by Lee Smolin
This state IN BETWEEN is an example of a correlated state. We call it that because the properties of the two systems are correlated. The state of the atom is uncertain, but if we know what state the atom is in, we can deduce which state the Geiger counter will be in.
But if we then open the box and look inside, we never see a superposition. Looking inside is a measurement which is governed by Rule 2. We see either that the Geiger counter has clicked, so the atom has decayed, or that the atom is still excited and the counter has yet to click.
This seems downright weird. Here are some of the questions it raises.
Why are there two rules for how quantum systems change in time, rather than one?
Why do we treat measurements and observations differently from other processes? Certainly a measurement device is just a machine made out of atoms. Shouldn’t there just be one rule for how things change in time, which applies in all cases?
And just what is it about measuring devices that makes them different? Is it just the size or complexity of the device? Is it the vast number of atoms making it up? Or is it the fact that it can be used to gain information?
When does the collapse to a definite state happen? Is it when the atom meets the detector? Or when the signal is amplified? Or is it not until we become conscious of the information?
These questions are all aspects of the measurement problem.
The simplest answer is that, one way or another, it must be this way. We never observe large things to be indefinite: in our world there are no Geiger counters that both have and have not clicked. Every question we ask has a definite answer. But we need superpositions to explain atoms and radiation.
To emphasize how strange this all is, Schrödinger put a cat in the box along with the atom and the Geiger counter. He wired up the signal from the Geiger counter to a transformer, whose output was clipped to the cat’s ears. When the Geiger counter signaled its detection of the photon, the cat got a fatal pulse of electricity.
(Of course Schrödinger didn’t actually do this. This is a thought experiment intended to shock us, not the cat.)
We wait a half-life and then open the box. Do we apply Rule 1 or Rule 2? Let’s discuss what each of the two rules would predict.
Assume first that Rule 1 applies to the whole system inside the box, including the cat. That system consists of the atom, the Geiger counter, and the cat. There are again two states with easy interpretations. One of these is the initial state
INITIAL = EXCITED and NO and ALIVE
This is the state in which the atom is excited, the Geiger counter has detected nothing, and the cat is alive. After a long time we can be sure the atom has decayed and the cat has died.
FINAL = GROUND and YES and DEAD
This state is the result of the decay and features a stable atom in the ground state, a detector that clicked, and a dead cat.
In between, the state is a superposition of these two possibilities.
IN BETWEEN = (EXCITED and NO and ALIVE) or (GROUND and YES and DEAD)
But a cat is a mammal, with a brain and perhaps a conscious mind. It is nearly as complex as we are. So why does it make sense for the cat to be in a superposition of alive and dead? If it doesn’t make sense for us to exist in a superposition, it surely doesn’t for the cat either. If we apply Rule 2 to our observation, we should also apply it to the cat, who in essence observes the signal from the detector.
So we’d better apply Rule 2. When we open the box, the system makes a choice and jumps into a definite state. We find either a live cat or a dead cat.
So Rule 1 alone does not apply to humans or cats. But does it apply to Geiger counters? And where is the line? Why does it apply to atoms and not to big collections of atoms like detectors, cats, and humans?
FIGURE 4. The Schrödinger’s cat thought experiment. A detector is constructed to respond with an electrical pulse to a photon that would be emitted by an atom decaying and jumping down from an excited state into the ground state. The cat is connected to the circuit, so when the pulse comes it will electrocute him. After a short time, the atom is in a superposition of its excited and decayed states. Rule 1 applied to this case predicts that the cat inside the closed box is then in a superposition of two states, alive and dead.
This conversation is called the puzzle of Schrödinger’s cat. A measure of the fecundity of the human imagination is the number of responses that have been offered to this puzzle.
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A FEW YEARS AFTER Bell published his restriction, an even more powerful result was published which further limits the options for realist quantum theories. To describe it we go back to Bell.
One way to put what is surprising about Bell’s result is that the answers that Anna gives to the questions she is posed have to depend on the questions that Beth is asked. This is shocking because Anna and Beth are separated, so locality would preclude such a dependence. But notice that they don’t need to be separated for the conclusion to apply. Then the result that Anna’s answers depend on the questions Beth is asked is surprising for another reason.
Earlier we talked about pairs of measurements which are mutually incompatible, like a particle’s position and momentum. In these cases it seems that the act of measuring one quantity interferes with or disturbs the value of the other. We described this by saying that the order in which the two measurements are made matters.
But notice that the case of Anna and Beth is not like this. Questions asked of Beth are completely compatible with questions asked of Anna. The order in which they are questioned doesn’t matter. This was true when the two friends were far apart when they were questioned, but it would remain true if they were standing next to each other.
