Chapter 7 Chapter 6 Unification

one When the young and energetic Heisenberg was overcoming obstacles in Göttingen, Erwin.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 dynamics of solids and crystals. He was offered a contract for one year as a professor of physics.And from 1924, Schrödinger became interested in quantum mechanics and statistical theory, so he 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 also recall 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 original paper with the utmost interest and finally mastered it.I first learned about it from section 8 of your second paper on degenerate gases.The idea that every particle is regarded as a wave-like idea was extremely fascinating to Schrödinger, and he soon applied this theory in the statistical mechanics of gases, and published a paper entitled "On Einstein's Theory of Gases". paper.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.As always, Schrodinger came to the place he used to go to: Arosa (Arosa) at an altitude of 1,700 meters.Since he and Anne Marie.After Annemarie Bertel got married in 1920, the two often came here for vacation.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 in an astonishingly creative and insightful state for the next twelve 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 the assumption of a discrete energy level, and Heisenberg used his huge matrix to derive this result after complicated 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 Schrödinger wave function famous throughout the history of physics in the twentieth 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.The triangle △ is called the Laplace operator, which represents some kind of differential operation. h is the well-known Planck constant. E is the total energy of the system, and 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)=, the answer will be a set of values, x can be, 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. Think of 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 twenty centimeters, then obviously the length of the string can only be twenty centimeters, forty centimeters, sixty centimeters and not fifty 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 orbit of an electron, 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 the eigenfunction (Eigenfunction), and the discrete solution obtained is called the 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, thus completely establishing another new Mechanics 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 internal.

