Introduction:
Schrödinger equation
What does it tell us?
This equation treats matter not as particles, but as waves, and describes how such waves propagate.
Why is it important?
Schrödinger's equation is the basis of quantum mechanics, and together with general relativity it constitutes the most effective theory of the physical universe today.
What does it bring?
A radical revision of the physics that describes the world at extremely small scales, where every particle has a "wave function" that describes a probability cloud of possible states. At this level, the world is inherently uncertain. It attempted to connect the microscopic quantum world with the macroscopic classical world, leading to philosophical questions that still have an impact today. But experimentally, quantum theory works so well that without it, there would be no computer chips and lasers today.
｜Written
Lao Jia | Translation
In 1900, the great physicist Lord Kelvin declared that "two dark clouds" loomed over the latest thermal and optical theories, which were thought to be almost complete descriptions of nature.
The first dark cloud concerned the question: How could the Earth pass through an elastic solid? For example, this is basically the case with luminous ether. The second dark cloud was the Maxwell-Boltzmann theory of the distribution of energy. Kelvin's nose for important questions was extremely accurate. This article will see how the second question led to another important pillar of modern physics - quantum theory. .
The quantum world is notoriously weird. Many physicists believe that if you can't understand how weird it is, you don't understand it at all. There are many opinions on this idea, because the quantum world is so different from the human-scale world we know that even the simplest concepts become completely unrecognizable. For example, this is a world where light is both a particle and a wave. It's a world where a cat in a box can be alive and dead at the same time... until you open the box, that is, and the unfortunate animal's wave function suddenly "collapses" into one state or another. In the quantum multiverse, there exists a copy of our universe where Hitler lost World War II, and another copy where he won the war. We happen to live in (that is, exist as a quantum wave function in) the first copy. Other versions of us live in copies of another universe, equally real but invisible to us.
Quantum mechanics is definitely weird. Whether it's that weird, though, is another matter entirely.
It all starts with the light bulb. This makes sense, too, since this was one of the most dazzling applications to emerge from the emerging disciplines that Maxwell so brilliantly unified: electricity and magnetism. In 1894, an electrical company hired a German physicist named Max Planck to design the most efficient light bulb that would produce the most light and use the least power. He saw that the key to the problem lay in a fundamental question in physics that had been posed in 1859 by another German physicist, Gustav Kirchhoff. It involves a theoretical construct called a "black body" that absorbs all electromagnetic radiation that falls on it. The big question is: How does such an object emit radiation? It cannot store all the radiation, and some will definitely be emitted again. In particular, what is the relationship between the intensity of the emitted radiation and its frequency and the temperature of the object?
Thermodynamics has given an answer, which regards the black body as a box with the walls as perfect mirrors. Electromagnetic radiation bounces back and forth between mirrors. When the system stabilizes to equilibrium, how is the frequency distribution of the energy in the box? In 1876, Boltzmann proved the "equalization of energy theorem": energy is distributed equally to each independent component of motion. These components are like the fundamental wave on a violin string: normal modes.
There's just one problem with this answer: it can't be true. This means that the total power radiated at all frequencies must be infinite. This paradoxical conclusion has been called the "UV catastrophe": "UV" because it is the beginning of the high-frequency range, and "catastrophe" is indeed the right word. No real object can emit unlimited power.
Although Planck was aware of this problem, he did not take it seriously because he did not believe in the equipartition of energy theorem. Ironically, his work solved the paradox and eliminated the UV scourge, but he only noticed it later. He used experimental observations of the relationship between energy and frequency and fit mathematical formulas to the data. His formula was derived in early 1900 and initially had no physical basis. But this formula just works. But later that year, he tried to fit his formula with the classical thermodynamic formula and believed that the energy level of a blackbody resonator could not change continuously as thermodynamics assumed. Instead, these energy levels must be discrete—with tiny gaps between them. In fact, for any given frequency, the energy must be an integer multiple of that frequency, multiplied by a very small constant. We now call this number "Planck's constant", represented by . Its value (in joules·second) is
，
The numbers in parentheses may be inaccurate. This value is derived from the theoretical relationship between Planck's constant and other more easily measured quantities. The first such measurement was made by Robert Millikan using the photoelectric effect described later. Tiny packets of energy are now called "quanta" (plural of quantum), from the Latin quantus (how much).