Still, even if the order in which we question the two friends is irrelevant, so that questions to one are compatible with questions to the other, it remains the case that the answers Anna gives depend on the choice of which questions Beth is asked.
This dependence is called contextuality, because the answers Anna gives turn out to depend on the overall context, even to the point that they depend on choices made about which other questions will be asked. It turns out to be widely true of quantum mechanical systems. Contextuality occurs in situations in which our system is described by at least three properties, which we can call A, B, and C. A is compatible with both B and C, so A may be measured simultaneously with either B or C. But B and C are not compatible with each other, so we can measure only one at a time.
So we can measure A and B or we can measure A and C. We make a series of experiments in which we make both choices, and we record all the answers. When we do we will find—assuming that quantum mechanics is correct—that the answers to A depend on whether we chose to measure B or C along with A. The conclusion is that nature is contextual. This is the case with quantum mechanics, and experiments have been done which confirm this prediction of the theory. So it must be true in any deeper theory which will replace quantum mechanics.
This result was first proved by John Bell in the early 1960s, before he published his result on nonlocality. He submitted it to a journal but the paper was apparently lost for two years, “on the editor’s desk,” so it wasn’t published till 1966. By then the result had been proved again by two mathematicians, Simon Kochen and Ernst Specker, so the result that quantum mechanics is contextual is often attributed to them, but it ought properly to be called the Bell-Kochen-Specker theorem.4
Quantum mechanics was invented in order to explain certain puzzling experimental results concerning light, radiation, and atoms. The three new phenomena we discussed in this chapter—entanglement, nonlocality, and contextuality—are a far distance more puzzling. Each is so weird that they were for a time used to argue that quantum mechanics must be wrong, till experiments confirmed that they are indeed all aspects of the natural world. This was certainly not anticipated. Entan
glement, nonlocality, and contextuality each emerged from the study of quantum systems, and it is very fair to say that they were each predictions of the quantum theory which, very surprisingly, turned out to be true.
These three aspects of quantum physics present severe challenges to realism. Indeed, they rule out large classes of realist theories. In particular, nonlocal entanglement is incompatible with all theories whose beables influence each other only through local forces, whose actions propagate at the speed of light or slower. Any realist theory which can mimic quantum mechanics must then describe a world which violates this condition and so openly embraces nonlocality. This is why Einstein talked of “spooky action at a distance.” The choice we face is simple: we may give up realism and accept quantum mechanics as the final word, or we can move ahead and seek to understand how nature violates locality while still managing to make sense at all.
FIVE
What Quantum Mechanics Doesn’t Explain
Quantum mechanics doesn’t answer every question we can ask about the atomic world, but it gets a lot right. This is a good time to sum up what we’ve learned about what quantum mechanics does and does not explain.
Roughly speaking, quantum mechanics predicts and explains two kinds of properties: properties of individual systems, and averages taken over many individual systems. These are very different.
When we can attribute a definite value to a quantity—as we can when we make a measurement—this is a property of the individual system that has been measured. But often the uncertainty principle forbids us from discussing anything other than averages.
To what do these averages refer? Because of the uncertainty principle it can happen that two atoms, prepared identically in the same initial state, give different values when measured later. For example, atoms prepared in the same starting position will tend to spread out, and be found in different places later. When the final answers vary we can still measure their average value. Quantum mechanics tells us these averages are taken over many runs of an experiment. An experiment requires us to prepare many copies of a system, wait and then measure each copy, and then take the average of the results.
A collection of atoms which are similar in some way but different in others is called an ensemble. Quantum mechanics deals with ensembles. These may be defined by fixing one quantity, such as energy, to have some definite value, while other parameters vary over a range of values, as required by the uncertainty principle. When we speak of averages or probability in quantum mechanics, we are usually referring to something that can be measured by taking an average over the members of an ensemble consisting of many copies of the atom in question.
That is often easy to do because many experiments deal with a collection of atoms, such as a gas. These are real ensembles, because the atoms in the collection are real. Sometimes, though, the ensemble exists only in the theorist’s imagination.
It is normal to explain the results of averaging over many copies of an individual system in terms of the properties of those individual systems. However, in quantum mechanics it is often the other way around, and a property of an individual atom will be explained in terms of averages over many atoms. But how can the collective determine the individual? These kinds of cases are at the heart of what is most mysterious about the quantum world.
One of the individual properties that quantum mechanics can discuss is the energy of an atom or molecule. It turns out that in quantum mechanics the energies of many systems come in certain discrete values, called the spectrum. The spectrum is a property of individual atoms, as it can be observed in experiments involving just one atom. Atoms, molecules, and various materials all have spectra, and in all these cases they are correctly predicted by quantum mechanics. More than that, quantum mechanics explains why these systems can have only these energies. It accomplishes this by making use of the wave-particle duality. This is one place where averages over many systems are used to explain what happens in an individual system.
