Chapter 6 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 obviously had an obvious affection for this outstanding young man, also wrote to tell him that he had won a prize of one thousand dollars from the International Education Foundation (IEB) funded by the Rockefeller consortium, thus It 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 ten visiting scholars, six of whom were funded by IEB, and the number soon began to surge, making the three-story building It soon became crowded and had to be expanded.Heisenberg arrived in Copenhagen on September 17, 1924 after finishing his summer trip. He and Dr. King from the United States lived in the home of a professor who had just passed away. Their diet and daily life are taken care of by the widowed wife.For Heisenberg, the place was more like a language school, and his poor English and Danish skills 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 ground-based Maxwell theory on which the latter relies 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 corresponding principle, so that after quantization, they still retain It has all the advantages of classical wave theory (actually, it wants to further consider 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 just a statistical method. the average situation.This price was too high, which was strongly opposed by Einstein. Under his influence, Pauli also quickly changed his attitude. 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 negated 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 it was really a 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, Copenhagen's free spirit and academic atmosphere are almost unmatched in Europe, and all of this and Niels.Bohr, the godfather of quantum theory, is closely related.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 electron's orbit 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. From this, the basic principle of the quantum system should be found, right?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.Something is wrong with physics, he exclaimed, everything is too hard for me, I'd rather be a movie comedian and never hear what physics is! (As an aside, Pauli said that he would rather be a comedian because he is 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 of Matrix that people have never heard of, and they dare not even imagine before. 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.Science students cannot escape the linear algebra class, engineers cannot do without MatLab software, and beautiful girls often miss Keno.Reeves, there is no way.