As soon as Schrödinger's equation was published, physicists almost all over the world cheered for it.Planck called it an epoch-making work, and Einstein said: Your idea comes from a real genius.Your quantum equation has already taken a decisive step.Ehrenfest said: I am fascinated by your theory and the new ideas it brings.For the past two weeks, our group has spent hours each day at the blackboard trying to make sense of it from all angles.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 implications 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. The German word Mist is basically equivalent to bullshit or crap in English. Schrödinger was also unceremonious. In his paper, he said: My theory was inspired by de Broglie and I don't know of any inheritance from Heisenberg.Of course I know Heisenberg's theory, which is a very difficult method of super algebra without visualization.I'm at least dismayed, if not entirely dismissed, by this 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 power behind them, the Pope behind the scenes, is undoubtedly the great Niels Einstein in Copenhagen.Bohr.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 already been connected together under 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. Waves, and waves alone, are the only reality.Schrödinger affirmed that whether it is electrons, photons, or any particles, they are just bubbles on the surface of waves.They are all waves in essence, and the basic motion can be expressed by the wave equation. Absolutely beg to differ.Heisenberg countered that the fundamental 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 experiments, from photoelectric phenomena, to the jumping of electrons in atoms between energy levels, it is irrefutable that nature is discontinuous.Your wave equation is certainly a welcome achievement in mathematics, but we have to realize that we can't understand it in the traditional way. It doesn't mean that. Quite the opposite.That's what Schrödinger said.The wave function ψ (pronounced psai) is continuous in all directions and can be thought of as a kind of vibration.In fact, we have to imagine the electron as a resident eigenvibration, and the so-called transition of the electron is just a change in its vibration mode.There are no orbits, no energy levels, just waves. Ha ha.Heisenberg laughed and said, I'm afraid you don't understand what your own ψ is, do you?It is just a virtual function in a certain virtual space, but you insist on imagining it as a real wave.In fact, we must not be misled by everyday visualizations. After all, you cannot deny the behavior of electrons as classical particles. That's right.Schrödinger still refuses to show weakness, and I don't deny that it does exhibit particle-like behavior.However, just like a coconut, if you knock open the hard shell of its particles, you will find that there is still a soft sap inside.Electrons are undoubtedly composed of sine waves, but this wave does not stretch very much on all scales and can be regarded as a wave packet.When this wave packet travels as a whole, it looks like a particle.However, in essence, it is still a wave, and particles are just a derivative of waves. 