Chapter 8 Chapter 7 Uncertainty

one Our history has come to this point, it is time to look back at the journey we have traveled.We have seen how the magnificent classical physics building suddenly collapsed, and we have seen how Planck's quantum hypothesis ignited the spark of a new revolution with the black body problem as the guide.After that, Einstein's light quantum theory endowed the newborn quantum with substantial power, allowing it to stand up for the first time to stand out from the crowd, and Bohr's atomic theory created a new world with the help of its infinite energy. Come. We have also mentioned how the two theories of particle and wave have been in constant confrontation about the nature of light since three hundred years ago.Starting from de Broglie, this essential contradiction has become the basic problem of physics, and Heisenberg created his matrix mechanics from discontinuity, and Schrödinger also discovered his wave along another continuous path equation.Although these two theories have been proved to be equivalent by mathematics, their physical meaning has aroused widespread debate. Bonn's probability explanation has pushed the determinism of hundreds of years onto the stage of doubt and has become the focus of the wave. .On the other hand, the war between fluctuations and particles has now reached the most critical time.

Next, some really weird things are going to happen in physics.It will transform people's philosophy into a specious madness, and turn physics itself into a maelstrom.The most famous debate of the twentieth century is about to unfold, with repercussions that continue to this day.We've come a long way, we're all exhausted and weary, but we can't turn around.Looking back, the white clouds blocked the way home, and it was impossible to return to the warm comfort of the classical theory. Before our eyes, there was only a long and rugged road leading to a distant and unknown place.Now, let us muster up the greatest courage and follow the physicists to go on and see what kind of scene is hidden at the end of this road.

Here we are back to that magical winter of February 1927.The past few months have been a nightmare for Heisenberg, as more and more people turn to Schrödinger and his damned wave theory, forgetting about his matrix.Heisenberg's original excellent papers are now being rewritten into alternative forms of the wave equation, which makes him especially intolerable.He later wrote to Pauli: For every paper on matrices, people rewrite it into conjugate wave form, which annoys me very much.I think they'd better learn both ways. But what saddened him the most was undoubtedly that Bohr also turned to his opposite.Bohr, the Bohr who he regarded as a strict teacher, loving father, and good friend, who they called the Pope of Quantum Theory behind his back, the commander-in-chief and spiritual leader of the Copenhagen Legion, actually opposed him now!This made Heisenberg feel extremely wronged and sad.Later, when Bohr criticized his theory again, Heisenberg actually cried tears.For Heisenberg, Bohr's position in his mind was unique. Without his support, Heisenberg felt like a child swimming in a river without an adult's arms, feeling isolated and helpless.

However, now that Bohr has gone to Norway for vacation, he is probably skiing?Remembering Bohr's poor skiing skills, Heisenberg couldn't help smiling.Bohr could no longer provide any help, and now he and Klein huddled together, concentrating on studying relativistic fluctuations.fluctuation!Heisenberg snorted, even if he killed him, he would not admit it. Electrons should be interpreted as fluctuations.But things weren't so bad, he still had at least a few comrades in arms: his old friend Pauli, Jordan of Göttingen, and Dirac, who was now visiting Copenhagen. Not long ago, Dirac and Jordan developed a transformation theory respectively, which made it easy for Heisenberg to use matrices to deal with some probability problems that had been dealt with by Schrödinger's equation.To Heisenberg's delight, discontinuity was taken as a basis in Dirac's theory, which further convinced him that Schrödinger's explanation was dubious.However, if discontinuity is the premise, some variables in this system are difficult to explain. For example, the trajectory of an electron is always continuous, right?

Heisenberg tried his best to recall the history of the creation of matrix mechanics, trying to see where the problem was.We still remember that Heisenberg's assumption at that time was: the entire physical theory can only be premised on observable quantities, and only these variables are definite and can form the basis of any system.But Heisenberg also remembers that Einstein didn't quite agree with this. He was too heavily influenced by classical philosophy and was a hopelessly transcendentalist. You don't really believe that only observable quantities qualify for physics, do you?Einstein once asked him this.

why not?Heisenberg said in surprise, when you created the theory of relativity, didn't you abandon it because absolute time was unobservable? Einstein laughed: You can't play a good trick twice.You need to know that, in principle, it is wrong to try to build a theory based on observable quantities alone.The truth is quite the opposite: it is the theory that determines what we can observe. Yeah?Theory determines what we observe?So how does theory explain the trajectory of an electron in a cloud chamber?In Schrödinger's view, this is a superposition of a series of eigenstates, but forget him!Heisenberg said to himself, let's explain it in terms of our more orthodox matrix.However, the matrix is ​​discontinuous, but the trajectory is continuous, and the so-called trajectory has long been discarded as an unobservable quantity when the matrix was created

The night was quiet outside the window, and Heisenberg was thinking hard but couldn't figure it out.He was full of worries, tossing and turning, and decided to get up and go for a walk in Faelled Park, not far from the Bohr Institute.The park was empty in the middle of the night, and the evening wind was still bitter and cold on the face, but it made people sober.Heisenberg's mind was full of matrices, large and small, and he remembered the strange multiplication rules of matrices: p×q≠q×p Theory determines what we observe?The theory says, p×q≠q×p, which determines what we observe?

What does I×II mean?Take Line I first and then transfer to Line II.So, what does p×q mean? p is momentum and q is position, it doesn't mean It seemed that a flash of lightning flashed across the night sky, and Heisenberg's mind suddenly became clear and clear. p×q≠q×p, doesn’t this mean that observing the momentum p first, and then observing the position q, is the result different from observing q first and then p? Wait, what does this mean?Suppose we have a small ball moving forward, so at every moment, aren't its momentum and position two definite variables?Why are the results different just because of the difference in the order of observation?Heisenberg's palms were sweating, he knew that there was an extremely important secret hidden here.How is this possible?If we want to measure the length and width of a rectangle, isn't it the same thing to measure the length first or the width first?

unless Unless the action of measuring the momentum p itself affects the value of q.In turn, the act of measuring q also affects the value of p.But, kidding, what if I measure p and q at the same time? Heisenberg suddenly seemed to have seen a divine revelation, and he suddenly became enlightened. p×q≠q×p, does our equation tell us that it is impossible to observe p and q at the same time?Theory not only determines what we can observe, it also determines what we cannot observe! However, I'm confused, what does it mean that p and q cannot be observed at the same time?Observation p affects q?Observation q affects p?What the hell are we talking about?If I say that a small ball is at time t, its position coordinates are ten meters, and its speed is five meters per second, is there any problem?

