Chapter 8 Chapter 6 Unification

one When the young and energetic Heisenberg was overcoming obstacles in Göttingen, Erwin Schrodinger was already a prestigious professor at the University of Zurich in Switzerland.Of course, compared to Heisenberg, Schrödinger can only be regarded as a late bloomer.The Austrian, who was born in Vienna, did not have the same luck as Heisenberg, studying in an environment full of top elite figures, and his academic research was hindered by several war service.But in any case, Schrödinger's physical genius is still well displayed. He has made outstanding contributions in optics, electromagnetism, molecular motion theory, and the dynamics of solids and crystals. Give him a contract to be a professor of physics.From 1924, Schrödinger became interested in quantum mechanics and statistical theory, and thus turned his research direction to this.

Unlike Bohr and Heisenberg, Schrödinger didn't want to fight hard in the extremely complex maze of spectral lines of the atom, hitting his head and bleeding.His inspiration comes directly from de Broglie's ingenious work.We still remember that in 1923, de Broglie's research revealed that with every moving electron, there is always a "phase wave" that goes with it.On the one hand, whether the nature of matter is a particle or a wave has covered a more mysterious veil, but at the same time it has provided the way to the final answer. Schrödinger also learned of de Broglie's work from Einstein's papers.He wrote to Einstein on November 3, 1925: "A few days ago I read de Broglie's ingenious paper with the utmost interest and finally mastered it. I have received from you It was first understood in Section 8 of the second treatise on degenerate gases." The idea of ​​treating every particle as a wave-like was so fascinating to Schrödinger that he was soon working on the statistical mechanics of gases. applied this theory in , and published a paper entitled "On Einstein's Theory of Gases".This was his last paper before founding wave mechanics, and it was only a month before that great moment.It can be seen from this that de Broglie's thinking has gained the trust of Schrödinger to the greatest extent, and he began to believe that only through this wave method can the goal that people are looking for be achieved.

Christmas in 1925 is coming soon, and the beautiful snow-covered Alps attract tourists and vacationers from all over the world.Schrodinger, as always, came to the place he used to go to: Arosa (Arosa) at an altitude of 1,700 meters.Ever since he and Annemarie Bertel married in 1920, the two have often visited here for holidays.Schrödinger's life has almost rigid rules, and he never let anything interfere with his vacation.And every time the couple came to Arosa, they always lived in Villa Herwig, a four-story cottage with a pointed roof. But in 1925, only Schrödinger came, and Anne stayed in Zurich.Their relationship was obviously extremely tense at the time, with more than one talk of breaking up and divorce.Schrödinger wrote to an "old girlfriend" in Vienna asking her to come to Arosa to keep him company.The identity of this mysterious girl has always been a mystery. After World War II, both science historians and gossip journalists tried their best to prove her true face, but they failed.Schrödinger's diary at the time has been lost, and from the clues left behind, she is not like any of Schrödinger's known lovers.But one thing is for sure: the mysterious woman was such an inspiration to Schrödinger that he remained amazingly creative and insightful for the next 12 months, and Six major papers on quantum mechanics were published in succession.Colleagues of Schrödinger always recall that Schrödinger's great work was done during a lusty period of his life.In a way, science has a small debt to thank for this unknown woman.

Back to more serious topics.After chewing on de Broglie's ideas, Schrödinger decided to use it in the description of atomic systems.We all know that the energy of electrons in an atom is not continuous, which is fully confirmed by the discrete spectral lines of the atom.In order to describe this phenomenon, Bohr imposed a "discrete energy level" assumption, and Heisenberg used his huge matrix to derive this result after complex calculations.Now it's Schrödinger's turn, he said, it doesn't need to be so complicated, and there is no need to introduce external assumptions, as long as we regard our electrons as de Broglie waves and use a wave equation to express it, that's fine.

Schrödinger initially wanted to start from the de Broglie equation based on the theory of relativity and extend it to bound particles.For this he derived an equation, which was not very satisfactory because it did not take into account the spin of the electron.At that time, spin was just discovered, and Schrödinger still knew a little about it.So, he turned around, starting from the Hamilton-Jacobi equation of classical mechanics, using the variational method and de Broglie formula, and finally found a non-relativistic wave equation, using the Greek letter ψ to represent the wave function , the final form is this:

△ψ+[8(π^2)m/h^2] (E - V)ψ = 0 This is the famous Schrödinger wave function throughout the history of physics in the 20th century.Of course, for ordinary readers, it is not necessary to explore the detailed meaning of mathematics, we only need to know the meaning of some symbols.Triangle △ is called "Laplace operator", which represents some kind of differential operation. h is the well-known Planck constant. E is the total energy of the system, V is the potential energy, which is -e^2/r in the atom.Solving this equation when the boundary conditions are determined, we can calculate the solution of E.

If we solve the equation sin(x)=0, the answer will be a set of values, x can be 0, π, 2π, or nπ. The function of sin(x) is continuous, but the solution of the equation is discontinuous and depends on the integer n.Similarly, when we solve E in the Schrödinger equation, we will also get a set of discrete answers, which contain the characteristics of quantization: the integer n.Our solution fits the experiment exactly, the mysterious spectrum of atoms is no longer exclusive to matrix mechanics, it can also be deduced naturally from the wave equation. Now, we can very vividly understand why electrons can only operate at certain energy levels.Electrons have a built-in vibrating frequency. Imagine a string on a guitar: when it is plucked, it vibrates.But because the two ends of the guitar string are fixed, it can only form an integer number of nodes.If a wavelength is 20 centimeters, then the length of the string can obviously only be 20 centimeters, 40 centimeters, 60 centimeters...not 50 centimeters.Because that contains half of the wave, thus contradicting its fixed two ends.If our strings form some kind of circular orbit, like the electron orbit, then obviously the size of this "orbit" can only be certain values.If a wavelength is 20 centimeters, the circumference of the orbit can only be an integer multiple of 20 centimeters, otherwise it will not be possible to connect the head to the tail.

Mathematically, this function is called "eigenfunction" (Eigenfunction), and the discrete solution obtained is called "eigenvalue" (Eigenvalue).So Schrödinger's paper is called "Quantization is an Eigenvalue Problem". From January to June 1926, he published four papers on this topic in a row, thus completely establishing another new mechanical system. - Wave mechanics.In the middle of these four papers, he also wrote a paper "Continuous Transition from Micro Mechanics to Macro Mechanics", proving that the ancient classical mechanics is only a special manifestation of the new wave mechanics, which is completely contained in the wave mechanics. Mechanics inside.