Planck's constant may indeed be small, but if the set of energy levels at a given frequency is discrete, the total energy is finite. So the UV catastrophe is a sign that continuum models fail to reflect nature. This means that, on extremely small scales, nature must be discrete. Planck did not think of this at first. He thought that his discrete energy level was a mathematical trick that could lead to a reasonable formula. In fact, Boltzmann had a similar idea in 1877, but did not go further. But when Einstein's fertile imagination bore fruit, everything changed and physics entered a new realm. In 1905, the same year he proposed his special theory of relativity, he studied the photoelectric effect, in which light strikes a suitable metal, causing it to emit electrons. Three years ago, Phillipe Lenard noticed that when the frequency of light is higher, the energy of electrons is also higher. But Maxwell fully confirmed the wave theory of light, suggesting that the energy of an electron should depend on the intensity of the light, not its frequency. Einstein realized that Planck's quanta would explain this difference. He believed that light was not a wave, but made up of tiny particles, which we now call "photons." The energy of a single photon of a given frequency should be the frequency times Planck's constant - just like one of Planck's quanta. Photons are quanta of light.
There is an obvious problem with Einstein's theory of the photoelectric effect: it assumes that light is a particle. But there is plenty of evidence that light is a wave. In addition, the photoelectric effect is incompatible with the fact that light is a wave. So is light a wave? Is it a particle?
yes.
It is both - or, in some respects, each. In some experiments, light appears like waves. In other cases, it behaves like particles. As physicists began to understand the universe at extremely small scales, they assumed that light wasn't the only thing with this strange dual nature—sometimes it was a particle, and sometimes it was a wave. This is true of all matter. They call it "wave-particle duality." The first person to understand this dual nature was Louis-Victor de Broglie, in 1924. He rewrote Planck's law in terms of momentum rather than energy, and suggested that the momentum of particles and the frequency of waves should be related: multiplying the two gives Planck's constant. Three years later, his theory was proven correct, at least for electrons. On the one hand, electrons are particles and we can observe their behavior; on the other hand, they diffract like waves. In 1988, sodium atoms were also discovered to be wavelike.
Matter is neither a particle nor a wave, but a combination of both - a wave particle.
Several more or less intuitive images have been devised of the dual nature of matter. In a picture, particles are local wave groups called "wave packets", as shown in Figure 14.1. The entire wave packet can behave like a particle, but some experiments can detect the wave-like structure inside it. Attention shifted from creating images of wave particles to understanding how they behave. The task was quickly accomplished, and the core equations of quantum theory emerged.
Figure 14.1 Wave packet
This equation was named after Erwin Schrödinger. In 1927, building on the work of several other physicists, notably Werner Heisenberg, he wrote a differential equation for an arbitrary quantum wave function. The equation looks like this:
Here 
describes the form of the wave, t is time (so taking 
it 
gives you its rate of change in time), is an expression called the Hamiltonian operator, and is 
, where is Planck's constant. So 
what? This is the strangest feature of the equation. It is the square root of -1. Schrödinger's equation applies to waves defined over complex numbers, not just real numbers like the wave equation we are familiar with.
Physicist Schrödinger (1887-1961). Source: Wikipedia
Where are the waves? The classical wave equation defines waves in space, the solutions of which are numerical functions of space and time. The same is true for the Schrödinger equation, but now the wave function takes complex values, not just real values. It's a bit like a 
wave with a height of . In many ways, 
the emergence of is the most mysterious and profound feature of quantum mechanics. It had appeared before 
in terms of solutions to equations and the methods for finding those solutions, but here it was part of the equation and a definite feature in the laws of physics.
One explanation for this is that quantum waves are a pair of related real waves, just like a complex ocean wave is actually two waves, one with a height of 2 and the other with a height of 3, and the directions of the two heights are at right angles to each other. . But it's not that simple because the two waves don't have a fixed shape. Over time, they cycle through a series of shapes with mysterious connections between them. It's a bit like the electric and magnetic components of light waves, but now electricity can and does "spun" into magnetism and vice versa. Two waves are two faces of a shape that rotates steadily about the unit circle in the complex plane. Both the real and imaginary parts of this rotated shape change in a very specific way: they combine sinusoidally. Mathematically, this leads to the idea that the quantum wave function has some special type of phase. The physical interpretation of this phase is similar but different from the role of phase in the classical wave equation.