There is a problem, and there is a big problem.Heisenberg clapped his hands.How can you know that at time t, the position of a certain ball is ten meters, and the speed is five meters per second?How do you know? By what?Needless to say?Observe, measure. The key is here!Measurement!Heisenberg banged his head and said, I now fully understand that the problem lies in the measurement behavior.The length and width of a rectangle are both fixed. When you measure its length, its width will never change, and vice versa.Let's talk about the classic ball again, how do you measure its position?You have to see it, or have some instrument to detect it, anyway, you have to touch it somehow, otherwise how do you know where it is?Take seeing for example, how can you see the position of a small ball?Some photon has to go from the light source, hit the ball, and bounce back into your eye, right?The point is, a classical sphere is a giant, and a photon hitting it is like an ant hitting an elephant, with negligible impact, never affecting its speed.Because of this, after we have measured its position, we can measure its speed calmly with negligible error.

But, we're talking electrons now!It is so small and light that the impact of photons on it must not be ignored.Measure the position of an electron?Well, we send a photon to perform this task, how does it report back?Yes, I got access to this electron, but it gave me a hard bump and went off somewhere, and I can't say anything about its current speed.See, in order to measure its position, we drastically changed its velocity, which is its momentum.We cannot know exactly where an electron is and its momentum at the same time. Heisenberg hurried back to the research institute, immersed himself in calculations, and finally came up with a formula: △p×△q >h/2π Δp and Δq are the errors of measuring p and measuring q respectively, and h is Planck's constant.Heisenberg found that the product of the error in measuring p and measuring q must be greater than a certain constant.If we measure p very precisely, that is to say, △p is very small, then correspondingly, △q must become very large, that is to say, our knowledge about q will become very vague and uncertain.Conversely, if we measure the position q very accurately, p will become wobbly, and the error will increase sharply. If we measure p with 100% accuracy, that is, △p=0, then △q will become infinite.That is to say, if we know all the information about the momentum p of an electron, then we lose all the information about its position q at the same time, we don’t know where it is at all, no matter how we arrange the experiment, we can’t do it better.You can't have both, either we know p precisely and let go of q, or we know q precisely and give up all knowledge of p, or we compromise and obtain a relatively vague p and a relatively vague q at the same time . p and q are like a pair of enemies in the previous life, they don't meet each other in life, they move like participating in a business, and they are in a state of having you but not me.Whenever we draw close to one, we simultaneously and dramatically alienate the other.This peculiar quantity is called a conjugate quantity, and we shall see later that there are many such quantities. This principle of Heisenberg was published in the "Journal of Physics" on March 23, 1927, and it is called the Uncertainty Principle.When it was first translated into Chinese, it was lovely translated as the Uncertainty Principle, but most of it is now changed to the more general Uncertainty Principle. two The Uncertainty Principle Unsure?Once again we have come across this nasty word.Again, this term is frowned upon in physics.If physics can't determine anything, what do we need it for?Born's probabilistic interpretation is annoying enough, given that all the conditions are not predictable.Now Heisenberg is doing even better, given all the conditions?This premise itself is impossible. Given some of the conditions, the other part of the conditions will become vague and undetermined.Given p, then we say goodbye to q. This is not pretty, there must be something wrong.We can't measure q if we measure p?I don't give up, I have to come and try to see if it works.Well, Heisenberg takes over, remember the Wilson cloud chamber?Didn't you worry about this problem in the first place?Through the cloud chamber we can see the trajectory of the electron movement, so of course we can calculate its instantaneous velocity by continuously measuring its position, so can we know its momentum at the same time? This question, Heisenberg said with a smile, I finally figured it out.What the electrons leave in the cloud chamber is not a fine track as we understand it, but in fact it is just a series of condensed water droplets.If you zoom in on it, it is discontinuous, a bunch of dotted lines, it is impossible to accurately get the concept of position, let alone violate the uncertainty principle. oh?That's right.Then let's be more careful, can't we just find out the fine trajectory of the electron?We can use a larger