As soon as Schrödinger's equation was published, physicists almost all over the world cheered for it.Planck called it "epoch-making work", and Einstein said: "...your idea originated from a real genius." "Your quantum equation has taken a decisive step." Ehrenfest Say, "I am fascinated by your theory and the new ideas it brings. For the past two weeks, our group has spent hours at the blackboard trying to understand it from every angle." Schrödinger's equation is a popular image, concise and easy to understand. When people look up from the strange maze of the matrix and see the familiar system expressed by differential equations again, they all seem to smell the fragrance of the soil in their hometown, and there is a kind of The urge to tears.However, this new system has obviously attracted the attention of the matrix, and those in Göttingen and Copenhagen, especially Heisenberg himself, are obviously not satisfied with this "popular" explanation.

Heisenberg wrote to Pauli: "The more I think about the physical meaning of Schrödinger's theory, the more disgusted I feel. Schrödinger's visual description of his theory is meaningless. In other words, it is purely a Mist." Mist is a German word, basically It is equivalent to bullshit or crap in English. Schrödinger was also unceremonious. In his paper, he said: "My theory was inspired by de Broglie... I don't know that it has any inheritance relationship with Heisenberg. Of course I know Heisenberg's theory, it is a kind of lack of visualization, very Difficult super-algebraic approach. I'm at least frustrated if not totally repelled by the theory."

Matrix mechanics, or wave mechanics?The new quantum theory was born less than a year ago, and soon it was facing civil war. two It is interesting to look back at the two very different paths that quantum theory has taken in its development.The idea of ​​the first method is to directly start from the observed atomic spectral lines, introduce the mathematical tool of matrix, and use this strange block to build the whole building of new mechanics.It emphasizes the separation and jumping of observations, and at the same time insists on taking mathematics as the only guide, and is not confused by the intuitive experience of daily life.However, if one looks at the fundamentals, the spectral lines it emphasizes and its non-continuous side can always see the faint figure of the particle force.The core figures of this theory are naturally Heisenberg, Bonn, and Jordan, and the spiritual force behind them, the "pope" behind the scenes, is undoubtedly the great Niels Bohr in Copenhagen.These closely related scientists pooled their resources and firepower to form a strong fighting group, made breakthroughs in a short period of time, and thus established the majestic fortress of Matrix Mechanics. Those who followed the other path were obviously much less organized.Roughly speaking, this is a faction based on de Broglie's theory and Schrödinger as the main general.Einstein, who played a key guiding role in the creation of wave mechanics, is the spiritual leader behind them.But the political point of view of this theory is also very clear: it emphasizes the continuous side of the electron as a wave, and its behavior is described by the wave equation.It enthusiastically embraces intuitive explanations and tries to restore the fine tradition of visualization in classical mechanics. It has a strong retro tendency, but the revolutionary sentiment is not as high as that of its opponents.To use an inappropriate analogy, the matrix advocates radical reforms, abandons the intuitiveness of old theories, and uses mathematics as the only basis. It is a revolutionary leftist.On the other hand, volatility is relatively conservative. It emphasizes inheritance and classical concepts, and attaches great importance to the visualization and physical meaning of the theory. It is a revolutionary right wing.The battle between the two factions will be intertwined in every step of the subsequent development of quantum theory, which will have a profound impact on the entire natural philosophy of mankind. In the previous section, we mentioned that Heisenberg and Schrödinger expressed an unabashed distaste for each other's theories (they, of course, had no personal animosity).They each believe that their own set of methods is the only correct one.This is a natural occurrence, since Matrix Mechanics and Wave Mechanics look so different, and both are known for their competitive and proud personalities.When the decaying Bohr theory withdrew from the stage of history, leaving a power vacuum, no doubt everyone wanted to occupy that supreme glory.However, in April 1926, at least on the surface, this confrontation eased. Schrödinger, Pauli, and Jordan all proved that the two mechanics are completely equivalent mathematically!In fact, we traced their respective family histories and found that they all come from the classic Hamiltonian function, except that one starts from the motion equation of particles and the other starts from the wave equation.And optics and