Remember Fourier’s technique for solving heat and wave equations? Some special solutions, such as Fourier's sines and cosines, have particularly pleasing mathematical properties. All other solutions, no matter how complex, are superpositions of these normal modules. We can use similar ideas to solve the Schrödinger equation, but now the basic pattern is more complex than sines and cosines. They are called eigenfunctions and we can distinguish them from all other solutions. The eigenfunction is not a general function of space and time, but a function that depends only on space, multiplied by a function that only depends on time. In jargon, space and time variables are separable. The eigenfunction depends on the Hamiltonian operator, which is a mathematical description of such a physical system. Different systems (an electron in a potential well, a pair of colliding photons, whatever) have different Hamiltonians and therefore different eigenfunctions.
For simplicity, consider a standing wave from the classical wave equation—a vibrating violin string with its ends pinned. The shape of the string is almost the same at all times, but the amplitude fluctuates: multiplied by a factor that varies sinusoidally with time. The complex phases of a quantum wave function are similar, but more difficult to draw. For any individual eigenfunction, the effect of a quantum phase change is just a change in the time coordinate. For the superposition of several eigenfunctions, we can decompose the wave function into these components, decompose each component into a pure space part times a pure time part; rotate the time part at the appropriate speed around the unit circle in the complex plane; and then Put the pieces back together. Each individual eigenfunction has a complex amplitude, and the amplitude fluctuates at its own specific frequency.
This may sound complicated, but if you don't decompose the wave function into eigenfunctions, it's a complete mess. Now at least you have a chance.
Despite these complexities, quantum mechanics is just a fancy version of the classical wave equation that results in two waves instead of one—but there's one weird thing that's puzzling. Classical waves can be observed to see what shape they are, even if it is a superposition of several Fourier modes. But in quantum mechanics, you can never observe the entire wave function. All you can observe at any given moment is a single component eigenfunction. Roughly speaking, if you try to measure two of these components at the same time, the measurement process of one component will interfere with the other component.
This immediately raises a difficult philosophical question. If the entire wave function cannot be observed, does it actually exist? Is it a real physical object, or just a convenient mathematical construct? Do unobservable quantities have scientific significance? It is here that Schrödinger's famous cat appears. It arises from a standard interpretation of quantum measurements called the Copenhagen interpretation. [Note 1]
Imagine a quantum system in some kind of superposition. For example, an electron's state is a mixture of spin-up and spin-down, which are pure states defined by eigenfunctions (it doesn't matter what spin-up and spin-down exactly mean). But when you look at the states, you either observe spin-up or spin-down. You cannot observe superposition. Furthermore, once you observe one of these (say spin up), that becomes the actual state of the electron. Your measurements seem to somehow force the superposition into specific component eigenfunctions. This is basically what the Copenhagen interpretation means: your measurement process collapses the original wave function into a single pure eigenfunction.
If you look at a large number of electrons, you'll see that sometimes the spin is up and sometimes the spin is down. You can deduce the probability that the electron is in one of these states. The wave function itself can therefore be interpreted as a probability cloud. It doesn't show the actual state of the electron: it shows the likelihood that when you measure it, you get a specific result. But that makes it a statistical pattern rather than a real thing. It doesn't prove that the wave function is real any more than Kettler's measurement of human height proves that the developing embryo has some sort of bell curve.
The Copenhagen interpretation is simple and straightforward; it reflects what happens in experiments and makes no detailed assumptions about what happens when a quantum system is observed. For these reasons, most working physicists are more than happy to use it. But in those early days when the theory was still being discussed over and over again, some people disagreed, and some still disagree today. One of these opponents was Schrödinger himself.
In 1935, Schrödinger interpreted Unrest for Copenhagen. He could see it working on a practical level for quantum systems such as electrons and photons. Even though the world around him was just a seething mass of quantum particles deep inside, it looked different. To find a way to make this distinction as obvious as possible, Schrödinger proposed a thought experiment in which quantum particles had an unexpectedly obvious effect on cats.
Imagine a box that is unaffected by any quantum interactions when closed. Place a radioactive atom, a radiation detector, a bottle of poison, and a live cat in the box. Now close the box and wait. At some point, a radioactive atom will decay and emit a radiation particle. A detector will detect it, which will trigger a mechanism that breaks the bottle and releases poison, which kills the cat.
In quantum mechanics, the decay of radioactive atoms is a random event. From the outside, no observer can tell whether an atom has decayed. If it decays, the cat is dead; if it doesn't decay, the cat is still alive. According to the Copenhagen Interpretation, it is a superposition of two quantum states - either decaying or not, until someone observed the atom. The same goes for the status of detectors, bottles, and cats. Therefore, the cat is in a superposition of two states: dead and alive.