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 was sitting below. Opposition attitude.Earlier, Kramer, Bohr's original assistant, accepted the offer from Utrecht University and left Copenhagen, so Heisenberg became the successor in this position. Now he can work in Bohr's university as he dreamed Worked around.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 picked him up at the train station.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 affable person on weekdays, once he was involved in this kind of physical debate, he looked like a human being. 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, the waves and the particles, the two old enemies for three hundred years, 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 play "Schrodinger's Girlfriend" (Schrodinger's Girlfriend) premiered at the famous Fort Mason Center in San Francisco.The comedy explores the relationship between love, sex, and quantum physics, set in the history of Schrödinger's founding of wave mechanics in 1926 by the company of his mysterious girlfriend in Arosa, and was generally well-received by critics .At the beginning of this year (2003), this 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 did not make much money at the time). Before he met Anne Marie, Schrödinger fell in love with four young girls in total , and mostly 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, also had a relationship with one of Schrödinger's best friends, Hermann Schödinger.Will (Hermann Weyl) maintains an affair (Will's own wife is infatuated with another guy, which is really 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, according to some popular parlance, was one of those amorous 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: the joys, sorrows, sorrows and joys of the past forty-one years have bound us together tightly, 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, and 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.Schrodinger also stopped breathing. three In 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, 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 quality 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.When everyone took a closer look, the big box seemed to be 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. Well, but I can't see anything, so how do I guess?people complained.Of course, of course.The host quickly said, we are not imitating Sun Wukong to