kinematics have long been linked together through the efforts of Hamilton himself, which is really called "the same root".Soon people have known that starting from the matrix, the expression form of the wave function can be derived, and conversely, our matrix can also be derived from the wave function. In 1930, Dirac published the classic textbook on quantum mechanics. The two mechanics were perfectly unified and presented to readers as different expressions of a theory. However, if anyone thinks that the world will be peaceful and everything will be fine from now on, then it is a big mistake.Although the two systems have been united in form, in terms of ideology deep down, the differences between them have become wider and wider, and an insurmountable gap has soon formed.Mathematical consistency does not prevent people from interpreting it differently, as far as the matrix is ​​concerned, it is intended to be granular and discontinuous.In terms of volatility, it is always talking about volatility and continuity.The wave-particle war has now reached its climax, and the two sides have found their respective governments that they can rely on, and have escalated the war to the level of explaining the entire physical law. "Wave, only wave is the only reality." Schrödinger said affirmatively, "Whether it is electrons, photons, or any particles, they are just bubbles on the surface of waves. They are all waves in essence, and can be Use the wave equation to express the basic mode of motion." "Absolutely disagree." Heisenberg retorted, "The basic phenomenon of the physical world is discreteness, or discontinuity. A large number of experimental facts have proved this point: from the spectrum of atoms to Compton's experiment, Discontinuities in nature have been shown irrefutably, from photoelectric phenomena, to the jumping of electrons between energy levels in atoms. Your wave equation is certainly a welcome achievement in mathematics, but we must realize that we It can’t be seen in the traditional way—it doesn’t mean that.” "On the contrary," said Schrödinger, "that's what it means. The wave function ψ (pronounced psai) is continuous in every direction, and it can be thought of as a kind of vibration. In fact, we have to think of the electron as a The resident eigenvibration, the so-called "transition" of the electron, is nothing but a change in its vibration mode. There are no 'orbits' and no 'energy levels', only waves." "Haha." Heisenberg said mockingly, "I'm afraid you don't understand what your own ψ is? It's just a virtual function in some virtual space, and you insist on imagining it as a kind of Real waves. In fact, we must not be misled by everyday figurative things, after all, you cannot deny the behavior of electrons as classical particles." "That's right." Schrödinger still refused to show weakness, "I don't deny that it does exhibit particle-like behavior. But, like a coconut, if you knock open the hard shell of its particle, you will find that there is still Fluctuating soft juice. Electrons are undoubtedly composed of sine waves, but such waves do not stretch much on all scales and can be seen as a 'wave packet.' When this wave packet advances as a whole, it looks It looks like a particle. But, in essence, it is still a wave, and a particle is just a derivative of a wave." As everyone had already guessed, neither could convince the other. In July 1926, Schrödinger was invited to give a lecture on his new mechanics at the University of Munich. Heisenberg sat below. He stood up and vehemently criticized Schrödinger's explanation, only to find sadly that the audience was against him.Earlier, Kramer, Bohr's original assistant, had accepted a letter of employment from Utrecht University and left Copenhagen, so Heisenberg became the successor in this position-now he can live in Bohr as he dreamed. Earl's side worked.Bohr was also disturbed by Schrödinger's theoretical view of returning to the classical tradition. In order to solve this problem, he invited Schrödinger to Copenhagen for an academic visit, trying to reach some kind of consensus in the exchange. At the end of September, Schrödinger arrived in Copenhagen, and Bohr went to the train station to pick him up.The debate started from that moment, day and night, without end, until Schrödinger finally left Copenhagen.Heisenberg later recalled this meeting in his book Parts and Wholes, saying that although Bohr was such an amiable person on weekdays, once he was involved in this kind of physical debate, he seemed to be very different. Like a paranoid fanatic, never willing to compromise a step.The debate is, of course, a matter of physics, but has largely become a philosophical one.Schrödinger just couldn't believe that an "unimaginable" theory had any practical significance.Bohr, on the other hand, insisted that the concept of imagery cannot be used in quantum processes, and it cannot be described in everyday language.They quarreled violently from day to night, and eventually Schrödinger was so exhausted that he soon fell ill and had to lie down in bed to be