Since the box is not affected by any quantum interactions, the only way to know if the atom has decayed and killed the cat is to open the box. The Copenhagen interpretation tells us that the instant we do this, the wave function collapses and the cat suddenly becomes pure: dead or alive. However, there is no difference between the inside of the box and the outside world, and we never observe a cat with a superimposed life/death state. So, before we open the box and look inside, the cat inside is either dead or alive.
Schrödinger considered this thought experiment a criticism of the Copenhagen Interpretation. Microscopic quantum systems follow the principle of superposition and can exist in mixed states; macroscopic systems cannot. By linking microscopic systems (atoms) to macroscopic systems (cats), Schrödinger pointed out what he saw as a flaw in the Copenhagen Interpretation: it was nonsense when applied to cats. He would have been shocked by the response from most physicists: "Yes, Erwin, you're absolutely right. The cat really is both dead and alive at the same time until someone opens the box." Especially when he discovers that even if someone opens the box Box, seeing a living or dead cat, he couldn't decide who was right. He might infer that the cat was already in that state before opening the box, but he couldn't be sure. The observable results are consistent with the Copenhagen interpretation.
Of course, we can also do this: put a film camera in the box and film what is actually happening. This concludes the conclusion. "Ah, no," the physicist replied, "you can only see what the camera captured when you open the box. Until then, the film is superimposed: a film of the cat alive, and a film of the cat dead. .”
The Copenhagen Interpretation frees physicists to calculate and figure out what quantum mechanics predicts, without having to face the difficult (if not impossible) problem of how the classical world is built on quantum foundations, an How unimaginably complex macroscopic devices at the quantum scale can measure quantum states. Since the Copenhagen interpretation works, they are not really interested in philosophical questions. So, all generations of physicists were taught was that Schrödinger invented his cat to prove that quantum superposition could also be extended to the macroscopic world—the exact opposite of what Schrödinger was originally trying to tell them.
It's not too surprising that matter behaves strangely at the level of electrons and atoms. We may initially resist the idea because of its unfamiliarity, but we can learn to accept it if an electron is really a small blob of waves rather than a small piece of solid matter. If this meant that the state of the electron itself was a little weird, not just spinning up or down on its axis, but both, we could live with that. If the limitations of the measuring equipment mean that we can never capture this behavior of electrons (any measurements we make will necessarily be of some pure state, up or down), then so be it. If the same applies to a radioactive atom, the state is either "decayed" or "undecayed" because the particles that make it up have states as elusive as electrons, and we might even accept that the entire atom itself is a superposition of those states , until we take measurements. But cats being cats, it seems like it takes some effort to imagine an animal that is simultaneously alive and dead, only to miraculously collapse into one state or another when we open the box. If quantum reality requires a superpositioned cat, why is it so shy that we don’t observe this state?
In the form of quantum theory, there were (until recently) good reasons for requiring that any measurement, i.e. any "observable", be an eigenfunction. There are even more solid reasons to believe why the state of a quantum system should be a wave and follow the Schrödinger equation. How can we change from one state to another? The Copenhagen Interpretation claims that the measurement process somehow (don't ask what) collapses the complex superposed wave function into a single component eigenfunction. Now that you have this formalism, your job as a physicist is to get on with things like measuring and calculating eigenfunctions and stop asking embarrassing questions. If success is measured by whether the answer is consistent with the experiment, it works surprisingly well. If the Schrödinger equation allowed the wave function to behave this way, all would be well, but it doesn't. In his book The Hidden Reality , Brian Greene puts it this way: "Even polite inquiry reveals disturbing features... The instantaneous collapse of a wave... Impossible Emerged from the mathematics of Schrödinger.” Instead, the Copenhagen Interpretation is a pragmatic adjunct to the theory, a way of dealing with measurement without having to understand or confront what it is all about.
