Quantum Mechanics

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The question of how to decide between them is an open one. Quantum mechanics was developed in the early twentieth century in response to several puzzles concerning the predictions of classical pre th century physics. Classical electrodynamics, while successful at describing a large number of phenomena, yields the absurd conclusion that the electromagnetic energy in a hollow cavity is infinite. It also predicts that the energy of electrons emitted from a metal via the photoelectric effect should be proportional to the intensity of the incident light, whereas in fact the energy of the electrons depends only on the frequency of the incident light.

Taken together with the prevailing account of atoms as clouds of positive charge containing tiny negatively charged particles electrons , classical mechanics entails that alpha particles fired at a thin gold foil should all pass straight through, whereas in fact a small proportion of them are reflected back towards the source. In response to the first puzzle, Max Planck suggested in that light can only be emitted or absorbed in integral units of h n , where n is the frequency of the light and h is a constant.

This is the hypothesis that energy is quantized —that it is a discrete rather than continuous quantity—from which quantum mechanics takes its name. This hypothesis can be used to explain the finite quantity of electromagnetic energy in a hollow cavity. In Albert Einstein proposed that the quantization of energy can solve the second puzzle too; the minimum amount of energy that can be transferred to an electron from the incident light is h n , and hence the energy of the emitted electrons is proportional to the frequency of the light.

Again, energy is quantized. The model has the additional benefit of explaining the spectrum of light emitted from excited atoms; since only certain energies are allowed, only certain wavelengths of light are possible when electrons jump between these levels, and this explains why the spectrum of the light consists of discrete wavelengths rather than a continuum of possible wavelengths. But the quantization of energy raises as many questions as it answers.

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Among them: Why are only certain energies allowed? What prevents the electrons in an atom from losing energy continuously and spiraling in towards the nucleus, as classical physics predicts? In Louis de Broglie suggested that electrons are wave-like rather than particle-like, and that the reason only certain electron energies are allowed is that energy is a function of wavelength, and only certain wavelengths can fit without remainder in the electron orbit for a given energy.

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This theory has been astonishingly successful. This success has continued. Quantum mechanics in the form of quantum electrodynamics correctly predicts the magnetic moment of the electron to an accuracy of about one part in a trillion, making it the most accurate theory in the history of science.

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And so far its predictive track record is perfect: no data contradicts it. But on a descriptive and explanatory level, the theory of quantum mechanics is less than satisfactory. Typically when a new theory is introduced, its proponents are clear about the physical ontology presupposed—the kind of objects governed by the theory. Superficially, quantum mechanics is no different, since it governs the evolution of waves through space.

But there are at least two reasons why taking these waves as genuine physical entities is problematic. First, although in the case of electron interference the number of electrons arriving at a particular location can be explained in terms of the propagation of waves though the apparatus, each electron is detected as a particle with a precise location, not as a spread-out wave.

As Max Born noticed in , the intensity squared amplitude of the quantum wave at a location gives the probability that the particle is located there; this is the Born rule for assigning probabilities to measurement outcomes.

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The second reason to doubt the reality of quantum waves is that the quantum waves do not propagate through ordinary three-dimensional space, but though a space of 3 n dimensions, where n is the number of particles in the system concerned. Hence it is not at all clear that the underlying ontology is genuinely of waves propagating through space. Indeed, the standard terminology is to call the quantum mechanical representation of the state of a system a wavefunction rather than a wave, perhaps indicating a lack of metaphysical commitment: the mathematical function that represents a system has the form of a wave, even if it does not actually represent a wave.

So quantum mechanics is a phenomenally successful theory, but it is not at all clear what, if anything, it tells us about the underlying nature of the physical world. Quantum mechanics, perhaps uniquely among physical theories, stands in need of an interpretation to tell us what it means. Four kinds of interpretation are described in detail below and some others more briefly. The first two—the Copenhagen interpretation and the many-worlds interpretation—take standard quantum mechanics as their starting point. The third and fourth—hidden variable theories and spontaneous collapse theories—start by modifying the theory of quantum mechanics, and hence are perhaps better described as proposals for replacing quantum mechanics with a closely related theory.

The earliest consensus concerning the meaning of quantum mechanics formed around the work of Niels Bohr and Werner Heisenberg in Copenhagen during the s, and hence became known as the Copenhagen interpretation. Rather, quantum mechanics is an extremely effective tool for predicting measurement results that takes the configuration of the measuring apparatus described classically as input, and produces probabilities for the possible measurement outcomes described classically as output.

It is sometimes claimed that the Copenhagen interpretation is a product of the logical positivism that flourished in Europe during the same period. The logical positivists held that the meaningful content of a scientific theory is exhausted by its empirical predictions; any further speculation into the nature of the world that produces these measurement outcomes is quite literally meaningless.

This certainly has some resonances with the Copenhagen interpretation, particularly as described by Heisenberg. However, Bohr thinks we can say little else about the micro-world. Bohr, like Kant, thinks that we can only conceive of things in certain ways, and that the world as it is in itself is not amenable to such conceptualization.

If this is correct, it is inevitable that our fundamental physical theories are unable to describe the world as it is, and the fact that we can make no sense of quantum mechanics as a description of the world should not concern us.

However, the motivation for adopting a Copenhagen-style interpretation can be made independent of any overarching philosophical position. Since the intensity of the wavefunction at a location gives the probability of the particle occupying that location, it is natural to regard the wavefunction as a reflection of our knowledge of the system rather than a description of the system itself.