looked after by Bohr's wife, Margaret.Even so, Bohr was still reluctant. He rushed into the ward and stood by Schrödinger's bedside to continue arguing with him.Of course, in the end everything was in vain, and no one was convinced by the other party. The air in physics has become very hot.The classic theory has collapsed, and now two buildings of matrix mechanics and wave mechanics have risen from the ground. They are connected with each other by some kind of bridge, and they should be regarded as one in theory.However, the foundations of the two buildings are still not related to each other, which makes the apparent goodwill somewhat tinged with duplicity.Moreover, waves and particles, the two 300-year-old enemies are still fighting bitterly, refusing to take a step back from their own territory.Both sides still claim that they have all sovereignty over light, electricity, and various physical phenomena, while their opponents are illegal armed forces and anti-government organizations.Now Schrödinger has joined the wave camp, and he even provided a complete constitution for the wave, which is his wave equation.In Schrödinger's view, volatility represents the glory of the old empire from Huygens, Young to Maxwell, and this noble tradition must be preserved and carried forward in the new country.Schrödinger believed that the concept of the simple image of a wave would once again rule the physical world, reducing everything to a unified image. Unfortunately, Schrödinger guessed wrong.Volatility will soon discover that their constitution turns out to have something much deeper meaning.Between the lines, we can read some hidden meanings. It says that the world is a public, and no one side can monopolize it. The two sides must negotiate peace and then form a coalition government to rule.It also revealed an even more astonishing secret: the two parties had an inseparable blood relationship.In the end, like the oracle of the priests of the temple of Artemis, it prophesied that under this union, physics would be very different: more wondrous, more mysterious, more prosperous. What a wonderful prophecy. *********** Gossip after dinner: Schrödinger's girlfriend In November 2001, playwright Matthew Wells' new work "Schrodinger's Girlfriend" (Schrodinger's Girlfriend) premiered at the famous Fort Mason Center in San Francisco.Set in the history of Schrödinger's founding of wave mechanics in 1926 by the company of his mysterious girlfriend in Arosa, the comedy explores the relationship between love, sex and quantum physics and was generally well-received by critics.At the beginning of this year (2003), the play was moved to the East Coast for performances, and it was also well received.In recent years, there has been a trend of drama creation based on scientific figures and the history of science. In addition to this "Schrödinger's Girlfriend", I am afraid that the Tony Award winner, Michael Frayn's "Copenhagen" is more famous. However, it is really difficult to count how many girlfriends Schrödinger has.The morality of this physicist is obviously far from that of ordinary people, and his weird behavior has always been rejected by people. In 1912, he almost gave up academics for a girl he liked, and switched to running his own family company (teaching at the university didn’t make much money at the time). Before he met Anne Marie, Schrödinger had fallen in love with 4 young girls in total, and mainly It's a spiritual relationship.In this regard, Walter Moore, one of Schrödinger's main biographers, argued that it should not be simply regarded as an act of indulgence. If all the above are considered normal, Schrödinger after marriage has a bit of a wild flavor of informality.His marriage with Anne was never stable and harmonious, and the two had no children in their lives.Schrödinger probably didn't do less about being promiscuous outside, and he didn't hide this from his wife.Anne, in turn, has an affair with one of Schrödinger's best friends, Hermann Weyl (Weyl's own wife is obsessed with someone else, it's dark).The two discussed divorce, but Anne's Catholic faith and expensive fees effectively prevented that from happening. "Schrödinger's Girlfriend" joked: "Is the wave-particle duality more difficult, or is the wife-lover duality more difficult?" Schrödinger was, according to some popular parlance, one of those "passionate seeds."He invited others to be his assistants, but in fact he fell in love with his wife.This woman (Hilde March) later bore him a daughter, whom Anne, surprisingly, took great pleasure in taking care of.Schrödinger and these two women cohabited openly, and in fact lived a life of monogamy and one concubine (this concubine was also someone else's legal wife), which was so shocking that it couldn't stand in Oxford and Princeton, so he had to leave.There is also a long list of his romantic history, including female students, actors, OLs, and several illegitimate children.But Schrödinger was not simply venting his desires. He had a strong romantic impulse in his heart. According to Duan Zhengchun, when he was