play with partitions and guess objects, and besides, it is definitely not a tattered clock, but it is a real treasure related to the whole physics.Well, that's right, although we can't 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.The following immediately chattered: energy?frequency?speed?distance?time?charge?quality?The host had to raise his voice and shout: Be quiet, be quiet, we've just started, don't make random guesses.From now on, whoever guesses wrong will be disqualified.So there was an instant silence. good.The host said with satisfaction, so 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, it can be seen from the construction of ψ that this is a dimensionless function.But it also has some connection with the position of the electron, and for each electron, it expands in a virtual three-dimensional space.Speaking of which, many people are already confused, only a few with particularly quick thinking are still thinking nervously. All in all, ψ accompanies every electron like a shadow, and spreads out like a cloud in its place.The cloud is sometimes thick and sometimes thin, but it evolves in a certain way.And, I repeat, this diffusion and its evolution are classical, continuous, and definite.So everyone fell into deep thought, without a clue. Yes, clouds, what a metaphor.At this time, a man with a thin face and pince-nez stood up with a smile and said.The host quickly 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 with bated breath to what he had to say. Well, things are already obvious, ψ is a spatial distribution function.Schrödinger said with confidence that when it is multiplied by the charge of an electron, it represents the actual distribution of the charge in space.Clouds, dear ones, an electron is not a particle, it is a mass of waves that spread out around space like a cloud.Our wave function describes exactly this extension and its behavior.The electron has no specific location, nor does it have a specific path, because it is a cloud, a wave, which extends in every direction although it decays quickly, which makes it roughly look like a particle.Ladies and gentlemen, I think the biggest significance of this discovery is that we have to get rid of all false images about particles, whether it is electrons, photons, or whatever, they are not the kind Particles in the traditional sense.Pull them out to magnify them, examine them closely, and you'll see it melt away in space, becoming a superposition of countless vibrations.Yes, an electron, it is smeared, like butter on bread, it is usually curled up so tightly that we all regard it as a small ball, but this has been proved by our wave function ψ that it is not real.After years of physics going astray, our