That's all well and good, but that's not what Schrödinger meant to point out. He introduced a cat rather than an electron or an atom because it brought the main issue in his mind most prominently. Cats belong to the macroscopic world in which we live, where matter does not behave as quantum mechanics requires. We don't see superimposed cats. [Note 2] Schrödinger asked why the "classical" universe we are familiar with does not resemble the underlying quantum reality. If everything that makes up the world can exist in a superposition, then why does the universe appear to be classical? Many physicists have performed wonderful experiments showing that electrons and atoms do indeed behave in accordance with quantum and Copenhagen extrapolations. But that misses the point: you have to experiment with cats. Theorists have wondered whether the cat could observe itself, or if someone else could secretly open the box and take note of what was inside. Following Schrödinger's logic, they concluded that if a cat observed its own state, then there would be a dead cat in the box that committed suicide by observing itself, plus a live cat that observed itself alive, until it was legal The observer (physicist) opens the box. Then the whole thing collapses into a certain state. Likewise, the friend becomes a superposition of two friends: one sees a dead cat and the other sees a live cat, until the physicist opens the box, causing the friend's state to collapse. This logic can continue until the state of the entire universe is the superposition of a universe with dead cats and a universe with live cats, and then when physicists open the box, the state of the universe collapses.
It's a little embarrassing. Physicists can go on with their work without figuring it out, they can even deny that there is something to figure out, but something is still missing. For example, what would happen if an alien physicist on the planet Aperobetne III opened the box? Will we suddenly discover that we have actually perished after an asteroid hit the Earth and have been living on borrowed time ever since?
The measurement process is not the nice, neat mathematical operation envisioned by the Copenhagen Interpretation. When asked to describe how this device makes a decision, the Copenhagen Interpretation would reply: "It just does." The image of the wave function collapsed into a single eigenfunction describes the input and output of the measurement process, but does not describe how to get from one state to another. become another state. But when you actually make measurements, you can't wave a magic wand and make the wave function collapse against Schrödinger's equation. Instead, from a quantum perspective, you have done something so complex that modeling it realistically is obviously hopeless. For example, to measure the spin of an electron, you let it interact with a suitable device that has a pointer that moves to an "up" or "down" position. Either through a digital display, or by sending a signal to a computer... This device derives one state, and only one state. You won't see the pointer stacked up and down.
We're pretty used to this because that's how the classic world works. But behind this world there should be a quantum world. If the cat is replaced by a device that measures spin, this device should indeed have a superposition state. It would be extremely complex if it were viewed as a quantum system. It contains a large number of particles—roughly estimated to be between and. From the interaction of a single electron with these astronomical numbers of particles, measurements somehow emerge. The professionalism of the company that built this instrument absolutely deserves endless admiration, and it's almost mind-boggling to extract anything meaningful from such a messy pile. It's like trying to figure out the size of a man's shoes by running him through a city. But if you're smart (arrange for him to meet a shoe store), you can get a meaningful result, and a smart instrument designer can get meaningful measurements of the electron's spin. However, detailed modeling of how such a device would work as a true quantum system remains elusive. There are so many details that even the world's largest computer would be defeated. This makes it very difficult to use the Schrödinger equation to analyze the real measurement process.
Even so, we have some idea of how our classical world arises from the quantum world behind it. Let me start with a simple version: light shining on a mirror. The classic answer - Snell's law states that reflected light will be reflected at the same angle as the incident light. In his book about quantum electrodynamics, QED: The Curious Theory of Light and Matter, physicist Richard Feynman explains that this is not what happens in the quantum world. Light is actually a stream of photons, and each photon can bounce anywhere. But if you add up everything a photon can do, you get Snell's law. The vast majority of photons bounce back at angles very close to the angle of incidence. Feynman even managed to show why without using any complicated mathematics, but behind this calculation is a universal mathematical idea: the stationary phase principle. If you add all the quantum states of an optical system together, you get the classical result that light follows the shortest path measured in time. You can even add some fancy stuff and embellish the light path with some classic wave optics diffraction fringes.
This example shows very clearly that the superposition of all possible worlds (in this optical framework) gives the classical world. The most important feature is not the detailed geometry of the ray, but that it produces only one world on a classical level. In the quantum details of a single photon, you can observe superposition, eigenfunctions, everything. But on a human scale, those all cancel out - well, add up - to get a clean, classic world.
The other part of the explanation is called decoherence. We have seen that quantum waves have phase and amplitude. This is a very funny aspect, it's a plural number, but it's still a phase anyway. This phase is critical to any superposition. If you take two superimposed states, change the phase of one of them, and add them back together, the result is unrecognizable. If you do the same thing with many components, the reassembled wave can be almost anything. Losing phase information destroys any Schrödinger's cat-like superposition. Not only do you lose information about whether it's alive or dead, you can't even tell it's a cat. Decoherence occurs when quantum waves no longer have a good phase relationship - they start to behave more like classical physics, and superposition loses all meaning. What causes them to decohere is interactions with surrounding particles. Presumably this is how the instrument measures the electron spin and obtains a specific and unique result.