with every woman, he was so desperate that he wanted to take his heart out and write a song for it. Lots of love poetry.I hope that everyone will not think that I am too gossip. In fact, the analysis of love history is an important part of Schrödinger's research. It helps us understand the scientist's extremely complex inner psychology and unique personality with personal color. The most surprising thing is that such a Schrödinger marriage almost got a perfect ending.Despite experiencing all kinds of storms and crossing many dangerous shoals, he and Annie finally grew old together, as they said in the oath: to have and to hold, in sickness and in health, till death parts us.In the last period of Schrödinger's life, the two had already reached an understanding. Anne said: "In the past 41 years, we have been tightly bound together by joys, sorrows, sorrows and joys, and we don't want to be separated in the last few years." When Schrödinger was dying, Annie stood by his bed and held his hand, Schrödinger said: "Now I have you again, everything is fine again." Schrödinger was buried in Alpbach after his death, and his cemetery was soon covered with snow.Four years later, Anne-Marie Schrödinger also stopped breathing. three In mid-1926, although the Matrix faction and the Wave faction were still dissatisfied with each other in their hearts, they were at least superficially united by mathematics.Moreover, not surprisingly, Schrödinger's wave equation was popular with most physicists for its catchy, concise and easy-to-learn features, and soon gained the upper hand in form.Although Heisenberg and his squatting square matrix were not happy, they had no choice but to accept the reality.Facts have proved that, except that the matrix has an advantage when dealing with a few problems about spin, the wave equation steals almost all the popularity at other times.In fact, physicists are quite different from what the public imagines. Few people like the kind of perverted mathematics that is difficult and weird. Since the two systems have been proved to be mathematically equivalent, everyone is happy to choose the one that looks Simple and familiar. Even within the Matrix school, the wave equation gained popularity.First was Sommerfeld, Heisenberg's teacher, and then one of the core figures in the establishment of matrix mechanics, another mentor of Heisenberg, Max Bonn.Born, who enthusiastically praised Schrödinger's achievement shortly after its release, called the wave equation "the most profound form of quantum law."It is said that Heisenberg felt very sad about this "betrayal" in Bonn. However, Heisenberg was overthinking. Bonn's approval of Schrödinger's equation did not mean that he chose to stand in the same trench as Schrödinger.Because although the equation is fixed, how to interpret it is a very different problem.The first thing people have to ask is, what does Schrödinger's wave function ψ (remind again, this Greek word is pronounced as psai) physically represent? We may wish to review Schrödinger's idea of ​​creating the wave equation: he started from the classic Hamiltonian equation, constructed a new function ψ of the system and substituted it, and then quoted the de Broglie relation and variational method, and finally found the equation And its solution, which is quite different from the physics we have in mind.Usually we think that the definition of physical quantities comes first, and then we can talk about finding their mathematical relationship.For example, we understand the concept of force F, acceleration a and mass m, and then we will understand the meaning of F=ma.But the path of modern physics may often be the opposite. For example, a physicist may first define a certain function F, let F=ma, and then search for the physical meaning of F, and find that it turns out to be a measure of force.Schrödinger's ψ is a certain distribution function defined in space, but people don't know what its physical meaning is. This looks amusing, because physicists have to sit back and do charades, too.Now let's relax and imagine that we are at a party and the host has arranged a fun quiz show for everyone's entertainment. "Ladies and gentlemen," he announced cheerfully, "let's play a guessing game. Whoever guesses what is hidden in this box first will get the highest honor at the party." The big box seemed heavy, and it really seemed to be hiding something good. On the lid of the box, there were a few large characters written in antique style: "Schrödinger's Equation". "Okay, but you can't see anything, so how do you guess?" people complained. "Of course, of course." The host quickly said, "We're not imitating Monkey King to guess things through partitions, and besides, it's definitely not a tattered clock, it's a treasure that is really related to the whole physics. Well, yes In this way, although we cannot see it, some of its properties can be known, and I will keep reminding everyone to see who can guess it first." The crowd clamored for a while, and the game began. "We don't know the name of this thing, but it's called