Translated in a mathematical sense, Matrix is ​​translated as a matrix in Chinese, which is essentially a two-dimensional table.For example, the following 2*2 matrix is ​​actually a 2*2 square table: 1 2 3 4 It can also be rectangular, such as this 2*3 matrix: 1 2 3 4 5 6 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 of 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 image of the orbit is familiar to people and can be compared to the orbit of a planet, but unlike planets, is there any way for people to actually see such an orbit of an electron and actually measure the energy represented by an orbit?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 from the quantum formula E1-E2=hν, we know that this is the result of the transition of electrons between two energy levels.But, argues Heisenberg, you still haven't solved my doubts.There is no actual observation to prove what energy level a certain orbital represents, and each spectral line only represents the energy difference between two energy levels.Therefore, only the energy level difference or the orbital difference can be directly observed, but the energy level and the 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 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, no matter how far it is, the price is the same, and the fare is no longer 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 fee for this car follows the old tradition of our nostalgic era. Instead of paying two 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 one yuan, and the distance between station B and C is 0.5 yuan. There is still one dollar between C and D, while D and E are far away, two dollars.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 pay 9.512 = 3.5 yuan as the fare.Conversely, if I get on the train from station D to station A, the reason is the same: 10.51 = 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 thus deduced: 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 starting station minus the coordinates of the final station, our coordinates can actually be regarded as a kind of fare energy level, and all situations can be fully included in the following In the form: Website coordinates (fare level) A 0 B1 C 1.5 D 2.5 E 4.5 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 unaware, 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 coordinates of station C are 1.5, or that the coordinates of station D are 2.5.As a passenger, the only thing I can observe and experience is that it costs one yuan to get from station C to station D, which is the most reliable and solid thing.Our fare rules can only be based on such facts, rather than 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: ABCDE A 011.52.54.5 B 100.51.53.5 C 1.50.5013 D 2.51.5102 E 4.53.5320 Here, the vertical one is the starting station, and the horizontal one is the terminal station.Now every digit in this table is actually observable and verifiable.