Both approaches lead to the same conclusion: if you make human-scale observations of a very complex quantum system, you're observing classical physics. Special experimental methods and special devices may preserve some quantum effects and make them stand out in our comfortable classical existence, but as we move to larger behavioral scales, universal quantum systems will soon no longer embody quantumness. .
This is one solution to the poor cat's fate. The experiment could produce a superposed cat only if the box was completely immune to quantum decoherence, and yet such a box does not exist. What did you use to make this box?
But there is another way, an opposite extreme. I said before: "This logic can continue until the state of the entire universe is a superposition." In 1957, Hugh Everett Jr. pointed out that in a sense, You must do this. The only way to provide an accurate quantum model of a system is to consider its wave function. Everyone is happy to do this, whether the system is an electron, an atom, or (more controversially) a cat. Everett turned this system into an entire universe.
He believes that if that's what you want to model, you have no choice. Only the universe can be truly isolated. Everything interacts with everything else. He discovered that if you get this far, the cat problem, and the paradoxical relationship between quantum and classical reality, is easily solved. The quantum wave function of the universe is not a pure eigenmode, but a superposition of all possible eigenmodes. While we can't calculate these things (not even cats can, and the universe is a little more complicated), we can reason. In fact, in a quantum mechanical sense, we express the universe as the combination of all possible things the universe can do.
As a result, the cat's wave function does not necessarily collapse and lead to a single classical observation. It can remain completely unchanged and does not violate the Schrödinger equation. Instead, there are two coexisting universes. In one universe the cat dies; in the other, it doesn't. When you open the box, there are two you and two boxes. One of them belongs to the wave function of the universe with dead cats, and the other belongs to another wave function of the universe with live cats. What we have is not one unique classical world that somehow arises from the superposition of quantum possibilities, but many, many classical worlds, each corresponding to a quantum possibility.
Everett's original version (which he called "relative state construction") attracted attention in the 1970s when Bryce DeWitt gave it a more catchy name: A many-worlds interpretation of quantum mechanics. It is often dramatized from a historical perspective. For example, there is a universe in which Adolf Hitler won World War II, and another universe in which he did not win. The universe in which I am writing this book is the latter, but elsewhere in the quantum realm, Ian Stewart is writing a very similar book, but in German, reminding his readers that They were in a universe where Hitler had won. Mathematically, Everett's interpretation can be seen as the logical equivalent of traditional quantum mechanics, and (in a more restricted interpretation) leads to an efficient way to solve physical problems. Therefore, his form will be able to withstand any experimental test that traditional quantum mechanics is subjected to. So, does this mean that these parallel universes, what Americans call "alternate worlds", really exist? In a world where Hitler wins, is there another me typing happily on a computer keyboard? Or is this just a convenient mathematical construct?
There's an obvious question: How can we be sure that a computer like the one I'm using would exist in a world ruled by Hitler's fantasy Thousand-Year Reich? Clearly, there must be many more universes than just two, in which events must follow reasonable classical patterns. Maybe Stewart-2 doesn't exist, but Hitler-2 does. A common description of the formation and evolution of parallel universes talks about them "splitting" when there is a choice of quantum states. Green points out that this picture is wrong: there is no division. The wave function of the universe has been and will always be fragmented. Its component eigenfunctions are just there: when one of them is chosen, we imagine a split, but the key to Everett's explanation is that nothing in the wave function actually changes.
Despite this problem, a surprising number of quantum physicists accept the many-worlds interpretation. Schrödinger's cat is both alive and dead. One version of us lives in one of those universes, and the other doesn't. This is what the math says. It's not an explanation, but a convenient way of arranging calculations. It's as real as you and me. It is you and me.
This argument does not convince me. It's not the overlay that bothers me, though. I don’t think the existence of a parallel world is inconceivable or impossible. But I do vehemently disagree with the idea that you can isolate quantum wave functions based on human-scale historical narratives. Mathematical separation occurs at the level of the quantum states of the constituent particles. Most combinations of particle states make no sense for the human narrative. A simple substitute for a dead cat is not a live cat. It's a dead cat with one electron in a different state. There are far more complex alternatives than a live cat. They include a cat that suddenly explodes for no apparent reason, a cat that turns into a vase, a cat that is elected President of the United States, and a cat that survives despite the poison released by radioactive atoms. Those stand-in cats are useful as rhetoric, but not representative. Most of the surrogates are not cats; in fact, they defy description in classical terms. If so, most of Stewart's replacements don't look human at all—no shape at all, in fact—and almost all of them exist in a world that makes no sense at all to humans. Therefore, the possibility that another version of me happens to live in another world with human narrative meaning is extremely slim.