ψ." The host cleared his throat and said, "What I can tell you is that it represents a certain function of electrons in the atomic system." Gossip: "Energy? Frequency? Speed? Distance? Time? Charge? Quality?" The host had to raise his voice and shouted: "Quiet, quiet, we're just getting started, don't make random guesses. Who will guess from now on? If you make a mistake, you will be disqualified." Then there was an instant silence. "Okay." The host said with satisfaction, "Then let's continue. The second condition is this: Through my observation, I found that this ψ is a continuous thing." Everyone dared not speak this time, but Everyone quickly ruled it out in their hearts.Since it is continuous, the conditions of quantization known to us are all ruled out.For example, we already know that the energy levels of electrons are not continuous, so ψ does not look like this. "Next, through the construction of ψ, it can be seen that this is a dimensionless function. But it also has some relationship with the position of the electron. For each electron, it expands in a virtual three-dimensional space Go." At this point, many people are already confused, only a few with particularly quick thinking are still thinking nervously. "In a word, ψ follows every electron like a shadow, and spreads out in its place like a cloud. This cloud is sometimes thick and sometimes thin, but it evolves in a definite way. And, Let me emphasize again that this diffusion and its evolution are classic, continuous, and definite." So everyone fell into deep thought, without any clue. "Yes, clouds, this metaphor is wonderful." At this moment, a man with a thin face and pince-nez stood up with a smile and said.The host hurriedly introduced: "Ladies and gentlemen, this is Mr. Schrödinger, who is also the discoverer of this treasure chest." Everyone applauded for a while, and then listened intently to what he had to say. "Well, things are already obvious. ψ is a spatial distribution function." Schrödinger said confidently, "When it is multiplied by the charge of an electron, it represents the actual distribution of the charge in space. Clouds, my dear ones , the electron is not a particle, it is a wave that spreads out around space like a cloud. Our wave function describes exactly this expansion and its behavior. The electron has no specific location, nor does it have a specific The path, because it is a cloud, is a wave, and it goes out in every direction—although it decays so quickly that it roughly looks like a particle. Ladies and gentlemen, I think the greatest significance of this discovery is , we must get rid of all false images about particles from our minds, whether it is electrons, photons, or what kind of particles, they are not particles in the traditional sense. Pull them out and zoom in, carefully Examine them, and you will find that it melts in space and becomes a superposition of countless vibrations. Yes, an electron, it is spread out, like butter on bread, it is usually curled up so tightly, So much so that we all think of it as small balls, but this has been proven not to be true by our wave function ψ. Over the years physics has gone astray and our heads have been messed with spectral lines, transitions, energy levels, matrices, and all that weird stuff It's been a mess, and now, it's time to return to the classics." "This treasure chest," Schrödinger said excitedly, pointing to the big box, "is an inheritance, which was entrusted to us by King Solomon of the legendary empire in the past. It always reminds us not to be tempted by crooked ways, and we will never turn back. The fork in the road. Physics needs to be reformed, but the confusion of thought cannot be allowed. We have heard enough weird talk, such as electrons jumping around like fleas in atoms, like a drunkard who can't predict his direction at all. And that mystical so-called matrix that no one knows what physical meaning it contains, while it keeps shouting that it is the orthodoxy of physics. No, now let's go back to the solid ground where the giants fought This land, this land that once built such a majestic structure, this land is full of pride and glorious history. Simplicity, clarity, elegance, intuition, continuity, and visualization are the sticks of victory in the kingdom of physics , which has been handed down from generation to generation, leading us to victory. I have no doubt that new mechanics will be made on a continuous wave basis, bringing everything down to simple images, and carrying on the lineage of the old royal family. This is by no means conservative, because This lineage is also the soul that has carried modern science for 300 years. This is a symbol of physics, and its sacred status must not be shaken, not by anyone." Schrödinger's eloquent speech undoubtedly deeply infected most of the audience present, because there was a burst of warm applause and cheers from the crowd.But wait, there was a person who kept shaking his head and seemed disapproving. Schrödinger quickly recognized that it was Bonn in Göttingen, Heisenberg's teacher.Hadn't he just praised his own equation?Could it be that this kid Heisenberg used some method to win him over? "Well, Mr. Schrödinger," Bonn cleared his throat and stood up, "First of all, I would like to express my sincere admiration for your discovery. This is undoubtedly a rare treasure, and not everyone is so lucky to make such a great achievement Yes." Schrödinger nodded, feeling a little more relaxed. "But," Bonn went on, "may I ask you a question? Although you found it, have you ever actually opened the box yourself to see what's inside?" 