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 certain 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 the money needed to go from A to B is the same as from B to A.This may not be the case. It costs one yuan to go from A to B, but it is likely to cost one and a half yuan to go 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 digit where B is the abscissa and A is 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 sounded at the same time, it doesn't matter if they are mixed together.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: Line I (Matrix I): AB A 12 B 31 Line II (Matrix II): AB A 13 B 41 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 one in the first row and first column of matrix I means that if you take bus line I, depart from point A, and get off at point A, the fare will cost one yuan (ah? Why do you have to pay one yuan if you stay still? What about money? On the one hand, this is just a metaphor. Besides, you can think of one yuan as a starting fee. Besides, in most cities’ subways, if you go in and get out immediately, you will indeed have to deduct a little money from the electronic card. ).Similarly, the 2 in the first row and second column of the matrix I means that you need two 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 number three 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 an ordinary number, 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? 1 2 1 3 I: X II: =? 3 1 4 1 After-dinner gossip: Physics for boys In 1925, when Heisenberg made his breakthrough contribution, he was just twenty-four 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 that I didn'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 twenty-eight years old when he proposed his atomic structure in 1913.When De Broglie proposed Xiangbo in 1923, he was thirty-one 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 twenty-five years old, Di Lacker was twenty-three, Uhlenbeck was twenty-five, Gudschmidt was twenty-three, Jordang was twenty-three.Compared with them, the thirty-six-year-old Schrödinger and the forty-three-year-old Bonn are indeed 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. 1 3 1 3 X=? 3 1 4 1 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. 