The explanation involves two steps. The first is to use the relation between energy and frequency, which is the foundation of the wave-particle duality. A spectrum of discrete values of energy corresponds to a spectrum of discrete frequencies. The second step exploits the picture of a quantum state as a wave. A wave ringing at a definite frequency is like a bell or a guitar string producing sound. The string resonates when plucked, as does the bell when struck, ringing at a definite frequency.
We then use the equation for quantum states changing in time to predict the resonant frequencies of the system. The equation takes as input the masses of the particles involved in the system and the forces between them, and gives as output the spectrum of resonant frequencies. These are then translated into resonant energies.
This works well. For example, if we input that the system is made of an electron and a proton, bound together by their electrical attraction, the equation outputs the spectrum of the hydrogen atom.
In most cases, there is a state of lowest energy, which is called the ground state. States of higher energy are called excited states. You excite the ground state by adding the energy needed to bring it up to the level of one of these excited states. This causes the state to transition from the ground state to the excited state. The added energy is often delivered by photons. Excited states tend to be unstable, because they can drop back down to the ground state by radiating away the excess energy in the form of a photon. The ground state has no state below it to decay to, and so it is stable. Most systems spend most of the time in their ground states.
This method has been tested on a great many systems, including atoms, molecules, nuclei, and solids. In all cases the predicted spectra are observed. In addition to getting the spectrum of possible energies right, quantum mechanics makes predictions for averaged quantities, such as average values of the positions of the particles making up the system.
For each resonant frequency, the equation that defines quantum mechanics can be solved to yield the corresponding wave. We then use Born’s rule (that the square of the wave is proportional to the probability of finding the particle) to predict probabilities for the particle to be found different places.
The states of definite energy have indefinite positions. Suppose we prepare a million different hydrogen atoms, all in the ground state. In each of these, we measure the position of the electron (relative to the proton, which is held fixed in the center of the atom). Each individual measurement results in a different position. Measuring a million different atoms gives us a million different positions. Some will be far from the proton, but most will be clustered around the proton in the center. The array of possible positions makes up a statistical distribution and it is this distribution, rather than a definite position, that quantum mechanics predicts.
According to the uncertainty principle, the position of any one of the electrons cannot be predicted. But the statistical distribution of positions, which results from measuring a great many cases, can be found. These statistical distributions are computed by squaring the wave.
To summarize, quantum mechanics makes two kinds of predictions. It makes predictions for the discrete spectra of energies, or other quantities, a system can have. And it also makes predictions for statistical distributions of quantities such as positions of particles.
In every case I know of, these two kinds of predictions have been confirmed by experiment. This is exceedingly impressive.
But does quantum mechanics explain how individual atoms work? Is a successful prediction always the same as an explanation?
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IT IS EQUALLY IMPRESSIVE what quantum mechanics does not do. It does not describe or predict where a particular individual electron will be found. Because it deals in averages, quantum mechanics has little to tell us about what goes on in individual systems.
There are lots of cases where we deal with averages. We have no problem measuring the average height
of Canadians. This is because each Canadian is some definite number of centimeters tall. We add all those centimeters up, divide by the number of Canadians we measured, and we get the average.
In cases like this, the average is made up of individual heights, which are properties of individuals. We could choose to work with the whole list of heights, but for many purposes, such as designing furniture or cars, the averaged value is all we need. If we need anything else, it is likely to be the standard deviation, which tells us the typical range of variations of height. Using the average and standard deviation, an airline could (if it wanted to) build airplane seats in which 95 percent of Canadians would be comfortable.
In these cases, the information which we ignore when we use averages is really present in the world, but we choose to suppress it in favor of the averages. The uncertainties which arise from our use of probabilities are purely due to our ignorance.
But suppose that each time we measured someone’s height, we got a different result. There is then an element of genuine randomness, because there is no way for us to know how tall someone might be the next time they are measured. That is closer to the case we deal with in quantum theory. What does the average signify, and what does it explain, when there is no story about individual cases?
Quantum mechanics makes correct predictions for averages, in spite of having nothing definite to say about individual cases. We seem to lack the kind of explanation we usually expect in cases like height, where the basis of an average is found in the fact that the average is composed of individual cases.
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ONE OF THE MOST UNEXPECTED ASPECTS of quantum mechanics is that a system can change over time in two ways. I described these in chapter 3. Most of the time the quantum state evolves deterministically under Rule 1. But when we make a measurement of the system it evolves in a very different way under Rule 2. The measurement will produce one number out of a range of possible values. Just after the measurement, the quantum state jumps into a state corresponding to the definite value which was measured in the experiment.