Speakable and unspeakable in quantum mechanics pdf download

It provides very clear presentation of the principles of quantum mechanics for the physics student without any previous background. This is how one would learn quantum mechanics in a standard university course. There are a great many textbooks available for studying quantum mechanics.

Here are a few especially important ones with some notes to guide choices among them. It is good to work with two or three texts when learning QM.

No text is perfect and differences in approach can illuminate the subject from different angles. This book is not recommended for beginners, and not recommended as a textbook. It is recommended once one has some technical background to deepen understanding of the fundamental concepts of quantum mechanics. This is a brief, but elegant introduction.

This is a classic, beautiful book that remains one of the clearest presentations of quantum mechanics. Even a beginner will be able to follow the presentation. This is a comprehensive, encyclopedic text. This is a standard undergraduate text for a first course in QM, and I would recommend it as a starting point for beginners. It is concise and very easy to read. There is an emphasis on conceptual development. Unfortunately, there are no worked examples in the book, and the answers to the problems are available only to instructors.

It is easy to find and has recently been updated. This is a nicely designed book, relatively well-written. It is a good starting point for beginners, but not at comprehensive as Shankar.

This is a standard graduate text in the US, not recommended for beginners, but quite good at an advanced level. This is generally used as a graduate text. The material is introduced at a higher level than Griffiths and Shankar, with lots of mathematics.

There is a wealth of problems, but unfortunately few solutions are provided, making it most useful in a classroom setting or in conjunction with a book that contains worked examples and derivations. This book is extremely mathematical in emphasis.

There is less emphasis on conceptual development, and it is best used after one has acquired a conceptual understanding of QM and wants to see the mathematical development. The approach is very revealing. It is a difficult text, in part because some of the formalism is abstract and unconventional, but it is well worth the effort to comprehend.

The problems throughout are excellent, but again unfortunately, solutions are not included in the text. This book is highly recommended as a starting point. It starts from ground zero, developing the mathematical tools needed to understand quantum mechanics. It is well written, and friendlier than Griffiths for students who are learning the subject on their own.

QM is not introduced until page The introductory chapter on linear algebra is very good. At pages, it is comprehensive. It covers Feynman path integrals more thoroughly than other books, and contains solved problems.

If you buy one book on QM, this is a good choice. This is a very good book as well. It covers theory and problem solving in an integrated way. It is easy to follow and full of problems and solutions that are related to the experimental basis of the theory.

Even a seasoned teacher will find himself from time to time reaching for them:. The last three decades have been a golden age for studying foundations of quantum Mechanics. Most of the active research is published in journals. The discussion surrounding standard non-relativistic quantum mechanics has stabilized in a way that makes it possible to survey.

Three recent books absorb and organize the work of these decades. This is a recent text on the history and conceptual foundations of quantum mechanics.

It will serve an excellent primary text on the foundations of quantum mechanics for philosophy students, and will also make an excellent supplement to the standard quantum physics texts of physics students.

It provides is an up-to-date survey of the landscape with sophisticated analysis and commentary. The book is well-suited for use in or for the layperson with a serious interest in foundations.

The discussion is sophisticated without undue technicality and manages philosophical analysis in a jargon-free way. This is an excellent, if challenging introduction to quantum foundations. The book is unparalleled in clarity and uncompromising in its insistence on ontological intelligibility.

The author makes no bones about where his own sympathies lie, but it will reward the study of any beginning student or seasoned practitioner. The book is a gripping tale of a turbulent time in the history of physics, when personalities clashed as deeply as philosophical sympathies. This is a lively development and well-written defense of the Everettian viewpoint that looks beyond standard non-relativistic theory and argues that the real lesson of quantum conundra and their reconciliation of quantum mechanics with General Relativity is the recognition that space-time is not fundamental.

Terminology 2. Mathematics 2. Quantum Mechanics 4. Vector Addition. Bibliography Books Useful For Beginners Here are some recent books that will be especially useful to beginners. Travis Norsen, T. Susskind, L. Quantum Mechanics Textbooks There are a great many textbooks available for studying quantum mechanics. Ballentine, L. Basdevant, J. Dalibard, , Quantum Mechanics , Berlin: Springer. Set 1 in parameter Singapore Tel: 65 Fax: 65 www.

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It is a merit of the de Broglie-Bohm version to bring this out so explicitly that it cannot be ignored. Thus the nonlocal velocity relation in the guiding equation is but one aspect of the nonlocality of Bohmian mechanics. There is also the nonlocality, or nonseparability, implicit in the wave function itself, which is present even without the structure—actual configurations—that Bohmian mechanics adds to orthodox quantum theory.

As Bell has shown, using the connection between the wave function and the predictions of quantum theory about experimental results, the elimination of this nonlocality, if possible at all, is extremely difficult see Section 2. The nonlocality of Bohmian mechanics can be appreciated perhaps most efficiently, in all its aspects, by focusing on the conditional wave function.

Suppose, for example, that in an EPR-Bohm experiment particle 1 passes through its Stern-Gerlach magnet before particle 2 arrives at its magnet. You can dictate the kind of spin eigenstate produced for particle 2 by appropriately choosing the orientation of an arbitrarily distant magnet.

As to the future behavior of particle 2, in particular how its magnet affects it, this of course depends very much on the character of its conditional wave function; hence the choice of orientation of the distant magnet strongly influences that behavior. This nonlocal effect upon the conditional wave function of particle 2 follows from combining the standard analysis of the evolution of the wave function in the EPR-Bohm experiment with the definition of the conditional wave function.