1 3 1 3 a b X = 3 1 4 1 c d 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 represents the product of the toll for Line I and the toll for Line II.However, the situation is not so 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 of matrix I is A and whose ordinate is B, that is, the two or two dollars in the second column of the first row.Well, then we transferred from point B to line II and returned to point A. The fare here corresponds to the four 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 also meets the conditions for us to 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 one piece (first row, first column of matrix I) and three pieces (first row, second column of matrix II), so one×three=three.Another way is to take line I to go from A to B, charge two yuan (the first row and second column of matrix I), and then transfer to line II at the same place at B, and charge one yuan (the second column of matrix II) Row 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: 1 2 1 3 9 5 X = 3 1 4 1 7 10 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: 1 3 1 3 a b X = 3 1 4 1 c d 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.Corresponding to one × one.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), one block.Then turn to Line I and get on the bus at A and get off at A (the first row and first column of matrix I), which is also one block. 1×1=1.There is also a possibility that I take Line II and get on the bus at A and get off at B (the first row and second column of matrix II), three yuan.Then turn to Line I at B and return to A from B (the second row and first column of matrix II), three 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 an absurd result: I×II≠II×I. Let's calculate the result in its entirety: 1 2 1 3 9 5 X = 3 1 4 1 7 10 1 2 1 3 10 5 X = 3 1 4 1 7 9 Indeed, I×II≠II×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 mathematics says I×II≠II×I, then we have to think so, even if the world ridicules us in a 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 do no more verbal quarrels, and in the summer of 1925 he was caught by a fever and had to leave Göttingen for 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 three years, unimaginable three years in the history of physics. The golden age of theoretical physics will finally radiate its most dazzling brilliance, adorning the entire twentieth century as 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. 是的，曙光已經出現，太陽正從海平線上冉冉升起，萬道霞光染紅了海面和空中的雲彩，在天地間流動著奇幻的輝光。在高高的石崖頂上，海森堡面對著壯觀的日出景象，他腳下碧海潮生，一直延伸到無窮無盡的遠方。是的，他知道，this is the moment，他已經作出生命中最重要的突破，而物理學的黎明也終於到來。 