microscope to do this job, is it not impossible in theory? By the way, the microscope!Heisenberg said enthusiastically, I was just about to talk about the microscope.Let's do a thought experiment (Gedanken-experiment), imagine that we have an extremely powerful microscope.However, no matter how powerful a microscope is, it also has its basic principles. You must know that no matter what, if we use a wave to observe things smaller than its wavelength, it will not be accurate at all, just like using a coarse wave. It's like the pen can't draw a thin line.If we want to see something as tiny as an electron, we have to use very short wavelengths of light.Ordinary light is not enough, you need to use ultraviolet rays, X-rays, or even gamma rays. Well, since thought experiments don't cost money anyway, let's just assume that higher-ups have allocated huge sums of money for the first time to build us a state-of-the-art gamma-ray microscope.So, can't we now see exactly where the electrons are? But, Heisenberg pointed out, have you forgotten?Any wave that detects electrons must cause disturbances in the electrons themselves.The shorter the wavelength of a wave, the higher its frequency, right?Everyone should still remember Planck's formula E = hν. The higher the frequency, the higher the energy will be. This will disturb the electron more severely, and at the same time, we will be even more unable to understand its momentum.You see, this satisfies the uncertainty principle perfectly. You are sophistry.Well let's accept the fact that every time we use a photon to detect the electron's position, it will cause a strong disturbance to it, causing it to change direction and speed, and fly in another direction.However, we can still use some clever, roundabout ways to achieve our goals.For example, we can measure the directional velocity of the bounced photon, so as to deduce how it affects the electron, and then export the directional velocity of the electron itself.What, doesn't this break your trick? Or not.Heisenberg shook his head and said that in order to achieve such high sensitivity, our microscope must have a lens with a large diameter.You know, the lens gathers light from all directions into one focal point, so we can't tell where the photons that bounce back come from.If we shrink the diameter of the lens to ensure that the photons are not focused, then the sensitivity of the microscope becomes too poor for the job.So your cleverness still doesn't work. It's evil.So, what about looking at the bounce of the microscope itself? In the same way, to observe such a subtle effect, it is necessary to use light with a short wavelength, so its energy is large, and it will cause disturbance to the microscope itself to erase everything Wait, we don't give up.Well, we admit that our observation equipment is very rough, our fingers are clumsy, our civilization is only a few thousand years old, and modern science has only been established for less than three hundred years.We admit that with our current state of technology, we cannot simultaneously observe the position and momentum of a tiny electron, because our instruments are stupid and clumsy.However, this does not mean that electrons do not have position and momentum at the same time. Maybe in the future, even in the distant future, we will develop a cutting-edge technology, and we will invent extremely sophisticated instruments to accurately measure the position and momentum of electrons. What about momentum?You can't deny the possibility. That's not what it says.The problem here, Heisenberg muses, is that theory limits what we can observe, not that experiments introduce errors.Simultaneously measuring exact momentum and position is impossible in principle, no matter how advanced the technology.Just as you can never build a perpetual motion machine, you can never build a microscope that can detect p and q at the same time.No matter what theories we create in the future, they must obey the uncertainty principle. This is a basic principle, and all subsequent theories must be under its supervision to gain legitimacy. Isn't Heisenberg's conclusion too overbearing?Moreover, wouldn't physicists lose all face in this way?Imagine the public: what, you're a physicist?Oh, I'm so sorry for you, you don't even know the momentum and position of an electron!At least our Tommy knows how to handle his ball. However, we still have to present the facts, reason, and convince others with virtue.One thought experiment after another has been proposed, but we just can't precisely measure the electron's momentum and at the same time get its position precisely.The product of the errors of the two must be greater than that constant, that is, h divided by 2π.Fortunately, we all remember that h is very small, only 6.626×10^-34 joule seconds, so if △p and △q are of similar magnitude, they are both on the order of 10^-17.We can now reassure the uninformed masses that things are not so bad, and the effect only becomes apparent at the scale of electrons and photons.For Tommy's ball, 10^-17 is so