这令薛定谔大大地尴尬，他踟躇了好一会儿才回答：“说实话，我也没有真正看见过里面的东西，因为我没有箱子的钥匙。”众人一片惊诧。 “如果是这样的话，”波恩小心翼翼地说，“我倒以为，我不太同意您刚才的猜测呢。” “哦？”两个人对视了一阵，薛定谔终于开口说：“那么您以为，这里面究竟是什么东西呢？” “毫无疑问，”波恩凝视着那雕满了古典花纹的箱子和它上面那把沉重的大锁，“这里面藏着一些至关紧要的事物，它的力量足以改变整个物理学的面貌。但是，我也有一种预感，这股束缚着的力量是如此强大，它将把物理学搞得天翻地覆。当然，你也可以换个词语说，为物理学带来无边的混乱。” “哦，是吗？”薛定谔惊奇地说，“照这么说来，难道它是潘多拉的盒子？” “嗯。”波恩点了点头，“人们将陷入困惑和争论中，物理学会变成一个难以理解的奇幻世界。老实说，虽然我隐约猜到了里面是什么，我还是不能确定该不该把它说出来。” 薛定谔盯着波恩：“我们都相信科学的力量，在于它敢于直视一切事实，并毫不犹豫地去面对它，检验它，把握它，不管它是什么。何况，就算是潘多拉盒子，我们至少也还拥有盒底那最宝贵的东西，难道你忘了吗？” “是的，那是希望。”波恩长出了一口气，“你说的对，不管是祸是福，我们至少还拥有希望。只有存在争论，物理学才拥有未来。” “那么，你说这箱子里是……？”全场一片静默，人人都不敢出声。 波恩突然神秘地笑了：“我猜，这里面藏的是……” “……骰子。” Four dice?骰子是什么东西？它应该出现在大富翁游戏里，应该出现在澳门和拉斯维加斯的赌场中，但是，物理学？不，那不是它应该来的地方。骰子代表了投机，代表了不确定，而物理学不是一门最严格最精密，最不能容忍不确定的科学吗？ 可以想象，当波恩于1926年7月将骰子带进物理学后，是引起了何等的轩然大波。围绕着这个核心解释所展开的争论激烈而尖锐，把物理学加热到了沸点。这个话题是如此具有争议性，很快就要引发20世纪物理史上最有名的一场大论战，而可怜的波恩一直要到整整28年后，才因为这一杰出的发现而获得诺贝尔奖金——比他的学生们晚上许多。 不管怎么样，我们还是先来看看波恩都说了些什么。骰子，这才是薛定谔波函数ψ的解释，它代表的是一种随机，一种概率，而决不是薛定谔本人所理解的，是电子电荷在空间中的实际分布。波恩争辩道，ψ,或者更准确一点，ψ的平方，代表了电子在某个地点出现的“概率”。电子本身不会像波那样扩展开去，但是它的出现概率则像一个波，严格地按照ψ的分布所展开。 我们来回忆一下电子或者光子的双缝干涉实验，这是电子波动性的最好证明。当电子穿过两道狭缝后，便在感应屏上组成了一个明暗相间的图案，展示了波峰和波谷的相互增强和抵消。但是，正如粒子派指出的那样，每次电子只会在屏上打出一个小点，只有当成群的电子穿过双缝后，才会逐渐组成整个图案。 现在让我们来做一个思维实验，想象我们有一台仪器，它每次只发射出一个电子。这个电子穿过双缝，打到感光屏上，激发出一个小亮点。那么，对于这一个电子，我们可以说些什么呢？很明显，我们不能预言它组成类波的干涉条纹，因为一个电子只会留下一个点而已。事实上，对于这个电子将会出现在屏幕上的什么地方，我们是一点头绪都没有的，多次重复我们的实验，它有时出现在这里，有时出现在那里，完全不是一个确定的过程。 不过，我们经过大量的观察，却可以发现，这个电子不是完全没有规律的：它在某些地方出现的可能性要大一些，在另一些地方则小一些。它出现频率高的地方，恰恰是波动所预言的干涉条纹的亮处，它出现频率低的地方则对应于暗处。现在我们可以理解为什么大量电子能组成干涉条纹了，因为虽然每一个电子的行为都是随机的，但这个随机分布的总的模式却是确定的，它就是一个干涉条纹的图案。这就像我们掷骰子，虽然每一个骰子掷下去，它的结果都是完全随机的，从1到6都有可能，但如果你投掷大量的骰子到地下，然后数一数每个点的数量，你会发现1到6的结果差不多是平均的。 关键是，单个电子总是以一个点的面貌出现，它从来不会像薛定谔所说的那样，在屏幕上打出一滩图案来。只有大量电子接二连三地跟进，总的干涉图案才会逐渐出现。其中亮的地方也就是比较多的电子打中的地方，换句话说，就是单个电子比较容易出现的地方，暗的地带则正好相反。如果我们发现，有9成的粒子聚集在亮带，只有1成的粒子在暗带，那么我们就可以预言，对于单个粒子来说，它有90％的可能出现在亮带的区域，10％的可能出现在暗带。但是，究竟出现在哪里，我们是无法确定的，我们只能预言概率而已。 我们只能预言概率而已。 但是，等等，我们怎么敢随便说出这种话来呢？这不是对于古老的物理学的一种大不敬吗？从伽利略牛顿以来，成千上百的先辈们为这门科学呕心沥血，建筑起了这样宏伟的构筑，它的力量统治整个宇宙，从最大的星系到最小的原子，万事万物都在它的威力下必恭必敬地运转。任何巨大的或者细微的动作都逃不出它的力量。星系之间产生可怕的碰撞，释放出难以想象的光和热，并诞生数以亿计的新恒星；宇宙射线以惊人的高速穿越遥远的空间，见证亘古的时光；微小得看不见的分子们你推我搡，喧闹不停；地球庄严地围绕着太阳运转，它自己的自转轴同时以难以觉察的速度轻微地振动；坚硬的岩石随着时光流逝而逐渐风化；鸟儿扑动它的翅膀，借着气流一飞冲天。这一切的一切，不都是在物理定律的监视下一丝不苟地进行的吗？ 更重要的是，物理学不仅能够解释过去和现在，它还能预言未来。我们的定律和方程能够毫不含糊地预测一颗炮弹的轨迹以及它降落的地点；我们能预言几千年后的日食，时刻准确到秒；给我一张电路图，多复杂都行，我能够说出它将做些什么；我们制造的机器乖乖地按照我们预先制定好的计划运行。事实上，对于任何一个系统，只要给我足够的初始信息，赋予我足够的运算能力，我能够推算出这个体系的一切历史，从它最初怎样开始运行，一直到它在遥远的未来的命运，一切都不是秘密。是的，一切系统，哪怕骰子也一样。告诉我骰子的大小，质量，质地，初速度，高度，角度，空气阻力，桌子的质地，摩擦系数，告诉我一切所需要的情报，那么，只要我拥有足够的运算能力，我可以毫不迟疑地预先告诉你，这个骰子将会掷出几点来。 物理学统治整个宇宙，它的过去和未来，一切都尽在掌握。这已经成了物理学家心中深深的信仰。19世纪初，法国的大科学家拉普拉斯（Pierre Simon de Laplace）在用牛顿方程计算出了行星轨道后，把它展示给拿破仑看。拿破仑问道：“在你的理论中，上帝在哪儿呢？”拉普拉斯平静地回答：“陛下，我的理论不需要这个假设。” 是啊，上帝在物理学中能有什么位置呢？一切都是由物理定律来统治的，每一个分子都遵照物理定律来运行，如果说上帝有什么作用的话，他最多是在一开始推动了这个体系一下，让它得以开始运转罢了。在之后的漫长历史中，有没有上帝都是无关紧要的了，上帝被物理学赶出了舞台。 “我不需要上帝这个假设。”拉普拉斯站在拿破仑面前说。这可算科学最光荣最辉煌的时刻之一了，它把无边的自豪和骄傲播撒到每一个科学家的心中。不仅不需要上帝，拉普拉斯想象，假如我们有一个妖精，一个大智者，或者任何拥有足够智慧的人物，假如他能够了解在某一刻，这个宇宙所有分子的运动情况的话，那么他就可以从正反两个方向推演，从而得出宇宙在任意时刻的状态。对于这样的智者来说，没有什么过去和未来的分别，一切都历历在目。宇宙从它出生的那一刹那开始，就坠入了一个预定的轨道，它严格地按照物理定律发展，没有任何岔路可以走，一直到遇见它那注定的命运为止。就像你出手投篮，那么，这究竟是一个三分球，还是打中篮筐弹出，或者是一个air ball，这都在你出手的一刹那决定了，之后我们所能做的，就是看着它按照写好的剧本发展而已。 是的，科学家知道过去；是的，科学家明白现在；是的，科学家了解未来。只要掌握了定律，只要搜集足够多的情报，只要能够处理足够大的运算量，科学家就能如同上帝一般无所不知。整个宇宙只不过是一台精密的机器，它的每个零件都按照定律一丝不苟地运行，这种想法就是古典的，严格的决定论（determinism）。宇宙从出生的那一刹那起，就有一个确定的命运。我们现在无法了解它，只是因为我们所知道的信息太少而已。 那么多的天才前仆后继，那么多的伟人呕心沥血，那么多在黑暗中的探索，挣扎，奋斗，这才凝结成物理学在19世纪黄金时代的全部光荣。物理学家终于可以说，他们能够预测神秘的宇宙了，因为他们找到了宇宙运行的奥秘。他们说这话时，带着一种神圣而不可侵犯的情感，决不饶恕任何敢于轻视物理学力量的人。 可是，现在有人说，物理不能预测电子的行为，它只能找到电子出现的概率而已。无论如何，我们也没办法确定单个电子究竟会出现在什么地方，我们只能猜想，电子有90％的可能出现在这里，10％的可能出现在那里。这难道不是对整个物理历史的挑衅，对物理学的光荣和尊严的一种侮辱吗？ 我们不能确定？物理学的词典里是没有这个字眼的。在中学的物理考试中，题目给了我们一个小球的初始参数，要求t时刻的状态，你敢写上“我不能确定”吗？要是你这样做了，你的物理老师准会气得吹胡子瞪眼睛，并且毫不犹豫地给你亮个红灯。Not sure?不可能，物理学什么都能确定。诚然，有时候为了方便，我们也会引进一些统计的方法，比如处理大量的空气分子运动时，但那是完全不同的一个问题。科学家只是凡人，无法处理那样多的复杂计算，所以应用了统计的捷径。但是从理论上来说，只要我们了解每一个分子的状态，我们完全可以严格地推断出整个系统的行为，分毫不爽。 然而波恩的解释不是这样，波恩的意思是，就算我们把电子的初始状态测量得精确无比，就算我们拥有最强大的计算机可以计算一切环境对电子的影响，即便如此，我们也不能预言电子最后的准确位置。这种不确定不是因为我们的计算能力不足而引起的，它是深藏在物理定律本身内部的一种属性。即使从理论上来说，我们也不能准确地预测大自然。这已经不是推翻某个理论的问题，这是对整个决定论系统的挑战，而决定论是那时整个科学的基础。量子论挑战整个科学。 波恩在论文里写道：“……这里出现的是整个决定论的问题了。”（Hier erhebt sich der ganze Problematik des Determinismus.） 对于许多物理学家来说，这是一个不可原谅的假设。dice?不确定？Do not make jokes.对于他们中的好些人来说，物理学之所以那样迷人，那样富有魔力，正是因为它深刻，明晰，能够确定一切，扫清人们的一切疑惑，这才使他们义无反顾地投身到这一事业中去。现在，物理学竟然有变成摇奖机器的危险，竟然要变成一个掷骰子来决定命运的赌徒，这怎么能够容忍呢？ 不确定？ 一场史无前例的大争论即将展开，在争吵和辩论后面是激动，颤抖，绝望，泪水，伴随着整个决定论在20世纪的悲壮谢幕。 *********** 饭后闲话：决定论 可以说决定论的兴衰浓缩了整部自然科学在20世纪的发展史。科学从牛顿和拉普拉斯的时代走来，辉煌的成功使它一时得意忘形，认为它具有预测一切的能力。