This view, held by Einstein, suggests that quantum mechanics is incomplete, since it gives us only an instrumental recipe for calculating the probabilities of outcomes, rather than a description of the underlying state of the system that gives rise to those probabilities. But it was later proved as we shall see that given certain plausible assumptions, it is impossible to construct such a description of the underlying state. However, the Copenhagen interpretation has at least two major drawbacks.

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First, a good deal of the early evidence for quantum mechanics comes from its ability to explain the results of interference experiments involving particles like electrons. In Hugh Everett proposed a radically new way of interpreting the quantum state. His proposal was to take quantum mechanics as descriptive and universal; the quantum state is a genuine description of the physical system concerned, and macroscopic systems are just as well described in this way as microscopic ones.

An immediate problem facing such a realist interpretation of the quantum state is the provenance of the outcomes of quantum measurements.

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Recall that in the case of electron interference, what is detected is not a spread-out wave, but a particle with a well-defined location, where the wavefunction intensity at a location gives the probability that the particle is located there. How does Everett account for these facts? What he suggests is that we model the measurement process itself quantum mechanically.

It is by no means uncontroversial that measuring devices and human observers admit of a quantum mechanical description, but given the assumption that quantum mechanics applies to all material objects, such a description ought to be available at least in principle. So consider for simplicity the situation in which the wavefunction intensity for the electron at the end of the experiment is non-zero in only two regions of space, A and B.

The detectors at these locations can be modeled using a wavefunction too, with the result that the electron wavefunction component at A triggers a corresponding change in the wavefunction of the A-detector, and similarly at B. In the same way, we can model the experimenter who observes the detectors using a wavefunction, with the result that the change in the wavefunction of the A-detector causes a change in the wavefunction of the observer corresponding to seeing that the A-detector has fired, and the change in the wavefunction of the B-detector causes a change in the wavefunction of the observer corresponding to seeing that the B-detector has fired.

In sum, the wave structure of the electron-detector-observer system consists of two distinct branches, the A-outcome branch and the B-outcome branch. Since these two branches are relatively causally isolated from each other, we can describe them as two distinct worlds , in one of which the electron hits the detector at A and the observer sees the A-detector fire, and in the other of which the electron hits the detector at B and the observer sees the B-detector fire. This talk of worlds needs to be treated carefully, though; there is just one physical world, described by the quantum state, but because observers along with all other physical objects exhibit this branching structure, it is as if the world is constantly splitting into multiple copies.

It is not clear whether Everett himself endorsed this talk of worlds, but this is the understanding of his work that has become canonical; call it the many-worlds interpretation. According to the many-worlds interpretation, then, every physically possible outcome of a measurement actually occurs in some branch of the quantum state, but as an inhabitant of a particular branch of the state, a particular observer only sees one outcome. This explains why, in the electron interference experiment, the outcome looks like a discrete particle even though the object that passes through the interference device is a wave; each point in the wave generates its own branch of reality when it hits the detectors, so from within each of the resulting branches it looks like the incoming object was a particle.

The main advantage of the many-worlds interpretation is that it is a realist interpretation that takes the physics of standard quantum mechanics literally.

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It is often met with incredulity, since it entails that people along with other objects are constantly branching into innumerable copies, but this by itself is no argument against it. Still, the branching of people leads to philosophical difficulties concerning identity and probability, and these particularly the latter constitute genuine difficulties facing the approach.

Various solutions have been developed in the literature. One might follow Derek Parfit and bite the bullet here: what fission cases like this show is that strict identity is not a useful concept for describing the relationship between people and their successors. Or one might follow David Lewis and rescue strict identity by stipulating that a person is a four-dimensional history rather than a three dimensional object.

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  5. According to this picture, there are two people two complete histories present both before and after the fission event; they initially overlap but later diverge. Identity over time is preserved, since each of the pre-split people is identical with exactly one of the post-split people. Both of these positions have been proposed as potential solutions to the problem of personal identity in a many-worlds universe. A third solution that is sometimes mentioned is to stipulate that a person is the whole of the branching entity, so that the pre-split person is identical to both her successors, and despite our initial intuition otherwise the successors are identical to each other.

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    So the problem of identity admits of a number of possible solutions, and the only question is how one should try to decide between them. Indeed, one might argue that there is no need to decide between them, since the choice is a pragmatic one about the most useful language to use to describe branching persons. The problem of probability, though, is potentially more serious.

    As noted above, quantum mechanics makes its predictions in the form of probabilities: the square of the wavefunction amplitude in a region tells us the probability of the particle being located there. The striking agreement of the observed distribution of outcomes with these probabilities is what underwrites our confidence in quantum mechanics.

    But according to the many-worlds interpretation, every outcome of a measurement actually occurs in some branch of reality, and the well-informed observer knows this. It is hard to see how to square this with the concept of probability; at first glance, it looks like every outcome has probability 1, both objectively and epistemically. In particular, if a measurement results in two branches, one with a large squared amplitude and one with a small squared amplitude, it is hard to see why we should regard the former as more probable than the latter.

    But unless we can do so, the empirical success of quantum mechanics evaporates. It is worth noting, however, that the foundations of probability are poorly understood. When we roll two dice, the chance of rolling 7 is higher than the chance of rolling But there is no consensus concerning the meaning of chance claims, or concerning why the higher chance of 7 should constrain our expectations or behavior.

    So perhaps a quantum branching world is in no worse shape than a classical linear world when it comes to understanding probability. We may not understand how squared wavefunction amplitude could function as chance in guiding our expectations, but perhaps that is no barrier to postulating that it does so function.