insignificant that it can't be felt at all.Tommy could shoot his ball without worrying about losing it because he couldn't figure out where it was. But in the electronic-scale world, that's very different.At the end of the last chapter, we imagined that we were reduced to the size of electrons to explore the mysteries of atoms. At that time, our height was only 10^-23 meters.Now, Mom, worried about our naughty behavior, wanted to measure where we were, but they were doomed to be disappointed: the measurement was off by 10^-17 meters, a million times our own height!What does an error of one million times mean? If we are usually 1.75 meters tall, the error will reach 1.75 million meters, which is 1,750 kilometers. Look for us everywhere along the Shanghai Railway.It is impossible to predict that it will become worthy of the name. At all times, nature stubbornly sticks to this bottom line, never giving us any chance to get accurate values ​​​​of position and momentum at the same time.No matter how many tricks we can do, it will always be better than us, and it will beat our cleverness every time.Can't measure the electron's position and momentum?Let's design a very small and very small container, which can only accommodate one electron without leaving any extra space. How about this?Electronics can't move around, right?However, first of all, this kind of container cannot be manufactured, because it must also be composed of electrons, so it must have a position that fluctuates, causing the internal space to fluctuate.Taking a step back, even if it is possible, in this case, the electrons will mysteriously permeate through the container wall and appear outside the container, like the legendary Laoshan Taoist who passed through the wall.The uncertainty principle endows it with this magical ability to break through all constraints.Another way is to cool down.We all know that atoms are constantly vibrating, and temperature is the macroscopic manifestation of this vibration. When the temperature drops to absolute zero, the atoms are theoretically completely still.At that time, the momentum is determined to be zero, so we only need to measure the position, right?Unfortunately, absolute zero cannot be reached. No matter how hard we try, the atom still desperately retains the last bit of internal energy to prevent us from measuring its momentum.No matter who it is, it is impossible to make the atoms completely still, and neither can the legendary saints. They cannot overcome the uncertainty principle. Momentum p and position q, they are really life and death.Whenever one quantity appears in the universe, the other mysteriously disappears.Or, both appear in an indistinct guise.Heisenberg soon discovered another pair of similar enemies, energy E and time t.As long as the energy E is measured more accurately, the time t becomes more blurred; conversely, the more accurately the time t is measured, the energy E begins to fluctuate on a large scale.Moreover, their relationship obeys the same uncertainty rules: △E×△t >h/2π Ladies and gentlemen, our universe has become very strange.All kinds of physical quantities follow Heisenberg's uncertainty principle, one after another, like bubbles rising and bursting in the mysterious sea.In the eyes of the ancients, emptiness means nothingness.But later people learned that there are also countless molecules in the invisible air, and the void should refer to the vacuum that evacuates the air.Later, people felt that various fields, from gravitational field to electromagnetic field, should also be excluded from the concept of space, and it should only refer to space itself. But now, the concept is getting muddled again.First of all, Einstein's theory of relativity tells us that space itself can also be warped and deformed. In fact, gravity is just its curvature.And Heisenberg's uncertainty principle presents a more exotic scenario: we know that the more accurately t is measured, the more uncertain E becomes.So for a very, very short instant, a very definite instant, there can be huge fluctuations in energy even in a vacuum.This kind of energy appears out of nowhere completely relying on uncertainty, and it really violates the law of energy conservation!But this moment is very short, before people have time to discover it, it disappears mysteriously, making the law of energy conservation as a whole maintained.The shorter the interval, the more certain t is, the more uncertain E is, and the greater the energy that can appear out of thin air. Therefore, our vacuum is actually boiling all the time, and mysterious energies are produced and disappeared everywhere.Einstein told us that energy and matter can be converted into each other, so in the vacuum, there are actually some ghost substances constantly appearing, but they disappeared in another world before we caught them.Vacuum itself is the best medium to provide this fluctuation. Now if we talk about emptiness, we should say it clearly: there is no matter, no energy, no time, and no space.This is nothing, it is not imaginable at all (can you imagine what it would be like to have no space?).But many people say that this is not empty, because space and time themselves seem to be created out of nothing through some mechanism. I am really going crazy, so what is empty? Gossip after dinner: out of thin air Once upon a time, all scientists believed that it was absolutely impossible to create something out of nothing.Matter cannot be created out of thin air, nor can energy be created out of thin air, let alone space-time itself.But the emergence of the uncertainty principle shattered all these old concepts. Heisenberg told us that in a very small space and a very short time, anything is possible, because we are very certain about time, so in turn, we are very uncertain about energy.Energetic matter can escape the shackles of the laws of physics, appearing and disappearing freely.However, the price of this kind of freedom is that it can only be limited to a very short period of time. When the time comes, Cinderella will show her original shape, and these mysterious material energies will disappear, in order to maintain the law of conservation of mass and energy. Not destroyed on a large scale. However, at the end of the 1960s, someone thought of a possibility: the energy of gravity is a negative number (because gravity is suction, assuming that the potential energy at infinity is 0, then when the object approaches, the gravitational work makes its potential energy negative), Therefore, the material energy generated out of thin air in a short period of time can form a gravitational field between them, and the negative energy produced by it just offsets itself, so that the total energy remains 0, and the law of conservation is not violated.In this way matter is literally created from nothing. Many people believe that our universe itself arose through this mechanism.Quantum effects make a small piece of space-time suddenly arise from no space-time at all, and then due to the action of various forces, it suddenly expands exponentially, expanding to the scale of the entire universe in an instant. MIT scientist Alan.Starting from this idea, Alan Guth created the inflation theory of the universe (Inflation).In the very early days of the creation of the universe, each piece of space exploded at an unimaginably astonishing speed, which caused the total volume of the universe to increase many, many times.This can explain why its structure looks uniform in all directions today. There have been many versions of the theory of inflation since its inception, but it is difficult to confirm whether this theory is correct, because the universe is not like our laboratory, which can be observed and studied at will.But most physicists still prefer it as a promising theory.In 1998, Gus also published a popular book on inflation. His favorite sentence is: the universe itself is a free lunch.It means that the universe came from nothing. However, if it is harsher, this cannot be regarded as strict creation out of nothing.Because even if there is no matter, no time and space, we still have a premise: the existence of physical laws!How did the various rules of relativity and quantum theory, such as the uncertainty principle itself, emerge from nothing?Or are they self-evident?We're getting more and more mysterious, so let's stop here. three When Heisenberg completed his uncertainty principle, he immediately wrote to Pauli and Bohr in Norway to tell them his thoughts.After receiving Heisenberg's letter, Bohr immediately set off from Norway and returned to Copenhagen, ready to have an in-depth discussion with Heisenberg on this issue.Heisenberg probably thought that such a great discovery would surely move Bohr's heart and make him agree with his consistent ideas on quantum mechanics.However, he was very wrong. In Norway, Bohr thought about the wave-particle problem while skiing, and the new idea gradually took shape in his mind. .When he saw Heisenberg's paper, he naturally used this idea to confirm the whole conclusion.He asked Heisenberg whether this uncertainty came from the nature of the particle or from the nature of the wave?Heisenberg was taken aback for a moment, he didn't think about any waves at all.Of course it is a particle, since the photon hits the electron, the uncertainty of position and momentum is caused, isn't it obvious? Bohr shook his head seriously. He used the giant microscope imagined by Heisenberg to prove that the uncertainty to a large extent is not only caused by the discontinuous particle nature, but also by the wave nature.We have discussed the de Broglie wavelength formula λ=h/mv before, mv is the momentum p, so p=h/λ, for every momentum p, there is always a concept of wavelength accompanying it.For the relationship of Et, E=hν, there is still the fluctuation concept of frequency ν in it.Heisenberg flatly refused, and it was not easy for him to accept volatility. Bohr obviously became impatient with Heisenberg's stubbornness, and he clearly said to Heisenberg: Your microscope experiment is No, this made Heisenberg cry.The two had a