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1. Introduction

Later he modified it to say that it was the square of the wave function and still later he said the absolute value determined the probability. It was a postulate. Now the present generation of quantum theorists—some of them—find this unsatisfactory and want to produce a derivation.

If they are right it would be revolutionary but to me the whole enterprise is a solution looking for a problem. But also in classical mechanics, given the inevitable uncertainties in the initial conditions, the situation is actually similar, because only probabilities of the outcome at a later time can be predicted. The main difference with quantum mechanics is that in this theory there is a limit on the relative size of the initial uncertainties, e.

Quantum Methods in Social Science

We agree with Professor Weinberg that this is deeply unsatisfactory. According to the second option, the predictions of quantum theory are not quite correct, but only a very good approximation to the more correct predictions of the spontaneous collapse theory. Many experiments are being carried out in order to decide between spontaneous collapse theories and quantum mechanics. So far, there is no indication that quantum mechanics is wrong and its spectacular successes show that, if its predictions are indeed violated in some situations, this will not be easy to demonstrate.

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Weinberg presents the Copenhagen interpretation on the one hand, and many-worlds and spontaneous collapse theories on the other, as corresponding respectively to what he calls an instrumentalist and a realist approach to the wave function. In the instrumentalist approach the wave function is not regarded as something to be taken seriously as real or objective, but merely as a convenient tool for describing the behavior of measuring devices and the like.

In the realist approach, according to Weinberg, the wave function is not only real and objective but also exhaustive, providing a complete description of the physical state of affairs.


In other words, the alternatives for the wave function for Weinberg are either that it is nothing or it is everything. However, Weinberg does not mention a third possibility, the de Broglie—Bohm theory or Bohmian mechanics, in which the wave function is something but not everything. This theory, which we consider to be, by far, the simplest version of quantum mechanics, does not require any modification of the predictions of ordinary quantum mechanics, nor a bizarre to say the least multiplication of parallel universes.

It was proposed by Louis de Broglie in and rediscovered and developed by David Bohm in For several decades its main proponent was John Stewart Bell, the physicist who did more than any other to establish the existence of the quantum non-locality mentioned by Weinberg. But the quest to construct a complete theory of quantum gravity faces formidable hurdles, both technical and philosophical.

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Overcoming those obstacles is the full-time occupation of a large number of working physicists. There is one fairly direct way of addressing the conceptual issues associated with wave function collapse: Simply deny that it ever happens, and insist that ordinary smooth evolution of the wave function suffices to explain everything we know about the world.

To understand how it works, we need to take a detour into perhaps the most profound feature of quantum mechanics of all: entanglement. When we introduced the idea of a wave function we considered a very minimalist physical system, consisting of a single object a cat. We would obviously like to be able to move beyond that, to consider systems with multiple parts—perhaps a cat and also a dog.

So it would be the most natural thing in the world to guess that the correct quantum-mechanical description of a cat and a dog would simply be two wave functions, one for the cat and one for the dog. In quantum mechanics, no matter how many individual pieces make up the system you are thinking about, there is only one wave function. Other people enjoy the grandiosity for its own sake. As before, we imagine that when we look for Miss Kitty, there are only two places we can find her: on the sofa or under the table.

Dog: in the living room or out in the yard. But instead, quantum mechanics instructs us to consider every possible alternative for the entire system—cat plus dog—and assign an amplitude to every distinct possibility. Here, the first entry tells us where we see Miss Kitty, and the second where we see Mr. According to quantum mechanics, the wave function of the universe assigns every one of these four possibilities a distinct amplitude, which we would square to get the probability of observing that alternative. You may wonder what the difference is between assigning amplitudes to the locations of the cat and dog separately, and assigning them to the combined locations.

The answer is entanglement—properties of any one subset of the whole can be strongly correlated with properties of other subsets. Schematically, that means the state of the system must be of the form. This means there is a nonzero amplitude that the cat is under the table and the dog is in the living room, and also a nonzero amplitude that the cat is on the sofa and the dog is in the yard. An observation collapses the wave function onto one of the two possibilities, table, living room or sofa, yard , with equal probability, 50 percent each. Dog is doing, we would say that there is an equal probability for observing Miss Kitty under the table or on the sofa.

Dog is going to be before we look. Here is the kicker: Even though we have no idea where Mr. Dog is going to be before we look, if we first choose to look for Miss Kitty, once that observation is complete we know exactly where Mr.

Dog is going to be, even without ever looking for him! That means that, given the form of the wave function we started with, it must have collapsed onto the possibility sofa, yard. We therefore know with certainty assuming we were right about the initial wave function that Mr.