决定论认为，万物都已经由物理定律所规定下来，连一个细节都不能更改。过去和未来都像已经写好的剧本，宇宙的发展只能严格地按照这个剧本进行，无法跳出这个窠臼。 矜持的决定论在20世纪首先遭到了量子论的严重挑战，随后混沌动力学的兴起使它彻底被打垮。现在我们已经知道，即使没有量子论把概率这一基本属性赋予自然界，就牛顿方程本身来说，许多系统也是极不稳定的，任何细小的干扰都能够对系统的发展造成极大的影响，差之毫厘，失之千里。这些干扰从本质上说是不可预测的，因此想凭借牛顿方程来预测整个系统从理论上说也是不可行的。典型的例子是长期的天气预报，大家可能都已经听说过洛伦兹著名的“蝴蝶效应”，哪怕一只蝴蝶轻微地扇动它的翅膀，也能给整个天气系统造成戏剧性的变化。现在的天气预报也已经普遍改用概率性的说法，比如“明天的降水概率是20％”。 1986年，著名的流体力学权威，詹姆士?莱特希尔爵士（Sir James Lighthill，他于1969年从狄拉克手里接过剑桥卢卡萨教授的席位，也就是牛顿曾担任过的那个）于皇家学会纪念牛顿《原理》发表300周年的集会上发表了轰动一时的道歉： “现在我们都深深意识到，我们的前辈对牛顿力学的惊人成就是那样崇拜，这使他们把它总结成一种可预言的系统。而且说实话，我们在1960年以前也大都倾向于相信这个说法，但现在我们知道这是错误的。我们以前曾经误导了公众，向他们宣传说满足牛顿运动定律的系统是决定论的，但是这在1960年后已被证明不是真的。我们都愿意在此向公众表示道歉。” （We are all deeply conscious today that the enthusiasm of our forebears for the marvelous achievements of Newtonian mechanics led them to make generalizations in this area of predictability which, indeed, we may have generally tended to believe before 1960, but which we now recognize were false. We collectively wish to apologize for having misled the general educated public by spreading ideas about the determinism of systems satisfying Newtons laws of motion that, after 1960,were to be proved incorrect.） 决定论的垮台是否注定了自由意志的兴起？这在哲学上是很值得探讨的。事实上，在量子论之后，物理学越来越陷于形而上学的争论中。也许形而上学（metaphysics）应该改个名字叫“量子论之后”（metaquantum）。在我们的史话后面，我们会详细地探讨这些问题。 Ian Stewart写过一本关于混沌的书，书名也叫。这本书文字优美，很值得一读，当然和我们的史话没什么联系。我用这个名字，一方面是想强调决定论的兴衰是我们史话的中心话题，另外，毕竟爱因斯坦这句名言本来的版权是属于量子论的。 Fives 在我们出发去回顾新量子论与经典决定论的那场惊心动魄的悲壮决战之前，在本章的最后还是让我们先来关注一下历史遗留问题，也就是我们的微粒和波动的宿怨。波恩的概率解释无疑是对薛定谔传统波动解释的一个沉重打击，现在，微粒似乎可以暂时高兴一下了。 “看，”它嘲笑对手说，“薛定谔也救不了你，他对波函数的解释是站不住脚的。难怪总是有人说，薛定谔的方程比薛定谔本人还聪明哪。波恩的概率才是有道理的，电子始终是一个电子，任何时候你观察它，它都是一个粒子，你吵嚷多年的所谓波，原来只是那看不见摸不着的'概率'罢了。哈哈，把这个头衔让给你，我倒是毫无异议的，但你得首先承认我的正统地位。” 但是波动没有被吓倒，说实话，双方300年的恩怨缠结，经过那么多风风雨雨，早就练就了处变不惊的本领。“哦，是吗？”它冷静地回应道，“恐怕事情不如你想象得那么简单吧？我们不如缩小到电子那个尺寸，去亲身感受一下一个电子在双缝实验中的经历如何？” 微粒迟疑了一下便接受了：“好吧，让你彻底死心也好。” 那么，现在让我们也想象自己缩小到电子那个尺寸，跟着它一起去看看事实上到底发生了什么事。一个电子的直径小于一亿分之一埃，也就是10^-23米，它的质量小于10^-30千克，变得这样小，看来这必定是一次奇妙的旅程呢。 好，现在我们已经和一个电子一样大了，突然缩小了那么多，还真有点不适应，看出去的世界也变得模糊扭曲起来。不过，我们第一次发现，世界原来那么空旷，几乎是空无一物，这也情有可原，从我们的尺度看来，原子核应该像是远在天边吧？好，现在迎面来了一个电子，这是个好机会，让我们睁大眼睛，仔细地看一看它究竟是个粒子还是波?奇怪，为什么我们什么都看不见呢？啊，原来我们忘了一个关键的事实！ 要“看见”东西，必须有光进入我们的眼睛才行。但现在我们变得这么小，即使光——不管它是光子还是光波——对于我们来说也太大了。但是不管怎样，为了探明这个秘密，我们必须得找到从电子那里反射过来的光，凭感觉，我知道从左边来了一团光（之所以说“一团”光，是因为我不清楚它究竟是一个光粒子还是一道光波，没有光，我也看不到光本身，是吧？），现在让我们勇敢地迎上去，啊，秘密就要揭开了！ 随着“砰”地一声，我们被这团光粗暴地击中，随后身不由己地飞到半空中，被弹出了十万八千里。这次撞击使得我们浑身筋骨欲脱，脑中天旋地转，眼前直冒金星。我们忘了自己现在是个什么尺寸！要不是运气好，这次碰撞已经要了咱们的小命。当好不容易爬起来时，早就不知道自己身在何方，那个电子更是无影无踪了。 刚才真是好险，看来这一招是行不通的。不过，我听见声音了，是微粒和波动在前面争论呢，咱们还是跟着这哥俩去看个究竟。它们为了模拟一个电子的历程，从某个阴极射线管出发，现在，面前就是那著名的双缝了。 “嗨，微粒。”波动说道，“假如电子是个粒子的话，它下一步该怎样行动呢？眼前有两条缝，它只能选择其中之一啊，如果它是个粒子，它不可能两条缝都通过吧？” “嗯，没错。”微粒说，“粒子就是一个小点，是不可分割的。我想，电子必定选择通过了其中的某一条狭缝，然后投射到后面的光屏上，激发出一个小点。” “可是，”波动一针见血地说，“它怎能够按照干涉模式的概率来行动呢？比如说它从右边那条缝过去了吧，当它打到屏幕前，它怎么能够知道，它应该有90％的机会出现到亮带区，10％的机会留给暗带区呢？要知道这个干涉条纹可是和两条狭缝之间的距离密切相关啊，要是电子只通过了一条缝，它是如何得知两条缝之间的距离的呢？” 微粒有点尴尬，它迟疑地说：“我也承认，伴随着一个电子的有某种类波的东西，也就是薛定谔的波函数ψ，波恩说它是概率，我们就假设它是某种看不见的概率波吧。你可以把它想象成从我身上散发出去的某种看不见的场，我想，在我通过双缝之前，这种看不见的波场在空间中弥漫开去，探测到了双缝之间的距离，从而使我得以知道如何严格地按照概率行动。但是，我的实体必定只能通过其中的一条缝。” “一点道理也没有。”波动摇头说，“我们不妨想象这样一个情景吧，假如电子是一个粒子，它现在决定通过右边的那条狭缝。姑且相信你的说法，有某种概率波事先探测到了双缝间的距离，让它胸有成竹知道如何行动。可是，假如在它进入右边狭缝前的那一刹那，有人关闭了另一道狭缝，也就是左边的那道狭缝，那时会发生什么情形呢？” 微粒有点脸色发白。 “那时候，”波动继续说，“就没有双缝了，只有单缝。电子穿过一条缝，就无所谓什么干涉条纹。也就是说，当左边狭缝关闭的一刹那，电子的概率必须立刻从干涉模式转换成普通模式，变成一条长狭带。” “现在，我倒请问，电子是如何在穿过狭缝前的一刹那，及时地得知另一条狭缝关闭这个事实的呢？要知道它可是一个小得不能再小的电子啊，另一条狭缝距离它是如此遥远，就像从上海隔着大洋遥望洛杉矶。它如何能够瞬间作出反应，修改自己的概率分布呢？除非它收到了某种瞬时传播来的信号，怎么，你想开始反对相对论了吗？” “好吧，”微粒不服气地说，“那么，我倒想听听你的解释。” “很简单，”波动说，“电子是一个在空间中扩散开去的波，它同时穿过了两条狭缝，当然，这也就是它造成完美干涉的原因了。如果你关闭一个狭缝，那么显然就关闭了一部分波的路径，这时就谈不上干涉了。” “听起来很不错。”微粒说，“照你这么说，ψ是某种实际的波，它穿过两道狭缝，完全确定而连续地分布着，一直到击中感应屏前。不过，之后呢？之后发生了什么事？” “之后……”波动也有点语塞，“之后，出于某种原因，ψ收缩成了一个小点。” “哈，真奇妙。”微粒故意把声音拉长以示讽刺，“你那扩散而连续的波突然变成了一个小点！请问发生了什么事呢？波动家族突然全体罢工了？” 波动气得面红耳赤，它争辩道：“出于某种我们尚不清楚的机制……” “好吧，”微粒不耐烦地说，“实践是检验真理的唯一标准是吧？既然我说电子只通过了一条狭缝，而你硬说它同时通过两条狭缝，那么搞清我们俩谁对谁错不是很简单吗？我们只要在两道狭缝处都安装上某种仪器，让它在有粒子——或者波，不论是什么——通过时记录下来或者发出警报，那不就成了？这种仪器又不是复杂而不可制造的。” 波动用一种奇怪的眼光看着微粒，良久，它终于说：“不错，我们可以装上这种仪器。我承认，一旦我们试图测定电子究竟通过了哪条缝时，我们永远只会在其中的一处发现电子。两个仪器不会同时响。” 微粒放声大笑：“你早说不就得了？害得我们白费了这么多口水！怎么，这不就证明了，电子只可能是一个粒子，它每次只能通过一条狭缝吗？你还跟我唠叨个什么！”但是它渐渐发现气氛有点不对劲，终于它笑不出来了。 “怎么？”它瞪着波动说。 波动突然咧嘴一笑：“不错，每次我们只能在一条缝上测量到电子。但是，你要知道，一旦我们展开这种测量的时候，干涉条纹也就消失了……” ... 时间是1927年2月，哥本哈根仍然是春寒料峭，大地一片冰霜。玻尔坐在他的办公室里若有所思：粒子还是波呢？5个月前，薛定谔的那次来访还历历在目，整个哥本哈根学派为了应付这场硬仗，花了好些时间去钻研他的波动力学理论，但现在，玻尔突然觉得，这个波动理论非常出色啊。它简洁，明确，看起来并不那么坏。在写给赫维西（Hevesy）的信里，玻尔已经把它称作“一个美妙的理论”。尤其是有了波恩的概率解释之后，玻尔已经毫不犹豫地准备接受这一理论并把它当作量子论的基础了。 嗯，波动，波动。玻尔知道，海森堡现在对于这个词简直是条件反射似地厌恶。在他的眼里只有矩阵数学，谁要是跟他提起薛定谔的波他准得和谁急，连玻尔本人也不例外。事实上，由于玻尔态度的转变，使得向来亲密无间的哥本哈根派内部第一次产生了裂痕。海森堡……他在得知玻尔的意见后简直不敢相信自己的耳朵。现在，气氛已经闹得够僵了，玻尔为了不让事态恶化，准备离开丹麦去挪威度个长假。过去的1926年就是在无尽的争吵中度过的，那一整年玻尔只发表了一篇关于自旋的小文章，是时候停止争论了。 但是，粒子？波？那个想法始终在他脑中缠绕不去。 进来一个人，是他的另一位助手奥斯卡?克莱恩（Oskar Klein）。在过去的一年里他的成就斐然，他不仅成功地把薛定谔方程相对论化了，还在其中引进了“第五维度”的思想，这得到了老洛伦兹的热情赞扬。不管怎么说，他可算哥本哈根最熟悉量子波动理论的人之一了。有他助阵，玻尔更加相信，海森堡实在是持有一种偏见，波动理论是不可偏废的。 “要统一，要统一。”玻尔喃喃地说。克莱恩抬起头来看他：“您对波动理论是怎么想的呢？” “波，电子无疑是个波。”玻尔肯定地说。 “哦，那样说来……” “但是，”玻尔打断他，“它同时又不是个波。从BKS倒台以来，我就隐约地猜到了。” 克莱恩笑了：“您打算发表这一观点吗？” “不，还不是时候。” "why?" 玻尔叹了一口气：“克莱恩，我们的对手非常强大……非常强大，我还没有准备好……” （注：老的说法认为，互补原理只有在不确定原理提出后才成型。但现在学者们都同意，这一思想有着复杂的来源，为了把重头戏留给下一章，我在这里先带一笔波粒问题。）