big quarrel, and of course Klein helped Bohr, which made the atmosphere in Copenhagen very sharp.What started out as a matter of physics turned into an almost personal misunderstanding, to the point that Heisenberg had to send Pauli's letter back for clarification.In the end, Pauli himself went to Denmark, which finally calmed down the aftermath of the incident. Unfortunately for Heisenberg, he was wrong about the microscope.Heisenberg was probably born with some kind of microscopic phobia, and became dizzy when he touched a microscope.Back then, he couldn't figure out the most basic microscope resolution problem during his doctoral dissertation defense, and almost didn't get his degree.This time Bohr finally made him realize that uncertainty is based on the double basis of waves and particles, which is actually a kind of swing between waves and particles: the more you know about the properties of waves, the Attributes are less known.Heisenberg finally accepted Bohr's criticism and added a footnote to his paper stating that uncertainty is actually built on both continuity and discontinuity at the same time, and thanked Bohr for pointing this out. Bohr also learned something from this debate. He found that the general significance of the uncertainty principle was greater than he imagined.He thought that this was just a partial principle, but now he realizes that this principle is one of the core cornerstones of quantum theory.In a letter to Einstein, Bohr praised Heisenberg's theory, saying that he showed in a very beautiful way how uncertainty can be applied to quantum theory.After the long Easter holiday, both parties took a step back, and the situation finally brightened.In his letter to Pauli, Heisenberg regained his good mood, saying that he could once again simply discuss physics and forget about everything else.Indeed, brothers are fighting against the wall, and they must also defend themselves against their insults. The Copenhagen faction is now united like a solid rock. They will soon face greater challenges together and engrave the name of Copenhagen deeply in the history of physics. glorious history. But then again.Volatility, granularity, these two words have haunted us from the very beginning of our history until now.Well, uncertainty is built on volatility and granularity at the same time, but isn't that just nonsense?Our patience is limited, why don't we open the skylight and tell the truth, is this damn electron a particle or a wave? Particles or waves, it is really a topic with a lot of emotion.This is a three-hundred-year-old legend, with ups and downs and ups and downs, interspersed with the greatest names in the history of physics: Newton, Hooke, Huygens, Young, Fresnel, Foucault, Maxwell, Hertz, Tom Who can explain the grievances and grievances of Johnson, Einstein, Compton, and de Broglie?We are in a dilemma. On the one hand, the double-slit experiment and Maxwell's theory unambiguously reveal the wave nature of light, and on the other hand, the photoelectric effect and the Compton effect also clearly show that it is a particle.As far as electrons are concerned, Bohr's transition, the spectrum in atoms, and Heisenberg's matrix all emphasize its discontinuous side. It seems that the particle nature has the upper hand, but Schrödinger's equation exaggerates its continuity. Even put the label of volatility on its face. No matter how you look at it, there is no way that an electron is not a particle; no matter how you look at it, there is no way that an electron is not a wave. How should this be done? When encountering difficult problems, the best way is to ask our idol, the omnipotent Sherlock.Mr Holmes.He put it this way: My method is based on the assumption that when you eliminate all impossible conclusions, what remains, no matter how bizarre, must be the truth. ("New Detective: The Soldier with White Skin") What a wise saying.Then, it is impossible for the electron not to be a particle, and it is also impossible for it not to be a wave.The only remaining possibility is It is both a particle and a wave at the same time! But wait, is this too much?It's totally unacceptable.What does it mean to be both a particle and a wave at the same time?These two images are clearly mutually exclusive.Is it possible for a person to be both male and female (eunuchs and the like do not count)?Isn't this statement contradictory? However, if you want to believe in Holmes, you must also believe in Bohr, because Bohr thinks so.Undoubtedly, an electron must be interpreted from both particle and wave perspectives, and any unilateral description is incomplete.Only when the two concepts of particle and wave are organically combined, can an electron become a flesh and blood electron, and truly become a complete image.The electron without particle is blind, and the electron without wave is lame. This still doesn't convince us, is it both a particle and a wave?It's hard to imagine, is the electron like a ghost, emitting a strange wave around the particle at the same time, making itself a superposition of these two states?Has anyone ever witnessed such a nightmarish scenario?Come out to testify? No, you got it wrong.Bohr shook his head and said that whenever we observe an electron, of course it can only exhibit one property, either a particle or a wave.People who claim to see particle-wave hybrid superpositions are either presbyopic or just plain nonsense.However, as an overall concept of electron, it shows a wave-particle duality, it can show the side of particle, and also can show the side of wave, it all depends on how we observe it.We want to see a particle?Well, let it hit the screen as a dot.Behold, particles!We want to see a wave?