飯後閒話：矩陣 我們已經看到，海森堡發明了這種奇特的表格，I×II≠II×I，連他自己都沒把握確定這是個什麼怪物。當他結束養病，回到哥廷根後，就把論文草稿送給老師波恩，讓他評論評論。波恩看到這種表格運算大吃一驚，原來這不是什麼新鮮東西，正是線性代數裡學到的矩陣！回溯歷史，這種工具早在一八五八年就已經由一位劍橋的數學家Arthur Cayley所發明，不過當時不叫矩陣而叫做行列式（determinant，這個字後來變成了另外一個意思，雖然還是和矩陣關係很緊密）。發明矩陣最初的目的，是簡潔地來求解某些微分方程組（事實上直到今天，大學線性代數課還是主要解決這個問題）。但海森堡對此毫不知情，他實際上不知不覺地重新發明了矩陣的概念。波恩和他那精通矩陣運算的助教約爾當隨即在嚴格的數學基礎上發展了海森堡的理論，進一步完善了量子力學，我們很快就要談到。 數學在某種意義上來說總是領先的。Cayley創立矩陣的時候，自然想不到它後來會在量子論的發展中起到關鍵作用。同樣，黎曼創立黎曼幾何的時候，又怎會料到他已經給愛因斯坦和他偉大的相對論提供了最好的工具。 George.蓋莫夫在那本受歡迎的老科普書《從一到無窮大》（One, Two, ThreeInfinity）裡說，目前數學還有一個大分支沒有派上用場（除了智力體操的用處之外），那就是數論。古老的數論領域裡已經有許多難題被解開，比如四色問題，費馬大定理。也有比如著名的哥德巴赫猜想，至今懸而未決。天知道，這些理論和思路是不是在將來會給某個物理或者化學理論開道，打造出一片全新的天地來。 Four 從赫爾格蘭回來後，海森堡找到波恩，請求允許他離開哥廷根一陣，去劍橋講課。同時，他也把自己的論文給了波恩過目，問他有沒有發表的價值。波恩顯然被海森堡的想法給迷住了，正如他後來回憶的那樣：我對此著了迷海森堡的思想給我留下了深刻的印象，對於我們一直追求的那個體系來說，這是一次偉大的突破。於是當海森堡去到英國講 學的時候，波恩就把他的這篇論文寄給了《物理學雜誌》（Zeitschrift fur Physik），並於七月二十九日發表。這無疑標誌著新生的量子力學在公眾面前的首次亮相。 但海森堡古怪的表格乘法無疑也讓波恩困擾，他在七月十五日寫給愛因斯坦的信中說：海森堡新的工作看起來有點神秘莫測，不過無疑是很深刻的，而且是正確的。但是，有一天，波恩突然靈光一閃：他終於想起來這是什麼了。海森堡的表格，正是他從前所聽說過的那個矩陣！ 但是對於當時的歐洲物理學家來說，矩陣幾乎是一個完全陌生的名字。甚至連海森堡自己，也不見得對它的性質有著完全的瞭解。波恩決定為海森堡的理論打一個堅實的數學基礎，他找到泡利，希望與之合作，可是泡利對此持有強烈的懷疑態度，他以他標誌性的尖刻語氣對波恩說：是的，我就知道你喜歡那種冗長和複雜的形式主義，但你那無用的數學只會損害海森堡的物理思想。波恩在泡利那裡碰了一鼻子灰，不得不轉向他那熟悉矩陣運算的年輕助教約爾當（Pascual Jordan,再過一個禮拜，就是他一百零一年誕辰），兩人於是欣然合作，很快寫出了著名的論文《論量子力學》（Zur Quantenmechanik），發表在《物理學雜誌》上。在這篇論文中，兩人用了很大的篇幅來闡明矩陣運算的基本規則，並把經典力學的哈密頓變換統統改造成為矩陣的形式。傳統的動量p和位置q這兩個物理變數，現在成為了兩個含有無限資料的龐大表格，而且，正如我們已經看到的那樣，它們並不遵守傳統的乘法交換率，p×q≠q×p。 波恩和約爾當甚至把p×q和q×p之間的差值也算了出來，結果是這樣的： pq -qp ＝（h／2πi）I h是我們已經熟悉的普朗克常數，i是虛數的單位，代表-一的平方根，而I叫做單位矩陣，相當於矩陣運算中的一。波恩和約爾當奠定了一種新的力學矩陣力學的基礎。在這種新力學體系的魔法下，普朗克常數和量子化從我們的基本力學方程中自然而然地跳了出來，成為自然界的內在稟性。如果認真地對這種力學形式做一下探討，人們會驚奇地發現，牛頓體系裡的種種結論，比如能量守恆，從新理論中也可以得到。這就是說，新力學其實是牛頓理論的一個擴展，老的經典力學其實被包含在我們的新力學中，成為一種特殊情況下的表現形式。 這種新的力學很快就得到進一步完善。從劍橋返回哥廷根後，海森堡本人也加入了這個偉大的開創性工作中。十一月二十六日，《論量子力學II》在《物理學雜誌》上發表，作者是波恩，海森堡和約爾當。這篇論文把原來只討論一個自由度的體系擴展到任意個自由度，從而徹底建立了新力學的主體。現在，他們可以自豪地宣稱，長期以來人們所苦苦追尋的那個目標終於達到了，多年以來如此困擾著物理學家的原子光譜問題，現在終於可以在新力學內部完美地解決。《論量子力學II》這篇文章，被海森堡本人親切地稱呼為三人論文（Dreimannerarbeit）的，也終於註定要在物理史上流芳百世。 新體系顯然在理論上獲得了巨大的成功。泡利很快就改變了他的態度，在寫給克羅尼格（Ralph Laer Kronig）的信裡，他說：海森堡的力學讓我有了新的熱情和希望。隨後他很快就給出了極其有說服力的證明，展示新理論的結果和氫分子的光譜符合得非常完美，從量子規則中，巴爾末公式可以被自然而然地推導出來。非常好笑的是，雖然他不久前還對波恩咆哮說冗長和複雜的形式主義，但他自己的證明無疑動用了最最複雜的數學。 不過，對於當時其他的物理學家來說，海森堡的新體系無疑是一個怪物。矩陣這種冷冰冰的東西實在太不講情面，不給人以任何想像的空間。人們一再追問，這裡面的物理意義是什麼？矩陣究竟是個什麼東西？海森堡卻始終護定他那讓人沮喪的立場：所謂意義是不存在的，如果有的話，那數學就是一切意義所在。物理學是什麼？就是從實驗觀測量出發，並以龐大複雜的數學關係將它們聯繫起來的一門科學，如果說有什麼圖像能夠讓人們容易理解和記憶的話，那也是靠不住的。但是，不管怎麼樣，畢竟矩陣力學對於大部分人來說都太陌生太遙遠了，而隱藏在它背後的深刻含義，當時還遠遠沒有被發掘出來。特別是，p×q≠q×p，這究竟代表了什麼，令人頭痛不已。 一年後，當薛定諤以人們所喜聞樂見的傳統方式發佈他的波動方程後，幾乎全世界的物理學家都鬆了一口氣：他們終於解脫了，不必再費勁地學習海森堡那異常複雜和繁難的矩陣力學。