The universe is probably an incredibly complex superposition of alternative states. If you think quantum mechanics is fundamentally correct, then it must be. In 1983, physicist Stephen Hawking said that in this sense, the many-worlds interpretation was "obviously correct." But this does not mean that there is a superposition of universes in which the cat is alive or dead and Hitler wins or loses. There is no reason to think that mathematical components can be separated into sets suitable for creating human narratives. Hawking dismissed the narrative interpretation of the many-worlds form, saying: "Plainly speaking, it's all about calculating conditional probabilities - in other words, the probability of A happening when B happens. I think that's what the many-worlds interpretation is about . Some people put a lot of mysticism into it about the splitting of the wave function into different parts. But all you're calculating is conditional probabilities."
The story of the two Hitlers deserves comparison with the story of Feynman's light. Different from the previous story style, Feynman will tell us that there is a classical world in which light hitting a mirror will be reflected at the same angle as the incident angle; there is another classical world in which the reflection angle will be one degree different; and there is another world It will be two degrees different, etc. But he didn't. He tells us that there is a classical world that arises from the superposition of quantum possibilities. There may be an infinite number of parallel worlds at the quantum level, but these worlds do not correspond in any meaningful way to parallel worlds that can be described at the classical level. Snell's law applies to any classical world. If it wasn't, that world couldn't be canon. Just like Feynman's explanation of light, when you add up all the quantum possibilities, this classical world emerges. There is only one such superposition, so there is only one classical universe - ours.
Quantum mechanics is not limited to the laboratory. The entirety of modern electronics depends on it. Semiconductor technology is the basis of all integrated circuits, and silicon chips are quantum mechanically compliant. Without quantum physics, you wouldn't even think that such a device could work. Computers, cell phones, CD players, game consoles, cars, refrigerators, ovens, almost all modern household appliances have memory chips that store instructions to make these devices meet our needs. Many chips contain more complex circuits, such as microprocessors, which put an entire computer on a single chip. Most memory chips evolved from the first true semiconductor device, the transistor.
In the 1930s, American physicists Eugene Wigner and Frederick Seitz analyzed how electrons move through crystals, a problem that required quantum mechanics to solve. . They discovered some of the fundamental characteristics of semiconductors. Some materials are electrical conductors: electrons can flow easily through them. Metal is a good conductor, and copper wire is commonly used in our daily life. Insulators don't allow electrons to flow, so they block the flow of electricity: the plastic covering of a wire is the insulator and prevents us from getting shocked by the power cord of our TV. Semiconductors are a bit of both, depending on the situation. Silicon is the best-known semiconductor and is currently the most widely used, but several other elements such as antimony, arsenic, boron, carbon, germanium, and selenium are also semiconductors. Because semiconductors can switch from one state to another, they can be used to control electric current, which is the basis of all electronic circuits.
Wigner and Seitz discovered that the properties of a semiconductor depend on the energy levels of its internal electrons, and that these energy levels can be controlled by "doping," adding small amounts of specific impurities to the intrinsic semiconductor material. There are two important types of impurity semiconductors: n-type semiconductors, which carry electric current in the form of a flow of electrons; and p-type semiconductors, in which electric current flows in the opposite direction to the electron flow and is formed by "holes" (where there are fewer electrons than normal). )transmission. In 1947, John Bardeen and Walter Brattain at Bell Labs discovered that germanium crystals could be used as amplifiers. If current is fed into it, the output current is higher. William Shockley, head of the Solid State Physics Division, realized how important this was and launched a project to study semiconductors. This is where the transistor (short for "transfer resistor") was born. There are patents older than this, but no working devices or published papers. Since this initial breakthrough, many types of transistors have been invented. One type of transistor device is the JFET (junction field effect transistor, Figure 14.2). Texas Instruments produced the first silicon transistor in 1954. That same year, the U.S. military built a transistor-based computer, TRIDAC. It has a volume of 3 cubic feet and consumes as much power as a light bulb. The United States had a massive military program to develop replacements for vacuum tube electronics, which were too bulky, fragile, and unreliable for military use. This computer was an early step in the plan.