也行，讓它通過雙縫組成干涉圖樣。看，波！ 奇怪，似乎有哪裡不對，卻說不出來好吧，電子有時候變成電子的模樣，有時候變成波的模樣，嗯，不錯的變臉把戲。可是，撕下它的面具，它本來的真身究竟是個什麼呢？ 這就是關鍵！這就是你我的分歧所在了。玻爾意味深長地說，電子的真身？或者換幾個詞，電子的原型？電子的本來面目？電子的終極理念？這些都是毫無意義的單詞，對於我們來說，唯一知道的只是每次我們看到的電子是什麼。我們看到電子呈現出粒子性，又看到電子呈現出波動性，那麼當然我們就假設它是粒子和波的混合體。我一點都不關心電子本來是什麼，我覺得那是沒有意義的。事實上我也不關心大自然本來是什麼，我只關心我們能夠觀測到大自然是什麼。電子又是個粒子又是個波，但每次我們觀察它，它只展現出其中的一面，這裡的關鍵是我們如何觀察它，而不是它究竟是什麼。 玻爾的話也許太玄妙了，我們來通俗地理解一下。現在流行手機換彩殼，我昨天心情好，就配一個shining的亮銀色，今天心情不好，換一個比較有憂鬱感的藍色。咦奇怪了，為什麼我的手機昨天是銀色的，今天變成藍色了呢？這兩種顏色不是互相排斥的嗎？我的手機怎麼可能又是銀色，又是藍色呢？很顯然，這並不是說我的手機同時展現出銀色和藍色，變成某種稀奇的銀藍色，它是銀色還是藍色，完全取決於我如何搭配它的外殼。我昨天決定這樣裝配它，它就呈現出銀色，而今天改一種方式，它就變成藍色。它是什麼顏色，取決於我如何裝配它！ 但是，如果你一定要打破砂鍋地問：我的手機本來是什麼顏色？那可就糊塗了。假如你指的是它原裝出廠時配著什麼外殼，我倒可以告訴你。不過要是你強調是哲學意義上的本來，或者理念中手機的顏色到底是什麼，我會覺得你不可理喻。真要我說，我覺得它本來沒什麼顏色，只有我們給它裝上某種外殼並觀察它，它才展現出某種顏色來。它是什麼顏色，取決於我們如何觀察它，而不是取決於它本來是什麼顏色。我覺得，討論它本來的顏色是癡人說夢。 再舉個例子，大家都知道白馬非馬的詭辯，不過我們不討論這個。我們問：這匹馬到底是什麼顏色呢？你當然會說：白色啊。可是，也許你身邊有個色盲，他會爭辯說：不對，是紅色！大家指的是同一匹馬，它怎麼可能又是白色又是紅色呢？你當然要說，那個人在感覺顏色上有缺陷，他說的不是馬本來的顏色，可是，誰又知道你看到的就一定是正確的顏色呢？假如世上有一半色盲，誰來分辨哪一半說的是真相呢？不說色盲，我們戴上一副紅色眼鏡，這下看出去的馬也變成了紅色吧？它怎麼剛剛是白色，現在是紅色呢？哦，因為你改變了觀察方式，戴上了眼鏡。那麼哪一種方式看到的是真實呢？天曉得，莊周做夢變成了蝴蝶還是蝴蝶做夢變成了莊周？你戴上眼鏡看到的是真實還是脫下眼鏡看到的是真實？ 我們的結論是，討論哪個是真實毫無意義。我們唯一能說的，是在某種觀察方式確定的前提下，它呈現出什麼樣子來。我們可以說，在我們運用肉眼的觀察方式下，馬呈現出白色。同樣我們也可以說，在戴上眼鏡的觀察方式下，馬呈現出紅色。色盲也可以聲稱，在他那種特殊構造的感光方式觀察下，馬是紅色。至於馬本來是什麼色，完全沒有意義。甚至我們可以說，馬本來的顏色是子虛烏有的。我們大多數人說馬是白色，只不過我們大多數人採用了一種類似的觀察方式罷了，這並不指向一種終極真理。 電子也是一樣。電子是粒子還是波？那要看你怎麼觀察它。如果採用光電效應的觀察方式，那麼它無疑是個粒子；要是用雙縫來觀察，那麼它無疑是個波。它本來到底是個粒子還是波呢？又來了，沒有什麼本來，所有的屬性都是同觀察聯繫在一起的，讓本來見鬼去吧。 但是，一旦觀察方式確定了，電子就要選擇一種表現形式，它得作為一個波或者粒子出現，而不能再曖昧地混雜在一起。這就像我們可憐的馬，不管誰用什麼方式觀察，它只能在某一時刻展現出一種顏色。從來沒有人有過這樣奇妙的體驗：這匹馬同時又是白色，又是紅色。波和粒子在同一時刻是互斥的，但它們卻在一個更高的層次上統一在一起，作為電子的兩面被納入一個整體概念中。這就是玻爾的互補原理（Complementary Principle），它連同波恩的概率解釋，海森堡的不確定性，三者共同構成了量子論哥本哈根解釋的核心，至今仍然深刻地影響我們對於整個宇宙的終極認識。 第三次波粒戰爭便以這樣一種戲劇化的方式收場。而量子世界的這種奇妙結合，就是大名鼎鼎的波粒二象性。 Four 三百年硝煙散盡，波和粒子以這樣一種奇怪的方式達成了妥協：兩者原來是不可分割的一個整體。就像漫畫中教皇善與惡的兩面，雖然在每個確定的時刻，只有一面能夠體現出來，但它們確實集中在一個人的身上。波和粒子是一對孿生兄弟，它們如此苦苦爭鬥，卻原來是演出了一場物理學中的絕代雙驕故事，這教人拍案驚奇，唏噓不已。 現在我們再回到上一章的最後，重溫一下波和粒子在雙縫前遇到的困境：電子選擇左邊的狹縫，還是右邊的狹縫呢？現在我們知道，假如我們採用任其自然的觀測方式，它波動的一面就占了上風。這個電子於是以某種方式同時穿過了兩道狹縫，自身與自身發生干涉，它的波函數ψ按照嚴格的干涉圖形花樣發展。但是，當它撞上感應屏的一剎那，觀測方式發生了變化！我們現在在試圖探測電子的實際位置了，於是突然間，粒子性接管了一切，這個電子凝聚成一點，按照ψ的概率隨機地出現在螢幕的某個地方。 假使我們在某個狹縫上安裝儀器，試圖測出電子究竟通過了哪一邊，注意，這是另一種完全不同的觀測方式！ ! !我們試圖探測電子在通過狹縫時的實際位置，可是只有粒子才有實際的位置。這實際上是我們施加的一種暗示，讓電子早早地展現出粒子性。事實上，的確只有一邊的儀器將記錄下它的蹤影，但同時，干涉條紋也被消滅，因為波動性隨著粒子性的喚起而消失了。我們終於明白，電子如何表現，完全取決於我們如何觀測它。種瓜得瓜，種豆得豆，想記錄它的位置？好，那是粒子的屬性，電子善解人意，便表現出粒子性來，同時也就沒有干涉。不作這樣的企圖，電子就表現出波動性來，穿過兩道狹縫並形成熟悉的干涉條紋。 量子派物理學家現在終於逐漸領悟到了事情的真相：我們的結論和我們的觀測行為本身大有聯繫。這就像那匹馬是白的還是紅的，這個結論和我們用什麼樣的方法去觀察它有關係。有些看官可能還不服氣：結論只有一個，親眼看見的才是唯一的真實。色盲是視力缺陷，眼鏡是外部裝備，這些怎麼能夠說是看到真實呢？其實沒什麼分別，它們不外乎是兩種不同的觀測方式罷了，我們的論點是，根本不存在所謂真實。 好吧，現在我視力良好，也不戴任何裝置，看到馬是白色的。那麼，它當真是白色的嗎？其實我說這話前，已經隱含了一個前提：用人類正常的肉眼，在普通光線下看來，馬呈現出白色。再技術化一點，人眼只能感受可見光，波長在四百-七百六十納米左右，這些頻段的光混合在一起才形成我們印象中的白色。所以我們論斷的前提就是，在四百-七百六十納米的光譜區感受馬，它是白色的。 許多昆蟲，比如蜜蜂，它的複眼所感受的光譜是大大不同的。蜜蜂看不見波長比黃光還長的光，卻對紫外線很敏感。在它看來，這匹馬大概是一種藍紫色，甚至它可能繪聲繪色地向你描繪一種難以想像的紫外色。現在你和蜜蜂吵起來了，你堅持這馬是白色的，而蜜蜂一口咬定是藍紫色。你和蜜蜂誰對誰錯呢？其實都對。那麼，馬怎麼可能又是白色又是紫色呢？其實是你們的觀測手段不同罷了。對於蜜蜂來說，它也是親眼見到，人並不比蜜蜂擁有更多的正確性，離真相更近一點。話說回來，色盲只是對於某些頻段的光有盲點，眼鏡只不過加上一個濾鏡而已，本質上也是一樣的，也沒理由說它們看到的就是虛假。 事實上，沒有什麼客觀真相。討論馬本質上到底是什麼顏色，正如我們已經指出過的，是很無聊的行為。根本不存在一個絕對的所謂本色，除非你先定義觀測的方式。 玻爾也好，海森堡也好，現在終於都明白：談論任何物理量都是沒有意義的，除非你首先描述你測量這個物理量的方式。一個電子的動量是什麼？我不知道，一個電子沒有什麼絕對的動量，不過假如你告訴我你打算怎麼去測量，我倒可以告訴你測量結果會是什麼。根據測量方式的不同，這個動量可以從十分精確一直到萬分模糊，這些結果都是可能的，也都是正確的。一個電子的動量，只有當你測量時，才有意義。假如這不好理解，想像有人在紙上畫了兩橫夾一豎，問你這是什麼字。嗯，這是一個工字，但也可能是橫過來的H，在他沒告訴你怎麼看之前，這個問題是沒有定論的。現在，你被告知：這個圖案的看法應該是橫過來看。這下我們明確了：這是一個大寫字母H。只有觀測手段明確之後，答案才有意義。 測量！在經典理論中，這不是一個被考慮的問題。測量一塊石頭的重量，我用天平，用彈簧秤，用磅秤，或者用電子秤來做，理論上是沒有什麼區別的。在經典理論看來，石頭是處在一個絕對的，客觀的外部世界中，而我觀測者對這個世界是沒有影響的，至少，這種影響是微小得可以忽略不計的。你測得的資料是多少，石頭的客觀重量就是多少。但量子世界就不同了，我們已經看到，我們測量的物件都是如此微小，以致我們的介入對其產生了致命的干預。我們本身的擾動使得我們的測量中充滿了不確定性，從原則上都無法克服。採取不同的手段，往往會得到不同的答案，它們隨著不確定性原理搖搖擺擺，你根本不能說有一個客觀確定的答案在那裡。在量子論中沒有外部世界和我之分，我們和客觀世界天人合一，融和成為一體，我們和觀測物互相影響，使得測量行為成為一種難以把握的手段。在量子世界，一個電子並沒有什麼客觀動量，我們能談論的，只有它的測量動量，而這又和我們的測量手段密切相關。 各位，我們已經身陷量子論那奇怪的沼澤中了，我只希望大家不要過於頭昏腦脹，因為接下來還有無數更稀奇古怪的東西，錯過了未免可惜。我很抱歉，這幾節我們似乎沉浸於一種玄奧的哲學討論，而且似乎還要繼續討論下去。這是因為量子革命牽涉到我們世界觀的根本變革，以及我們對於宇宙的認識方法。量子論的背後有一些非常形而上的東西，它使得我們的理性戰戰兢兢，汗流浹背。但是，為了理解量子論的偉大力量，我們又無法繞開這些而自欺欺人地盲目前進。如果你從史話的一開始跟著我一起走到了現在，我至少對你的勇氣和毅力表示讚賞，但我也無法給你更多的幫助。假如你感到困惑彷徨，那麼玻爾的名言如果誰不為量子論而感到困惑，那他就是沒有理解量子論或許可以給你一些安慰。而且，正如我們以後即將描述的那樣，你也許應該感到非常自豪，因為愛因斯坦和你是一個處境。 但現在，我們必須走得更遠。上面一段文字只是給大家一個小小的喘息機會，我們這就繼續出發了。 如果不定義一個測量動量的方式，那麼我們談論電子動量就是沒有意義的？這聽上去似乎是一種唯心主義的說法。難道我們無法測量電子，它就沒有動量了嗎？讓我們非常驚訝和尷尬的是，玻爾和海森堡兩個人對此大點其頭。一點也不錯，假如一個物理概念是無法測量的，它就是沒有意義的。我們要時時刻刻注意，在量子論中觀測者是和外部宇宙結合在一起的，它們之間現在已經沒有明確的分界線，是一個整體。