斯坦详细地分析了这个理论，于是玻尔改变了看法。在回去的路上，玻尔先经过哥廷根，海森堡和约尔当站在站台上。同样的问题：怎么看待自旋？最后，当玻尔的火车抵达柏林，泡利又站在了站台上——他从汉堡一路赶到柏林，想听听玻尔一路上有了什么看法的变化。
人们后来回忆起那个年代，简直像是在讲述一个童话。物理学家们一个个都被洪流冲击得站不住脚：节奏快得几乎不给人喘息的机会，爆炸性的概念一再地被提出，每一个都足以改变整个科学的面貌。但是，每一个人都感到深深的骄傲和自豪，在理论物理的黄金年代，能够扮演历史舞台上的那一个角色。人们常说，时势造英雄，在量子物理的大发展时代，英雄们的确留下了最最伟大的业绩，永远让后人心神向往。
回到我们的史话中来。现在，花开两朵，各表一支。我们去看看量子论是如何沿着另一条完全不同的思路，取得同样伟大的突破的。