當然，人人都必須承認，矩陣力學本身的偉大含義是不容懷疑的。 但是，如果說在一九二五年，歐洲大部分物理學家都還對海森堡，波恩和約爾當的力學一知半解的話，那我們也不得不說，其中有一個非常顯著的例外，他就是保羅．狄拉克。在量子力學大發展的年代，哥本哈根，哥廷根以及慕尼克三地搶盡了風頭，狄拉克的崛起總算也為老牌的劍橋挽回了一點顏面。 Paul.埃德里安．Morris.狄拉克（Paul Adrien Maurice Dirac）於一九○二年八月八日出生於英國布里斯托爾港。他的父親是瑞士人，當時是一位法語教師，狄拉克是家裡的第二個孩子。許多大物理學家的童年教育都是多姿多彩的，比如玻爾，海森堡，還有薛定諤。但狄拉克的童年顯然要悲慘許多，他父親是一位非常嚴肅而刻板的人，給保羅制定了眾多的嚴格規矩。比如他規定保羅只能和他講法語（他認為這樣才能學好這種語言），於是當保羅無法表達自己的時候，只好選擇沉默。在小狄拉克的童年裡，音樂、文學、藝術顯然都和他無緣，社交活動也幾乎沒有。這一切把狄拉克塑造成了一個沉默寡言，喜好孤獨，淡泊名利，在許多人眼裡顯得geeky的人。有一個流傳很廣的關於狄拉克的笑話是這樣說的：有一次狄拉克在某大學演講，講完後一個觀眾起來說：狄拉克教授，我不明白你那個公式是如何推導出來的。狄拉克看著他久久地不說話，主持人不得不提醒他，他還沒有回答問題。 回答什麼問題？狄拉克奇怪地說，他剛剛說的是一個陳述句，不是一個疑問句。 一九二一年，狄拉克從布里斯托爾大學電機工程系畢業，恰逢經濟大蕭條，結果沒法找到工作。事實上，很難說他是否會成為一個出色的工程師，狄拉克顯然長於理論而拙於實驗。不過幸運的是，布里斯托爾大學數學系又給了他一個免費進修數學的機會，二年後，狄拉克轉到劍橋，開始了人生的新篇章。 我們在上面說到，一九二五年秋天，當海森堡在赫爾格蘭島作出了他的突破後，他獲得波恩的批准來到劍橋講學。當時海森堡對自己的發現心中還沒有底，所以沒有在公開場合提到自己這方面的工作，不過七月二十八號，他參加了所謂卡皮察俱樂部的一次活動。卡皮察（PL Kapitsa）是一位年輕的蘇聯學生，當時在劍橋跟隨盧瑟福工作。他感到英國的學術活動太刻板，便自己組織了一個俱樂部，在晚上聚會，報告和討論有關物理學的最新進展。我們在前面討論盧瑟福的時候提到過卡皮察的名字，他後來也獲得了諾貝爾獎。 狄拉克也是卡皮察俱樂部的成員之一，他當時不在劍橋，所以沒有參加這個聚會。不過他的導師福勒（William Alfred Fowler）參加了，而且大概在和海森堡的課後討論中，得知他已經發明了一種全新的理論來解釋原子光譜問題。後來海森堡把他的證明寄給了福勒，而福勒給了狄拉克一個複印本。這一開始沒有引起狄拉克的重視，不過大概一個禮拜後，他重新審視海森堡的論文，這下他把握住了其中的精髓：別的都是細枝末節，只有一件事是重要的，那就是我們那奇怪的矩陣乘法規則：p×q≠q×p。 飯後閒話：約爾當 Ernst.帕斯庫爾．約爾當（Ernst Pascual Jordan）出生於漢諾威。在我們的史話裡已經提到，他是物理史上兩篇重要的論文《論量子力學》I和II的作者之一，可以說也是量子力學的主要創立者。但是，他的名聲顯然及不上波恩或者海森堡。 這裡面的原因顯然也是多方面的，一九二五年，約爾當才二十二歲，無論從資格還是名聲來說，都遠遠及不上元老級的波恩和少年成名的海森堡。當時和他一起做出貢獻的那些人，後來都變得如此著名：波恩，海森堡，泡利，他們的光輝耀眼，把約爾當完全給蓋住了。 從約爾當本人來說，他是一個害羞和內向的人，說話有口吃的毛病，總是結結巴巴的，所以他很少授課或發表演講。更嚴重的是，約爾當在二戰期間站到了希特勒的一邊，成為一個納粹的同情者，被指責曾經告密。這大大損害了他的聲名。 約爾當是一個作出了許多偉大成就的科學家。除了創立了基本的矩陣力學形式，為量子論打下基礎之外，他同樣在量子場論，電子自旋，量子電動力學中作出了巨大的貢獻。他是最先證明海森堡和薛定諤體系同等性的人之一，他發明了約爾當代數，後來又廣泛涉足生物學、心理學和運動學。他曾被提名為諾貝爾獎得主，卻沒有成功。約爾當後來顯然也對自己的成就被低估有些惱火，一九六四年，他聲稱《論量子力學》一文其實幾乎都是他一個人的貢獻波恩那時候病了。這引起了廣泛的爭議，不過許多人顯然同意，約爾當的貢獻應當得到更多的承認。 five p×q≠q×p。如果說狄拉克比別人天才在什麼地方，那就是他可以一眼就看出這才是海森堡體系的精髓。那個時候，波恩和約爾當還在苦苦地鑽研討厭的矩陣，為了建立起新的物理大廈而努力地搬運著這種龐大而又沉重的表格式方磚，而他們的文章尚未發表。但狄拉克是不想做這種苦力的，他輕易地透過海森堡的表格，把握住了這種代數的實質。不遵守交換率，這讓我想起了什麼？狄拉克的腦海裡閃過一個名詞，他以前在上某一門動力學課的時候，似乎聽說過一種運算，同樣不符合乘法交換率。但他還不是十分確定，他甚至連那種運算的定義都給忘了。那天是星期天，所有的圖書館都關門了，這讓狄拉克急得像熱鍋上的螞蟻。第二天一早，圖書館剛剛開門，他就衝了進去，果然，那正是他所要的東西：它的名字叫做泊松括弧。 我們還在第一章討論光和菲涅爾的時候，就談到過泊松，還有著名的泊松光斑。泊松括弧也是這位法國科學家的傑出貢獻，不過我們在這裡沒有必要深入它的數學意義。總之，狄拉克發現，我們不必花九牛二虎之力去搬弄一個晦澀的矩陣，以此來顯示和經典體系的決裂。我們完全可以從經典的泊松括弧出發，建立一種新的代數。這種代數同樣不符合乘法交換率，狄拉克把它稱作q數（q表示奇異或者量子）。我們的動量、位置、能量、時間等等概念，現在都要改造成這種q數。而原來那些老體系裡的符合交換率的變數，狄拉克把它們稱作c數（c代表普通）。 look.狄拉克說，海森堡的最後方程當然是對的，但我們不用他那種大驚小怪，牽強附會的方式，也能夠得出同樣的結果。用我的方式，同樣能得出xy-yx的差值，只不過把那個讓人看了生厭的矩陣換成我們的經典泊松括弧[x,y]罷了。然後把它用於經典力學的哈密頓函數，我們可以順理成章地匯出能量守恆條件和玻爾的頻率條件。重要的是，這清楚地表明瞭，我們的新力學和經典力學是一脈相承的，是舊體系的一個擴展。 c數和q數，可以以清楚的方式建立起聯繫來。 