minds are cluttered with spectral lines, transitions, energy levels, matrices, and all that weird stuff, and now, it's time to go back 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 go on a fork where we cannot turn back.Physics needs to be reformed, but not the chaos of thought. We've heard enough of grotesque stories about electrons jumping around like fleas in atoms, like a drunk who can't foresee his direction at all.There is also the so-called matrix that is so mysterious. No one knows what physical meaning it contains, but it keeps clamoring that it is the orthodoxy of physics.No, let us now return to the solid ground, the land where the giants once fought, the land where such majestic structures were once built, the land that is full of pride and glorious history.Conciseness, clarity, elegance, intuition, continuity, and visualization are the sticks of victory in the kingdom of physics, passed down from generation to generation, leading us to victory.I have no doubts that the new mechanics will be made on a continuous wave basis, bringing everything down to simple graphics and carrying on the lineage of the old royal family.This is by no means conservative, because this bloodline is also the soul that has carried modern science for three hundred years.It is a symbol of physics whose sanctity 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 and said, 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.Schrödinger nodded, feeling a little more relaxed.But then Bonn said, can I ask you a question?While this is what you found, have you ever actually opened the box yourself to see what's inside? This made Schrödinger very embarrassed, and he hesitated for a while before answering: To be honest, I haven't really seen what's inside, because I don't have the key to the box.Everyone was surprised. If that's the case, Bonn said cautiously, I think I don't agree with your guess just now. oh?兩個人對視了一陣，薛定諤終於開口說：那麼您以為，這裡面究竟是什麼東西呢？ 毫無疑問，波恩凝視著那雕滿了古典花紋的箱子和它上面那把沉重的大鎖，這裡面藏著一些至關緊要的事物，它的力量足以改變整個物理學的面貌。但是，我也有一種預感，這股束縛著的力量是如此強大，它將把物理學搞得天翻地覆。當然，你也可以換個詞語說，為物理學帶來無邊的混亂。 Oh, is it so?薛定諤驚奇地說，照這麼說來，難道它是潘朵拉的盒子？ Um.波恩點了點頭，人們將陷入困惑和爭論中，物理學會變成一個難以理解的奇幻世界。老實說，雖然我隱約猜到了裡面是什麼，我還是不能確定該不該把它說出來。 薛定諤盯著波恩：我們都相信科學的力量，在於它敢於直視一切事實，並毫不猶豫地去面對它，檢驗它，把握它，不管它是什麼。何況，就算是潘朵拉盒子，我們至少也還擁有盒底那最寶貴的東西，難道你忘了嗎？ 是的，那是希望。波恩長出了一口氣，你說的對，不管是禍是福，我們至少還擁有希望。只有存在爭論，物理學才擁有未來。 那麼，你說這箱子裡是？全場一片靜默，人人都不敢出聲。 波恩突然神秘地笑了：我猜，這裡面藏的是 骰子。 Four 骰子？骰子是什麼東西？它應該出現在大富翁遊戲裡，應該出現在澳門和拉斯維加斯的賭場中，但是，物理學？不，那不是它應該來的地方。骰子代表了投機，代表了不確定，而物理學不是一門最嚴格最精密，最不能容忍不確定的科學嗎？ 可以想像，當波恩於一九二六年七月將骰子帶進物理學後，是引起了何等的軒然大波。圍繞著這個核心解釋所展開的爭論激烈而尖銳，把物理學加熱到了沸點。這個話題是如此具有爭議性，很快就要引發二十世紀物理史上最有名的一場大論戰，而可憐的波恩一直要到整整二十八年後，才因為這一傑出的發現而獲得諾貝爾獎金比他的學生們晚上許多。 不管怎麼樣，我們還是先來看看波恩都說了些什麼。骰子，這才是薛定諤波函數ψ的解釋，它代表的是一種隨機，一種概率，而決不是薛定諤本人所理解的，是電子電荷在空間中的實際分佈。