在經典理論中，我們脫離一個絕對客觀的外部世界而存在，我們也許不瞭解這個世界的某些因素，但這不影響其客觀性。可如今我們自己也已經融入這個世界了，對於這個物我合一的世界來說，任何東西都應該是可以測量和感知的。只有可觀測的量才是存在的！ Carl.薩根（Karl Sagan）曾經舉過一個很有意思的例子，雖然不是直接關於量子論的，但頗能說明問題。 我的車庫裡有一條噴火的龍！他這樣聲稱。 太稀罕了！他的朋友連忙跑到車庫中，但沒有看見龍。龍在哪裡？ 哦，薩根說，我忘了說明，這是一條隱身的龍。 朋友有些狐疑，不過他建議，可以撒一些粉末在地上，看看龍的爪印是不是會出現。但是薩根又聲稱，這龍是飄在空中的。 那既然這條龍在噴火，我們用紅外線檢測儀做一個熱掃描？ 也不行。薩根說，隱形的火也沒有溫度。 要麼對這條龍噴漆讓它現形？這條龍是非物質的，滑不溜手，油漆無處可粘。 反正沒有一種物理方法可以檢測到這條龍的存在。薩根最後問：這樣一條看不見摸不著，沒有實體的，飄在空中噴著沒有熱度的火的龍，一條任何儀器都無法探測的龍，和根本沒有龍之間又有什麼差別呢？ 現在，玻爾和海森堡也以這種苛刻的懷疑主義態度去對待物理量。不確定性原理說，不可能同時測准電子的動量p和位置q，任何精密的儀器也不行。許多人或許會認為，好吧，就算這是理論上的限制，和我們實驗的笨拙無關，我們仍然可以安慰自己，說一個電子實際上是同時具有準確的位置和動量的，只不過我們出於某種限制無法得知罷了。 但哥本哈根派開始嚴厲地打擊這種觀點：一個具有準確p和q的經典電子？這恐怕是自欺欺人吧。有任何儀器可以探測到這樣的一個電子嗎？沒有，理論上也不可能有。那麼，同樣道理，一個在臆想的世界中生存的，完全探測不到的電子，和根本沒有這樣一個電子之間又有什麼區別呢？ 事實上，同時具有p和q的電子是不存在的！p和q也像波和微粒一樣，在不確定原理和互補原理的統治下以一種此長彼消的方式生存。對於一些測量手段來說，電子呈現出一個準確的p，對於另一些測量手段來說，電子呈現出準確的q。我們能夠測量到的電子才是唯一的實在，這後面不存在一個客觀的，或者實際上的電子！ 換言之，不存在一個客觀的，絕對的世界。唯一存在的，就是我們能夠觀測到的世界。物理學的全部意義，不在於它能夠揭示出自然是什麼，而在於它能夠明確，關於自然我們能說什麼。沒有一個脫離於觀測而存在的絕對自然，只有我們和那些複雜的測量關係，熙熙攘攘縱橫交錯，構成了這個令人心醉的宇宙的全部。測量是新物理學的核心，測量行為創造了整個世界。 飯後閒話：奧卡姆剃刀 同時具有p和q的電子是不存在的。有人或許感到不理解，探測不到的就不是實在嗎？ 我們來問自己，這個世界究竟是什麼和我們在最大程度上能夠探測到這個世界是什麼兩個命題，其實質到底有多大的不同？我們探測能力所達的那個世界，是不是就是全部實在的世界？比如說，我們不管怎樣，每次只能探測到電子是個粒子或者是個波，那麼，是不是有一個實在的世界，在那裡電子以波-粒子的奇妙方式共存，我們每次探測，只不過探測到了這個終極實在於我們感觀中的一部分投影？同樣，在這個實在世界中還有同時具備p和q的電子，只不過我們與它緣慳一面，每次測量都只有半面之交，沒法窺得它的真面目？ 假設宇宙在創生初期膨脹得足夠快，以致它的某些區域對我們來說是如此遙遠，甚至從創生的一剎那以光速出發，至今也無法與它建立起任何溝通。宇宙年齡大概有一百五十億歲，任何信號傳播最遠的距離也不過一百五十億光年，那麼，在距離我們一百五十億光年之外，有沒有另一些實在的宇宙，雖然它們不可能和我們的宇宙之間有任何因果聯繫？ 在那個實在世界裡，是不是有我們看不見的噴火的龍，是不是有一匹具有實在顏色的馬，而我們每次觀察只不過是這種實在顏色的膚淺表現而已。我跟你爭論說，地球其實是方的，只不過它在我們觀察的時候，表現出圓形而已。但是在那個實在世界裡，它是方的，而這個實在世界我們是觀察不到的，但不表明它不存在。 如果我們運用奧卡姆剃刀原理（Occam's Razor），這些觀測不到的實在世界全都是子虛烏有的，至少是無意義的。這個原理是十四世紀的一個修道士威廉所創立的，奧卡姆是他出生的地方。這位奧卡姆的威廉還有一句名言，那是他對巴伐利亞的路易四世說的：你用劍來保衛我，我用筆來保衛你。 剃刀原理是說，當兩種說法都能解釋相同的事實時，應該相信假設少的那個。比如，地球本來是方的，但觀測時顯現出圓形。這和地球本來就是圓的說明的是同一件事。但前者引入了一個莫名其妙的不必要的假設，所以前者是胡說。同樣，電子本來有準確的p和q，但是觀測時只有一個能顯示，這和只存在具有p或者具有q的電子說明的也是同一回事，但前者多了一個假設，我們應當相信後者。存在但觀測不到，這和不存在根本就是一碼事。 同樣道理，沒有粒子-波混合的電子，沒有看不見的噴火的龍，沒有絕對顏色的馬，沒有一百五十億光年外的宇宙（一百五十億光年這個距離稱作視界），沒有隔著一釐米四維尺度觀察我們的四維人，沒有絕對的外部世界。Stephen.霍金在《時間簡史》中說：我們仍然可以想像，對於一些超自然的生物，存在一組完全地決定事件的定律，它們能夠觀測宇宙現在的狀態而不必干擾它。然而，我們人類對於這樣的宇宙模型並沒有太大的興趣。看來，最好是採用奧卡姆剃刀原理，將理論中不能被觀測到的所有特徵都割除掉。 你也許對這種實證主義感到反感，反駁說：一片無人觀察的荒漠，難道就不存在嗎？以後我們會從另一個角度來討論這片無人觀察的荒漠，這裡只想指出，無人的荒漠並不是原則上不可觀察的。 five 正如我們的史話在前面一再提醒各位的那樣，量子論革命的破壞力是相當驚人的。在概率解釋，不確定性原理和互補原理這三大核心原理中，前兩者摧毀了經典世界的因果性，互補原理和不確定原理又合力搗毀了世界的客觀性和實在性。新的量子圖景展現出一個前所未有的世界，它是如此奇特，難以想像，和人們的日常生活格格不入，甚至違背我們的理性本身。但是，它卻能夠解釋量子世界一切不可思議的現象。這種主流解釋被稱為量子論的哥本哈根解釋，它是以玻爾為首的一幫科學家做出的，他們大多數曾在哥本哈根工作過，許多是量子論本身的創立者。哥本哈根派的人物除了玻爾，自然還有海森堡、波恩、泡利、狄拉克、克萊默、約爾當，也包括後來的魏紮克和蓋莫夫等等，這個解釋一直被當作是量子論的正統，被寫進各種教科書中。 當然，因為它太過奇特，太教常人困惑，近八十年來沒有一天它不受到來自各方面的置疑、指責、攻擊。也有一些別的解釋被紛紛提出，這裡面包括德布羅意-玻姆的隱函數理論，埃弗萊特的多重宇宙解釋，約翰泰勒的系綜解釋、Ghirardi-Rimini-Weber的自發定域（Spontaneous Localization），Griffiths-Omn's-GellMann-Hartle的脫散歷史態（Decoherent Histories, or Consistent Histories），等等，等等。我們的史話以後會逐一地去看看這些理論，但是公平地說，至今沒有一個理論能取代哥本哈根解釋的地位，也沒有人能證明哥本哈根解釋實際上錯了（當然，可能有人爭辯說它不完備）。隱函數理論曾被認為相當有希望，可惜它的勝利直到今天還仍然停留在口頭上。因此，我們的史話仍將以哥本哈根解釋為主線來敘述，對於讀者來說，他當然可以自行判斷，並得出他自己的獨特看法。 哥本哈根解釋的基本內容，全都圍繞著三大核心原理而展開。我們在前面已經說到，首先，不確定性原理限制了我們對微觀事物認識的極限，而這個極限也就是具有物理意義的一切。其次，因為存在著觀測者對於被觀測物的不可避免的擾動，現在主體和客體世界必須被理解成一個不可分割的整體。沒有一個孤立地存在於客觀世界的事物（being），事實上一個純粹的客觀世界是沒有的，任何事物都只有結合一個特定的觀測手段，才談得上具體意義。物件所表現出的形態，很大程度上取決於我們的觀察方法。對同一個物件來說，這些表現形態可能是互相排斥的，但必須被同時用於這個物件的描述中，也就是互補原理。 最後，因為我們的觀測給事物帶來各種原則上不可預測的擾動，量子世界的本質是隨機性。傳統觀念中的嚴格因果關係在量子世界是不存在的，必須以一種統計性的解釋來取而代之，波函數ψ就是一種統計，它的平方代表了粒子在某處出現的概率。當我們說電子出現在x處時，我們並不知道這個事件的原因是什麼，它是一個完全隨機的過程，沒有因果關係。 有些人可能覺得非常糟糕：又是不確定又是沒有因果關係，這個世界不是亂套了嗎？物理學家既然什麼都不知道，那他們還好意思呆在大學裡領薪水，或者在電視節目上欺世盜名？然而事情並沒有想像的那麼壞，雖然我們對單個電子的行為只能預測其概率，但我們都知道，當樣本數量變得非常非常大時，概率論就很有用了。我們沒法知道一個電子在螢幕上出現在什麼位置，但我們很有把握，當數以萬億記的電子穿過雙縫，它們會形成干涉圖案。這就好比保險公司沒法預測一個客戶會在什麼時候死去，但它對一個城市的總體死亡率是清楚的，所以保險公司一定是賺錢的！ 傳統的電視或者電腦螢幕，它後面都有一把電子槍，不斷地逐行把電子打到螢幕上形成畫面。對於單個電子來說，我並不知道它將出現在螢幕上的哪個點，只有概率而已。不過大量電子疊在一起，組成穩定的畫面是確定無疑的。看，就算本質是隨機性，但科學家仍然能夠造出一些有用的東西。如果你家電視畫面老是有雪花，不要懷疑到量子論頭上來，先去檢查一下天線。 當然時代在進步，俺的電腦螢幕現在變成了薄薄的液晶型，那是另一回事了。 至於令人迷惑的波粒二象性，那也只是量子微觀世界的奇特性質罷了。我們已經談到德布羅意方程λ＝h／p，改寫一下就是λp＝h，波長和動量的乘積等於普朗克常數h。對於微觀粒子來說，它的動量非常小，所以相應的波長便不能忽略。但對於日常事物來說，它們品質之大相比h簡直是個天文數字，所以對於生活中的一個足球，它所伴隨的德布羅意波微乎其微，根本感覺不到。我們一點都用不著擔心，在世界盃決賽中，眼看要入門的那個球會突然化為一縷波，消失得杳然無蹤。 但是，我們還是覺得不太滿意，因為對觀測行為，我們似乎還沒有作出合理的解釋。一個電子以奇特的分身術穿過雙縫，它的波函數自身與自身發生了干涉，在空間中嚴格地，確定地發展。在這個階段，因為沒有進行觀測，說電子在什麼地方是沒有什麼意義的，只有它的概率在空間中展開。物理學家們常常擺弄玄虛說：電子無處不在，而又無處在，指的就是這個意思。然而在那以後，當我們把一塊感光屏放在它面前以測量它的位置的時候，事情突然發生了變化！電子突然按照波函數的概率分佈而隨機地作出了一個選擇，並以一個小點的形式出現在了某處。這時候，電子確定地存在於某點，自然這個點的概率變成了一百％，而別的地方的概率都變成了0。也就是說，它的波函數突然從空間中收縮，聚集到了這一個點上面，在這個點出現了強度為一的高峰。而其他地方的波函數都瞬間降為。 哦，上帝，發生了什麼事？為什麼電子的波函數在一剎那發生了這樣的巨變？原本形態優美，嚴格地符合薛定諤方程的波函數在一剎那轟然崩潰，變成了一個針尖般的小點。從數學上來說，這兩種狀態顯然是沒法互相推導的。在我們觀測電子以前，它實際上處在一種疊加態，所有關於位置的可能性疊合在一起，彌漫到整個空間中去。但是，當我們真的去看它的時候，電子便無法保持它這樣優雅而面面俱到的行為方式了，它被迫作出選擇，在無數種可能性中挑選一種，以一個確定的位置出現在我們面前。 波函數這種奇蹟般的變化，在哥本哈根派的口中被稱之為坍縮（collapse），每當我們試圖測量電子的位置，它那原本按照薛定諤方程演變的波函數ψ便立刻按照那個時候的概率分佈坍縮（我們記得ψ的平方就是概率），所有的可能全都在瞬間集中到某一點上。而一個實實在在的電子便大搖大擺地出現在那裡，供我們觀賞。 在電子通過雙縫前，假如我們不去測量它的位置，那麼它的波函數就按照方程發散開去，同時通過兩個縫而自我互相干涉。但要是我們試圖在兩條縫上裝個儀器以探測它究竟通過了哪條縫，在那一剎那，電子的波函數便坍縮了，電子隨機地選擇了一個縫通過。而坍縮過的波函數自然就無法再進行干涉，於是乎，干涉條紋一去不復返。 奇怪，非常奇怪。為什麼我們一觀測，電子的波函數就開始坍縮了呢？ 事實似乎是這樣的，當我們閉上眼睛不去看這個電子，它就不是