狄拉克把論文寄給海森堡，海森堡熱情地讚揚了他的成就，不過帶給狄拉克一個糟糕的消息：他的結果已經在德國由波恩和約爾當作出了，是通過矩陣的方式得到的。想來狄拉克一定為此感到很鬱悶，因為顯然他的法子更簡潔明晰。隨後狄拉克又出色地證明了新力學和氫分子實驗資料的吻合，他又一次鬱悶了泡利比他快了一點點，五天而已。哥廷根的這幫傢伙，海森堡，波恩，約爾當，泡利，他們是大軍團聯合作戰，而狄拉克在劍橋則是孤軍奮鬥，因為在英國懂得量子力學的人簡直屈指可數。但是，雖然狄拉克慢了那麼一點，但每一次他的理論都顯得更為簡潔、優美、深刻。而且，上天很快會給他新的機會，讓他的名字在歷史上取得不遜於海森堡、波恩等人的地位。 現在，在舊的經典體系的廢墟上，矗立起了一種新的力學，由海森堡為它奠基，波恩，約爾當用矩陣那實心的磚塊為它建造了堅固的主體，而狄拉克的優美的q數為它做了最好的裝飾。現在，唯一缺少的就是一個成功的廣告和落成典禮，把那些還在舊廢墟上唉聲嘆氣的人們都吸引到新大廈裡來定居。這個慶典在海森堡取得突破後三個月便召開了，它的主題叫做電子自旋。 我們還記得那讓人頭痛的反常塞曼效應，這種複雜現象要求引進1/2的量子數。為此，泡利在一九二五年初提出了他那著名的不相容原理的假設，我們前面已經討論過，這個規定是說，在原子大廈裡，每一間房間都有一個四位數的門牌號碼，而每間房只能入住一個電子。所以任何兩個電子也不能共用同一組號碼。 這個四位數的號碼，其每一位都代表了電子的一個量子數。當時人們已經知道電子有三個量子數，這第四個是什麼，便成了眾說紛紜的謎題。不相容原理提出後不久，當時在哥本哈根訪問的克羅尼格（Ralph Kronig）想到了一種可能：就是把這第四個自由度看成電子繞著自己的軸旋轉。他找到海森堡和泡利，提出了這一思路，結果遭到兩個德國年輕人的一致反對。因為這樣就又回到了一種圖像化的電子概念那裡，把電子想像成一個實實在在的小球，而違背了我們從觀察和數學出發的本意了。如果電子真是這樣一個帶電小球的話，在麥克斯韋體系裡是不穩定的，再說也違反相對論它的表面旋轉速度要高於光速。 到了一九二五年秋天，自旋的假設又在荷蘭萊頓大學的兩個學生，烏侖貝克（George Eugene Uhlenbeck）和古德施密特（Somul Abraham Goudsmit）那裡死灰復燃了。當然，兩人不知道克羅尼格曾經有過這樣的意見，他們是在研究光譜的時候獨立產生這一想法的。於是兩人找到導師埃侖費斯特（Paul Ehrenfest）徵求意見。埃侖費斯特也不是很確定，他建議兩人先寫一個小文章發表。於是兩人當真寫了一個短文交給埃侖費斯特，然後又去求教於老資格的洛侖茲。洛侖茲幫他們算了算，結果在這個模型裡電子錶面的速度達到了光速的十倍。兩人大吃一驚，風急火燎地趕回大學要求撤銷那篇短文，結果還是晚了，埃侖費斯特早就給Nature雜誌寄了出去。據說，兩人當時懊惱得都快哭了，埃侖費斯特只好安慰他們說：你們還年輕，做點蠢事也沒關係。 還好，事情並沒有想像的那麼糟糕。玻爾首先對此表示贊同，海森堡用新的理論去算了算結果後，也轉變了反對的態度。到了一九二六年，海森堡已經在說：如果沒有古德施密特，我們真不知該如何處理塞曼效應。一些技術上的問題也很快被解決了，比如有一個係數二，一直和理論所抵觸，結果在玻爾研究所訪問的美國物理學家湯瑪斯發現原來人們都犯了一個計算錯誤，而自旋模型是正確的。很快海森堡和約爾當用矩陣力學處理了自旋，結果大獲全勝，很快沒有人懷疑自旋的正確性了。 哦，不過有一個例外，就是泡利，他一直對自旋深惡痛絕。在他看來，原本電子已經在數學當中被表達得很充分了現在可好，什麼形狀、軌道、大小、旋轉種種經驗性的概念又幽靈般地回來了。原子系統比任何時候都像個太陽系，本來只有公轉，現在連自轉都有了。他始終按照自己的路子走，決不向任何力學模型低頭。事實上，在某種意義上泡利是對的，電子的自旋並不能想像成傳統行星的那種自轉，它具有一／二的量子數，也就是說，它要轉兩圈才露出同一個面孔，這裡面的意義只能由數學來把握。後來泡利真的從特定的矩陣出發，推出了這一性質，而一切又被偉大的狄拉克於一九二八年統統包含於他那相對論化了的量子體系中，成為電子內稟的自然屬性。 但是，無論如何，一九二六年海森堡和約爾當的成功不僅是電子自旋模型的勝利，更是新生的矩陣力學的勝利。不久海森堡又天才般地指出了解決有著兩個電子的原子氦原子的道路，使得新體系的威力再次超越了玻爾的老系統，把它的疆域擴大到以前未知的領域中。已經在迷霧和荊棘中彷徨了好幾年的物理學家們這次終於可以揚眉吐氣，把長久鬱積的壞心情一掃而空，好好地呼吸一下那新鮮的空氣。 但是，人們還沒有來得及歇一歇腳，欣賞一下周圍的風景，為目前的成就自豪一下，我們的快艇便又要前進了。物理學正處在激流之中，它飛流直下，一瀉千里，帶給人暈眩的速度和刺激。自牛頓起二百五十年來，科學從沒有在哪個時期可以像如今這般翻天覆地，健步如飛。量子的力量現在已經完全蘇醒了，在接下來的三年間，它將改變物理學的一切，在人類的智慧中刻下最深的烙印，並影響整個二十世紀的面貌。 當烏侖貝克和古德施密特提出自旋的時候，玻爾正在去往萊登（Leiden）的路上。當他的火車到達漢堡的時候，他發現泡利和斯特恩（Stern）站在月臺上，只是想問問他關於自旋的看法，玻爾不大相信，但稱這很有趣。到達萊登以後，他又碰到了愛因斯坦和埃侖費斯特，愛因斯坦詳細地分析了這個理論，於是玻爾改變了看法。在回去的路上，玻爾先經過哥廷根，海森堡和約爾當站在月臺上。同樣的問題：怎麼看待自旋？最後，當玻爾的火車抵達柏林，泡利又站在了月臺上他從漢堡一路趕到柏林，想聽聽玻爾一路上有了什麼看法的變化。 人們後來回憶起那個年代，簡直像是在講述一個童話。物理學家們一個個都被洪流衝擊得站不住腳：節奏快得幾乎不給人喘息的機會，爆炸性的概念一再地被提出，每一個都足以改變整個科學的面貌。但是，每一個人都感到深深的驕傲和自豪，在理論物理的黃金年代，能夠扮演歷史舞臺上的那一個角色。人們常說，時勢造英雄，在量子物理的大發展時代，英雄們的確留下了最最偉大的業績，永遠讓後人心神嚮往。 回到我們的史話中來。現在，花開兩朵，各表一支。我們去看看量子論是如何沿著另一條完全不同的思路，取得同樣偉大的突破的。