波恩爭辯道，ψ,或者更準確一點，ψ的平方，代表了電子在某個地點出現的概率。電子本身不會像波那樣擴展開去，但是它的出現概率則像一個波，嚴格地按照ψ的分佈所展開。 我們來回憶一下電子或者光子的雙縫干涉實驗，這是電子波動性的最好證明。當電子穿過兩道狹縫後，便在感應屏上組成了一個明暗相間的圖案，展示了波峰和波谷的相互增強和抵消。但是，正如粒子派指出的那樣，每次電子只會在屏上打出一個小點，只有當成群的電子穿過雙縫後，才會逐漸組成整個圖案。 現在讓我們來做一個思維實驗，想像我們有一台儀器，它每次只發射出一個電子。這個電子穿過雙縫，打到感光屏上，激發出一個小亮點。那麼，對於這一個電子，我們可以說些什麼呢？很明顯，我們不能預言它組成類波的干涉條紋，因為一個電子只會留下一個點而已。事實上，對於這個電子將會出現在螢幕上的什麼地方，我們是一點頭緒都沒有的，多次重複我們的實驗，它有時出現在這裡，有時出現在那裡，完全不是一個確定的過程。 不過，我們經過大量的觀察，卻可以發現，這個電子不是完全沒有規律的：它在某些地方出現的可能性要大一些，在另一些地方則小一些。它出現頻率高的地方，恰恰是波動所預言的干涉條紋的亮處，它出現頻率低的地方則對應於暗處。現在我們可以理解為什麼大量電子能組成干涉條紋了，因為雖然每一個電子的行為都是隨機的，但這個隨機分佈的總的模式卻是確定的，它就是一個干涉條紋的圖案。這就像我們擲骰子，雖然每一個骰子擲下去，它的結果都是完全隨機的，從一到六都有可能，但如果你投擲大量的骰子到地下，然後數一數每個點的數量，你會發現一到六的結果差不多是平均的。 關鍵是，單個電子總是以一個點的面貌出現，它從來不會像薛定諤所說的那樣，在螢幕上打出一灘圖案來。只有大量電子接二連三地跟進，總的干涉圖案才會逐漸出現。其中亮的地方也就是比較多的電子打中的地方，換句話說，就是單個電子比較容易出現的地方，暗的地帶則正好相反。如果我們發現，有九成的粒子聚集在亮帶，只有一成的粒子在暗帶，那麼我們就可以預言，對於單個粒子來說，它有九十％的可能出現在亮帶的區域，十％的可能出現在暗帶。但是，究竟出現在哪裡，我們是無法確定的，我們只能預言概率而已。 我們只能預言概率而已。 但是，等等，我們怎麼敢隨便說出這種話來呢？這不是對於古老的物理學的一種大不敬嗎？從伽利略牛頓以來，成千上百的先輩們為這門科學嘔心瀝血，建築起了這樣宏偉的構築，它的力量統治整個宇宙，從最大的星系到最小的原子，萬事萬物都在它的威力下必恭必敬地運轉。任何巨大的或者細微的動作都逃不出它的力量。星系之間產生可怕的碰撞，釋放出難以想像的光和熱，並誕生數以億計的新恒星；宇宙射線以驚人的高速穿越遙遠的空間，見證亙古的時光；微小得看不見的分子們你推我搡，喧鬧不停；地球莊嚴地圍繞著太陽運轉，它自己的自轉軸同時以難以覺察的速度輕微地振動；堅硬的岩石隨著時光流逝而逐漸風化；鳥兒撲動它的翅膀，借著氣流一飛沖天。這一切的一切，不都是在物理定律的監視下一絲不苟地進行的嗎？ 更重要的是，物理學不僅能夠解釋過去和現在，它還能預言未來。我們的定律和方程能夠毫不含糊地預測一顆炮彈的軌跡以及它降落的地點；我們能預言幾千年後的日食，時刻準確到秒；給我一張電路圖，多複雜都行，我能夠說出它將做些什麼；我們製造的機器乖乖地按照我們預先制定好的計畫運行。事實上，對於任何一個系統，只要給我足夠的初始資訊，賦予我足夠的運算能力，我能夠推算出這個體系的一切歷史，從它最初怎樣開始運行，一直到它在遙遠的未來的命運，一切都不是秘密。是的，一切系統，哪怕骰子也一樣。告訴我骰子的大小，品質，質地，初速度，高度，角度，空氣阻力，桌子的質地，摩擦係數，告訴我一切所需要的情報，那麼，只要我擁有足夠的運算能力，我可以毫不遲疑地預先告訴你，這個骰子將會擲出幾點來。 物理學統治整個宇宙，它的過去和未來，一切都盡在掌握。這已經成了物理學家心中深深的信仰。十九世紀初，法國的大科學家拉普拉斯（Pierre Simon de Laplace）在用牛頓方程計算出了行星軌道後，把它展示給拿破崙看。拿破崙問道：在你的理論中，上帝在哪兒呢？拉普拉斯平靜地回答：陛下，我的理論不需要這個假設。 是啊，上帝在物理學中能有什麼位置呢？一切都是由物理定律來統治的，每一個分子都遵照物理定律來運行，如果說上帝有什麼作用的話，他最多是在一開始推動了這個體系一下，讓它得以開始運轉罷了。在之後的漫長歷史中，有沒有上帝都是無關緊要的了，上帝被物理學趕出了舞臺。 我不需要上帝這個假設。拉普拉斯站在拿破崙面前說。這可算科學最光榮最輝煌的時刻之一了，它把無邊的自豪和驕傲播撒到每一個科學家的心中。不僅不需要上帝，拉普拉斯想像，假如我們有一個妖精，一個大智者，或者任何擁有足夠智慧的人物，假如他能夠瞭解在某一刻，這個宇宙所有分子的運動情況的話，那麼他就可以從正反兩個方向推演，從而得出宇宙在任意時刻的狀態。對於這樣的智者來說，沒有什麼過去和未來的分別，一切都歷歷在目。宇宙從它出生的那一剎那開始，就墜入了一個預定的軌道，它嚴格地按照物理定律發展，沒有任何岔路可以走，一直到遇見它那註定的命運為止。就像你出手投籃，那麼，這究竟是一個三分球，還是打中籃筐彈出，或者是一個air ball，這都在你出手的一剎那決定了，之後我們所能做的，就是看著它按照寫好的劇本發展而已。 是的，科學家知道過去；是的，科學家明白現在；是的，科學家瞭解未來。只要掌握了定律，只要搜集足夠多的情報，只要能夠處理足夠大的運算量，科學家就能如同上帝一般無所不知。整個宇宙只不過是一台精密的機器，它的每個零件都按照定律一絲不苟地運行，這種想法就是古典的，嚴格的決定論（determinism）。宇宙從出生的那一剎那起，就有一個確定的命運。我們現在無法瞭解它，只是因為我們所知道的資訊太少而已。 那麼多的天才前仆後繼，那麼多的偉人嘔心瀝血，那麼多在黑暗中的探索，掙扎，奮鬥，這才凝結成物理學在十九世紀黃金時代的全部光榮。物理學家終於可以說，他們能夠預測神秘的宇宙了，因為他們找到了宇宙運行的奧秘。他們說這話時，帶著一種神聖而不可侵犯的情感，決不饒恕任何敢於輕視物理學力量的人。 可是，現在有人說，物理不能預測電子的行為，它只能找到電子出現的概率而已。無論如何，我們也沒辦法確定單個電子究竟會出現在什麼地方，我們只能猜想，電子有九十％的可能出現在這裡，十％的可能出現在那裡。這難道不是對整個物理歷史的挑釁，對物理學的光榮和尊嚴的一種侮辱嗎？ 我們不能確定？物理學的詞典裡是沒有這個字眼的。