Dog will be in the yard if we look for him. We have collapsed Mr. Or, more correctly, we have collapsed the wave function of the universe, which has important consequences for Mr. Dog directly. This may or may not seem surprising to you. Nevertheless, entanglement can lead to consequences that—taken at face value—seem inconsistent with the spirit of relativity, if not precisely with the letter of the law. But there is a tension between them that makes people nervous. In particular, things seem to happen faster than the speed of light. Still, it rubs people the wrong way. Also, he is very adventurous, and lives in the future, when we have regular rocket flights to a space colony on Mars.

Introduction to quantum mechanics

Dog—in the alternative where he starts in the yard, not in the living room—runs away to the spaceport, stows away on a rocket, and flies to Mars, completely unobserved the entire time. Dog is actually observed, collapsing the wave function. But the implications are somewhat surprising. When Billy unexpectedly sees Mr.

Dog bounding out of the spaceship on Mars, he makes an observation and collapses the wave function. If he knew what the wave function was to begin with, featuring an entangled state of cat and dog, Billy immediately knows that Miss Kitty is on the sofa, not under the table. The wave function has collapsed to the possibility sofa, Mars. The important feature of the apparently instantaneous collapse of a wave function that is spread across immense distances is that it cannot be used to actually transmit any information faster than light. Once Billy observes Mr. Dog, we now have a percent chance of observing her to be on the sofa.

But so what? Dog we would find him in the living room. These are ideas that have yet to be fully understood, but the final theory of everything is likely to exhibit non-locality in some very dramatic ways. The leading contender for an alternative to the Copenhagen view of quantum mechanics is the so-called many-worlds interpretation.

The problem with this claim is that we appear to see wave functions collapsing all the time, or at least to observe the effects of the collapse. We can imagine arranging Miss Kitty in a quantum state that has equal amplitudes for finding her on the sofa or under the table; then we look for her, and see her under the table. And that way of thinking has empirical consequences, all of which have been successfully tested in real experiments.

The response of the many-worlds advocate is simply that you are thinking about it wrong. In particular, you have misidentified yourself in the wave function of the universe. After all, you are part of the physical world, and therefore you are also subject to the rules of quantum mechanics. There are three possible states you could be in: You could have seen her on the sofa, you could have seen her under the table, and you might not have looked yet.

This can be schematically portrayed like this:. Now you observe where she is. In the Copenhagen interpretation, we would say that the wave function collapses. But in the many-worlds interpretation, we say that your own state becomes entangled with that of Miss Kitty, and the combined system evolves into a superposition:.

There is no intrinsically quantum-mechanical arrow of time in this interpretation. For many reasons, this is an altogether more elegant and satisfying picture of the world than that provided by the Copenhagen picture.

Quantum Mechanics for Dummies

The problem, meanwhile, should be obvious: The final state has you in a superposition of two different outcomes! If you actually did make an observation of a system that was in a quantum superposition, after the observation you would always believe that you had observed some specific outcome.

Kitty on the sofa and another who saw her under the table, and they both honestly exist there in the wave function. Before we made an observation, the universe was described by a single wave function, which assigned a particular amplitude to every possible observational outcome; after the observation, the universe is described by a single wave function, which assigns a particular amplitude to every possible observational outcome. It has simply evolved in such a way that there are now a greater number of distinct subsets of the wave function describing individual conscious beings such as ourselves.

He left academic physics to work for the Defense Department, and eventually founded his own computer firm. In , theoretical physicist Bryce DeWitt who, along with Wheeler, was a pioneer in applying quantum mechanics to gravity took up the cause of the many-worlds interpretation, and helped popularize it among physicists. Everett lived to see a resurgence of interest in his ideas within the physics community, but he never returned to active research; he passed away suddenly of a heart attack in , at the age of fifty-one.

There remain unanswered questions, from the deep and conceptual—why are conscious observers identified with discrete branches of the wave function, rather than superpositions? But a great deal of progress has been made over the last few decades, especially involving an intrinsically quantum-mechanical phenomenon known as decoherence.

There are great hopes—although little consensus—that decoherence can help us understand why wave functions appear to collapse, even if the many-worlds interpretation holds that such collapse is only apparent. There are two alternatives, with equal amplitudes: the cat is under the table and the dog is in the living room, or the cat is on the sofa and the dog is in the yard:. We saw how, if someone observed the state of Mr. Dog, the wave function would in the Copenhagen language collapse, leaving Miss Kitty in some definite state. Dog, but we simply ignore him. Effectively, we throw away any information about the entanglement between Miss Kitty and Mr. Dog, and simply ask ourselves: What is the state of Miss Kitty all by herself? The problem is that interference—the phenomenon that convinced us we needed to take quantum amplitudes seriously in the first place—can no longer happen.