Figure 14.2 Structure of JFET. The source and drain are in the p-type layer at both ends, while the gate is the n-type layer that controls the flow of current. If you imagine the flow of electrons from source to drain as a hose, the gate is equivalent to squeezing the hose, increasing the pressure (voltage) on the drain
Because semiconductor technology is based on doped silicon or similar substances with impurities, it facilitates miniaturization. By bombarding the surface with desired impurities and etching away unwanted areas with acid, we can build circuits in layers on silicon substrates. The affected areas are defined by photographically generated masks, and these masks can be reduced to very small size using optics. The result is today's electronic products, including memory chips that can store billions of bytes of information and high-speed microprocessors that coordinate computer activity.
Another common application of quantum mechanics is lasers. This device emits a highly coherent beam of light—all light waves are in phase with each other. The device contains an optical cavity with a mirror at each end. The cavity is filled with materials that react to light of a specific wavelength and produce more of the same wavelength - optical amplifiers. Energy is pumped in to start the process, causing the light to travel back and forth along the cavity, amplifying it until it reaches a high enough intensity and is released. Gain media can be fluids, gases, crystals, or semiconductors. Different materials are used for different wavelengths. The amplification process depends on the quantum mechanics of atoms. Electrons in atoms can have different energy levels and can jump between energy levels by absorbing or emitting photons.
The word laser means "light amplification by stimulated emission of radiation". When the first laser was invented, many people laughed at it as looking for answers to problems. These people have a real lack of imagination: as soon as a solution is available, a series of suitable problems will quickly emerge. Generating coherent light beams is a fundamental technology that is bound to find uses, just as an improved hammer will find many uses of its own accord. When inventing general-purpose technology, you don't have to think about specific applications. Today, lasers are used in countless ways. It has more mundane uses, such as a laser pointer for lecturing or a laser beam for home repairs. CD players, DVD players and Blu-ray all use lasers to read information from small pits or marks on the disc. Surveyors use lasers to measure distances and angles. Astronomers use lasers to measure the distance from the Earth to the moon. Surgeons use lasers to cut delicate tissue. Laser eye treatment is also very common. It is used to repair detached retinas or reshape the surface of the cornea to correct vision so that patients do not need to wear glasses or contact lenses. The Star Wars anti-missile system was supposed to use powerful lasers to shoot down enemy missiles, and while it was never built, some of the lasers were. Lasers are currently being studied for military applications, similar to the light guns from bad science fiction. It might even be possible to send spacecraft riding powerful laser beams into space.
New uses for quantum mechanics emerge almost every week. Recently there has been the emergence of "quantum dots," tiny flakes of semiconductors whose electronic properties, including the light they emit, change depending on their size and shape. So we can customize it to get many good features. They already have many applications, such as bioimaging, where they can replace traditional (often toxic) dyes. They perform better and emit brighter light.
Some engineers and physicists are working on the basic components of quantum computers. In such devices, the binary states of 0 and 1 can be superimposed in any combination, effectively allowing computations to take on both values at the same time. This will allow it to perform many different calculations in parallel, giving it a huge speedup. Theoretical algorithms have been devised to perform tasks such as factoring numbers into prime factors. Conventional computers run into trouble when numbers exceed a hundred or so digits, but quantum computers should be able to easily factor larger numbers. The main obstacle to quantum computing is decoherence, which destroys superposition states. Schrödinger's cat is seeking revenge for the inhumane treatment it suffered.
This article is excerpted from "17 Equations That Changed the World", [English] written by Ian Stewart, translated by Lao Jia, published by People's Posts and Telecommunications Press in March 2023. "Mr. Sai" is authorized for publication.
Note:
1. It is generally believed that the Copenhagen interpretation originated from discussions among Niels Bohr, Werner Heisenberg, Max Born and others in the mid-1920s. It got its name because Bohr was Danish, but no physicist at the time used the term. Don Howard has suggested that the name and the ideas it contains first appeared in the 1950s, possibly by Heisenberg. See: D. Howard. "Who Invented the `Copenhagen Interpretation'? A Study in Mythology", Philosophy of Science 71 (2004) 669-682.
2. Our cat "Clown" can often observe the superposition state of "sleeping" and "snoring", but this may not count.
recommended reading
Author: [English] Ian Stewart
Translator: Lao Jia
The classic work of Ian Stewart, a famous British mathematics popularizer, has been translated into many languages.
Li Yongle recommends popular science books, "Eulerian Book Award" winning works
17 mathematical stories that changed the course of human civilization. Understand the underlying principles of how the world works and understand the laws of scientific development.