在中學的物理考試中，題目給了我們一個小球的初始參數，要求t時刻的狀態，你敢寫上我不能確定嗎？要是你這樣做了，你的物理老師準會氣得吹鬍子瞪眼睛，並且毫不猶豫地給你亮個紅燈。不能確定？不可能，物理學什麼都能確定。誠然，有時候為了方便，我們也會引進一些統計的方法，比如處理大量的空氣分子運動時，但那是完全不同的一個問題。科學家只是凡人，無法處理那樣多的複雜計算，所以應用了統計的捷徑。但是從理論上來說，只要我們瞭解每一個分子的狀態，我們完全可以嚴格地推斷出整個系統的行為，分毫不爽。 然而波恩的解釋不是這樣，波恩的意思是，就算我們把電子的初始狀態測量得精確無比，就算我們擁有最強大的電腦可以計算一切環境對電子的影響，即便如此，我們也不能預言電子最後的準確位置。這種不確定不是因為我們的計算能力不足而引起的，它是深藏在物理定律本身內部的一種屬性。即使從理論上來說，我們也不能準確地預測大自然。這已經不是推翻某個理論的問題，這是對整個決定論系統的挑戰，而決定論是那時整個科學的基礎。量子論挑戰整個科學。 波恩在論文裡寫道：這裡出現的是整個決定論的問題了。（Hier erhebt sich der ganze Problematik des Determinismus.） 對於許多物理學家來說，這是一個不可原諒的假設。骰子？不確定？Do not make jokes.對於他們中的好些人來說，物理學之所以那樣迷人，那樣富有魔力，正是因為它深刻，明晰，能夠確定一切，掃清人們的一切疑惑，這才使他們義無反顧地投身到這一事業中去。現在，物理學竟然有變成搖獎機器的危險，竟然要變成一個擲骰子來決定命運的賭徒，這怎麼能夠容忍呢？ 不確定？ 一場史無前例的大爭論即將展開，在爭吵和辯論後面是激動，顫抖，絕望，淚水，伴隨著整個決定論在二十世紀的悲壯謝幕。 飯後閒話：決定論 可以說決定論的興衰濃縮了整部自然科學在二十世紀的發展史。科學從牛頓和拉普拉斯的時代走來，輝煌的成功使它一時得意忘形，認為它具有預測一切的能力。決定論認為，萬物都已經由物理定律所規定下來，連一個細節都不能更改。過去和未來都像已經寫好的劇本，宇宙的發展只能嚴格地按照這個劇本進行，無法跳出這個窠臼。 矜持的決定論在二十世紀首先遭到了量子論的嚴重挑戰，隨後混沌動力學的興起使它徹底被打垮。現在我們已經知道，即使沒有量子論把概率這一基本屬性賦予自然界，就牛頓方程本身來說，許多系統也是極不穩定的，任何細小的干擾都能夠對系統的發展造成極大的影響，差之毫釐，失之千里。這些干擾從本質上說是不可預測的，因此想憑藉牛頓方程來預測整個系統從理論上說也是不可行的。典型的例子是長期的天氣預報，大家可能都已經聽說過洛倫茲著名的蝴蝶效應，哪怕一隻蝴蝶輕微地扇動它的翅膀，也能給整個天氣系統造成戲劇性的變化。現在的天氣預報也已經普遍改用概率性的說法，比如明天的降水概率是二十％。 一九八六年，著名的流體力學權威，詹姆士．萊特希爾爵士（Sir James Lighthill，他於一九六九年從狄拉克手裡接過劍橋盧卡薩教授的席位，也就是牛頓曾擔任過的那個）於皇家學會紀念牛頓《原理》發表三百周年的集會上發表了轟動一時的道歉： 現在我們都深深意識到，我們的前輩對牛頓力學的驚人成就是那樣崇拜，這使他們把它總結成一種可預言的系統。而且說實話，我們在一九六○年以前也大都傾向於相信這個說法，但現在我們知道這是錯誤的。我們以前曾經誤導了公眾，向他們宣傳說滿足牛頓運動定律的系統是決定論的，但是這在一九六○年後已被證明不是真的。我們都願意在此向公眾表示道歉。 （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 Newton's laws of motion that,after 1960, were to be proved incorrect.） 決定論的垮臺是否註定了自由意志的興起？這在哲學上是很值得探討的。事實上，在量子論之後，物理學越來越陷於形而上學的爭論中。也許形而上學（metaphysics）應該改個名字叫量子論之後（metaquantum）。在我們的史話後面，我們會詳細地探討這些問題。 Ian Stewart寫過一本關於混沌的書，書名也叫《上帝擲骰子嗎》。這本書文字優美，很值得一讀，當然和我們的史話沒什麼聯繫。我用這個名字，一方面是想強調決定論的興衰是我們史話的中心話題，另外，畢竟愛因斯坦這句名言本來的版權是屬於量子論的。 five 在我們出發去回顧新量子論與經典決定論的那場驚心動魄的悲壯決戰之前，在本章的最後還是讓我們先來關注一下歷史遺留問題，也就是我們的微粒和波動的宿怨。波恩的概率解釋無疑是對薛定諤傳統波動解釋的一個沉重打擊，現在，微粒似乎可以暫時高興一下了。 看，它嘲笑對手說，薛定諤也救不了你，他對波函數的解釋是站不住腳的。難怪總是有人說，薛定諤的方程比薛定諤本人還聰明哪。波恩的概率才是有道理的，電子始終是一個電子，任何時候你觀察它，它都是一個粒子，你吵嚷多年的所謂波，原來只是那看不見摸不著的概率罷了。哈哈，把這個頭銜讓給你，我倒是毫無異議的，但你得首先承認我的正統地位。 但是波動沒有被嚇倒，說實話，雙方三百年的恩怨纏結，經過那麼多風風雨雨，早就練就了處變不驚的本領。Oh, is it so?它冷靜地回應道，恐怕事情不如你想像得那麼簡單吧？我們不如縮小到電子那個尺寸，去親身感受一下一個電子在雙縫實驗中的經歷如何？ 微粒遲疑了一下便接受了：好吧，讓你徹底死心也好。 那麼，現在讓我們也想像自己縮小到電子那個尺寸，跟著它一起去看看事實上到底發生了什麼事。一個電子的直徑小於一億分之一埃，也就是10^-23米，它的品質小於10^-30千克，變得這樣小，看來這必定是一次奇妙的旅程呢。 好，現在我們已經和一個電子一樣大了，突然縮小了那麼多，還真有點不適應，看出去的世界也變得模糊扭曲起來。不過，我們第一次發現，世界原來那麼空曠，幾乎是空無一物，這也情有可原，從我們的尺度看來，原子核應該像是遠在天邊吧？好，現在迎面來了一個電子，這是個好機會，讓我們睜大眼睛，仔細地看一看它究竟是個粒子還是波？奇怪，為什麼我們什麼都看不見呢？啊，原來我們忘了一個關鍵的事實！ 要看見東西，必須有光進入我們的眼睛才行。但現在我們變得這麼小，即使光不管它是光子還是光波對於我們來說也太大了。但是不管怎樣，為了探明這個秘密，我們必須得找到從電子那裡反射過來的光，憑感覺，我知道從左邊來了一團光（之所以說一團光，是因為我不清楚它究竟是一個光粒子還是一道光波，沒有光，我也看不到光本身，是吧？），現在讓我們勇敢地迎上去，啊，秘密就要揭開了！ 隨著砰地一聲，我們被這團光粗暴地擊中，隨後身不由己地飛到半空中，被彈出了十萬八千里。這次撞擊使得我們渾身筋骨欲脫，腦中天旋地轉，眼前直冒金星。我們忘了自己現在是個什麼尺寸！要不是運氣好，這次碰撞已經要了咱們的小命。當好不容易爬起來時，早就不知道自己身在何方，那個電子更是無影無蹤了。 剛才真是好險，看來這一招是行不通的。不過，我聽見聲音了，是微粒和波動在前面爭論呢，咱們還是跟著這哥倆去看個究竟。它們為了類比一個電子的歷程，從某個陰極射線管出發，現在，面前就是那著名的雙縫了。 嗨，微粒。波動說道，假如電子是個粒子的話，它下一步該怎樣行動呢？眼前有兩條縫，它只能選擇其中之一啊，如果它是個粒子，它不可能兩條縫都通過吧？ Well, that's right.微粒說，粒子就是一個小點，是不可分割的。我想，電子必定選擇通過了其中的某一條狹縫，然後投射到後面的光屏上，激發出一個小點。 可是，波動一針見血地說，它怎能夠按照干涉模式的概率來行動呢？比如說它從右邊那條縫過去了吧，當它打到螢幕前，它怎麼能夠知道，它應該有九十％的機會出現到亮帶區，十％的機會留給暗帶區呢？要知道這個干涉條紋可是和兩條狹縫之間的距離密切相關啊，要是電子只通過了一條縫，它是如何得知兩條縫之間的距離的呢？ 微粒有點尷尬，它遲疑地說：我也承認，伴隨著一個電子的有某種類波的東西，也就是薛定諤的波函數ψ，波恩說它是概率，我們就假設它是某種看不見的概率波吧。你可以把它想像成從我身上散發出去的某種看不見的場，我想，在我通過雙縫之前，這種看不見的波場在空間中彌漫開去，探測到了雙縫之間的距離，從而使我得以知道如何嚴格地按照概率行動。但是，我的實體必定只能通過其中的一條縫。 There is no reason at all.波動搖頭說，我們不妨想像這樣一個情景吧，假如電子是一個粒子，它現在決定通過右邊的那條狹縫。姑且相信你的說法，有某種概率波事先探測到了雙縫間的距離，讓它胸有成竹知道如何行動。可是，假如在它進入右邊狹縫前的那一剎那，有人關閉了另一道狹縫，也就是左邊的那道狹縫，那時會發生什麼情形呢？ 微粒有點臉色發白。 那時候，波動繼續說，就沒有雙縫了，只有單縫。電子穿過一條縫，就無所謂什麼干涉條紋。也就是說，當左邊狹縫關閉的一剎那，電子的概率必須立刻從干涉模式轉換成普通模式，變成一條長狹帶。 現在，我倒請問，電子是如何在穿過狹縫前的一剎那，及時地得知另一條狹縫關閉這個事實的呢？要知道它可是一個小得不能再小的電子啊，另一條狹縫距離它是如此遙遠，就像從上海隔著大洋遙望洛杉磯。它如何能夠瞬間作出反應，修改自己的概率分佈呢？除非它收到了某種暫態傳播來的信號，怎麼，你想開始反對相對論了嗎？ 好吧，微粒不服氣地說，那麼，我倒想聽聽你的解釋。 很簡單，波動說，電子是一個在空間中擴散開去的波，它同時穿過了兩條狹縫，當然，這也就是它造成完美干涉的原因了。如果你關閉一個狹縫，那麼顯然就關閉了一部分波的路徑，這時就談不上干涉了。 Sounds good.微粒說，照你這麼說，ψ是某種實際的波，它穿過兩道狹縫，完全確定而連續地分佈著，一直到擊中感應屏前。不過，之後呢？之後發生了什麼事？ 之後波動也有點語塞，之後，出於某種原因，ψ收縮成了一個小點。 哈，真奇妙。微粒故意把聲音拉長以示諷刺，你那擴散而連續的波突然變成了一個小點！請問發生了什麼事呢？波動家族突然全體罷工了？ 波動氣得面紅耳赤，它爭辯道：出於某種我們尚不清楚的機制 好吧，微粒不耐煩地說，實踐是檢驗真理的唯一標準是吧？既然我說電子只通過了一條狹縫，而你硬說它同時通過兩條狹縫，那麼搞清我們倆誰對誰錯不是很簡單嗎？我們只要在兩道狹縫處都安裝上某種儀器，讓它在有粒子或者波，不論是什麼通過時記錄下來或者發出警報，那不就成了？這種儀器又不是複雜而不可製造的。 波動用一種奇怪的眼光看著微粒，良久，它終於說：不錯，我們可以裝上這種儀器。我承認，一旦我們試圖測定電子究竟通過了哪條縫時，我們永遠只會在其中的一處發現電子。兩個儀器不會同時響。 微粒放聲大笑：你早說不就得了？害得我們白費了這麼多口水！怎麼，這不就證明了，電子只可能是一個粒子，它每次只能通過一條狹縫嗎？你還跟我嘮叨個什麼！但是它漸漸發現氣氛有點不對勁，終於它笑不出來了。 how?它瞪著波動說。 波動突然咧嘴一笑：不錯，每次我們只能在一條縫上測量到電子。但是，你要知道，一旦我們展開這種測量的時候，干涉條紋也就消失了 時間是一九二七年二月，哥本哈根仍然是春寒料峭，大地一片冰霜。玻爾坐在他的辦公室裡若有所思：粒子還是波呢？五個月前，薛定諤的那次來訪還歷歷在目，整個哥本哈根學派為了應付這場硬仗，花了好些時間去鑽研他的波動力學理論，但現在，玻爾突然覺得，這個波動理論非常出色啊。它簡潔，明確，看起來並不那麼壞。在寫給赫維西（Hevesy）的信裡，玻爾已經把它稱作一個美妙的理論。尤其是有了波恩的概率解釋之後，玻爾已經毫不猶豫地準備接受這一理論並把它當作量子論的基礎了。 嗯，波動，波動。玻爾知道，海森堡現在對於這個詞簡直是條件反射似地厭惡。在他的眼裡只有矩陣數學，誰要是跟他提起薛定諤的波他準得和誰急，連玻爾本人也不例外。事實上，由於玻爾態度的轉變，使得向來親密無間的哥本哈根派內部第一次產生了裂痕。海森堡他在得知玻爾的意見後簡直不敢相信自己的耳朵。現在，氣氛已經鬧得夠僵了，玻爾為了不讓事態惡化，準備離開丹麥去挪威度個長假。過去的一九二六年就是在無盡的爭吵中度過的，那一整年玻爾只發表了一篇關於自旋的小文章，是時候停止爭論了。 但是，粒子？波？那個想法始終在他腦中纏繞不去。 進來一個人，是他的另一位助手奧斯卡．克萊恩（Oskar Klein）。在過去的一年裡他的成就斐然，他不僅成功地把薛定諤方程相對論化了，還在其中引進了第五維度的思想，這得到了老洛倫茲的熱情讚揚。不管怎麼說，他可算哥本哈根最熟悉量子波動理論的人之一了。有他助陣，玻爾更加相信，海森堡實在是持有一種偏見，波動理論是不可偏廢的。 要統一，要統一。玻爾喃喃地說。克萊恩抬起頭來看他：您對波動理論是怎麼想的呢？ 波，電子無疑是個波。玻爾肯定地說。 哦，那樣說來 但是，玻爾打斷他，它同時又不是個波。從BKS倒臺以來，我就隱約地猜到了。 克萊恩笑了：您打算發表這一觀點嗎？ 不，還不是時候。 Why? 玻爾嘆了一口氣：克萊恩，我們的對手非常強大非常強大，我還沒有準備好 （注：老的說法認為，互補原理只有在不確定原理提出後才成型。但現在學者們都同意，這一思想有著複雜的來源，為了把重頭戲留給下一章，我在這裡先帶一筆波粒問題。）