For simplicity, we ignore permutation symmetry. Before reaching any magnets the EPR-Bohm wave function is a sum of two terms, corresponding to nonvanishing values for two of the four possible joint spin components for the two particles. Each term is a product of an eigenstate for a component of spin in a given direction for particle 1 with the opposite eigenstate i.

Moreover, by virtue of its symmetry under rotations, the EPR-Bohm wave function has the property that any component of spin, i. This property is very interesting. The evolution of the particle-1 factor leads to a displacement along the magnetic axis in the direction determined by the sign of the spin component i.

Once this displacement has occurred and is large enough the conditional wave function for particle 2 will correspond to the term in the sum selected by the actual position of particle 1.

For a more explicit and detailed discussion see Norsen The nonlocality of Bohmian mechanics has a remarkable feature: it is screened by quantum equilibrium.

This follows from the fact that, given the quantum equilibrium hypothesis, the observable consequences of Bohmian mechanics are the same as those of orthodox quantum theory, for which instantaneous communication based on quantum nonlocality is impossible see Eberhard Valentini emphasizes the importance of quantum equilibrium for obscuring the nonlocality of Bohmian mechanics.

Valentini [a] has also suggested the possibility of searching for and exploiting quantum non-equilibrium. However, in contrast with thermodynamic non-equilibrium, we have at present no idea what quantum non-equilibrium, should it exist, would look like, despite claims and arguments to the contrary.

Like nonrelativistic quantum theory, of which it is a version, Bohmian mechanics is not compatible with special relativity, a central principle of physics: Bohmian mechanics is not Lorentz invariant. Nor can it easily be modified to accommodate Lorentz invariance. Configurations, defined by the simultaneous positions of all particles, play too crucial a role in its formulation, with the guiding equation defining an evolution on configuration space.

This difficulty with Lorentz invariance and the nonlocality in Bohmian mechanics are closely related. Since quantum theory itself, by virtue merely of the character of its predictions concerning EPR-Bohm correlations, is irreducibly nonlocal see Section 2 , one might expect considerable difficulty with the Lorentz invariance of orthodox quantum theory as well with Bohmian mechanics. For example, the collapse rule of textbook quantum theory blatantly violates Lorentz invariance. As a matter of fact, the intrinsic nonlocality of quantum theory presents formidable difficulties for the development of any many-particle Lorentz invariant formulation that avoids the vagueness of orthodox quantum theory see Maudlin Bell made a somewhat surprising evaluation of the importance of the problem of Lorentz invariance.

Those paradoxes are simply disposed of by the theory of Bohm, leaving as the question, the question of Lorentz invariance. So one of my missions in life is to get people to see that if they want to talk about the problems of quantum mechanics—the real problems of quantum mechanics—they must be talking about Lorentz invariance.

The most common view on this issue is that a detailed description of microscopic quantum processes, such as would be provided by a putative extension of Bohmian mechanics to the relativistic domain, must violate Lorentz invariance. In this view Lorentz invariance in such a theory would be an emergent symmetry obeyed by our observations—for Bohmian mechanics a statistical consequence of quantum equilibrium that governs the results of quantum experiments. This is the opinion of Bohm and Hiley , of Holland , and of Valentini However—unlike nonlocality—violating Lorentz invariance is not inevitable.

It should be possible, it seems, to construct a fully Lorentz invariant theory that provides a detailed description of microscopic quantum processes. Another possibility is that a fully Lorentz invariant account of quantum nonlocality can be achieved without the invocation of additional structure, exploiting only what is already at hand, for example, the wave function of the universe or light-cone structure. In the sort of theory discussed there, the wave function of the universe provides a covariant prescription for the desired foliation.

Such a theory would be clearly Lorentz invariant. But it is not so clear that it should be regarded as relativistic. Be that as it may, Lorentz invariant nonlocality remains somewhat enigmatic. The issues are extremely subtle. For example, Bell rightly would find.

Fundamental particles of the same kind, for example electrons, are treated in quantum mechanics as if they are somehow identical or indistinguishable. This treatment is reflected in the symmetry properties of the wave function of a many-particle system under permutations of the coordinates of those particles.

For electrons, and for other fermions , the wave function must change its sign when the coordinates of a pair of particles are exchanged; it must be antisymmetric. For photons and other bosons it must be symmetric, with no change at all. The justification usually given for this is that the only way to keep track of the individual particles and thereby retain their individuality is by following their trajectories, which of course one cannot and must not do, and in any case does not have, in standard quantum mechanics.

Like so many arguments used to justify various claims in quantum mechanics, this sort of argument, with its positivistic slant, is rather weak. Its conclusion, however, is quite solid. On this basis one often hears it maintained that Bohmian mechanics, because its particles have trajectories deterministic ones to boot , is unable to deal with identical particles. There is, however, no problem whatsoever in incorporating bosons and fermions into Bohmian mechanics.

One simply stipulates that the wave function in Bohmian mechanics has the same symmetry properties under permutations as in standard quantum mechanics. Once again, in Bohmian mechanics one does not change the wave function or its evolution equation; one merely adds to it the actual configuration of the particles and its guiding equation.

In fact, by taking the configuration seriously in Bohmian mechanics, one perhaps arrives even more naturally than in standard quantum mechanics at the classification of fundamental quantum particles as bosons and fermions, with wave functions of the appropriate symmetries.

It is natural, in other words, to regard the labelling we assign to particles as an unphysical convenience, and to use on the fundamental level unlabelled configurations rather than labelled ones. Wave functions on this natural configuration space are in effect symmetric, trivially so. Moreover the natural configuration space has a non-trivial topology. When this is taken into account the possibility of antisymmetric wave functions, of fermions, naturally emerges.

For a similar early analysis that is more traditionally quantum, see Leinaas and Myrheim A natural thought that one sometimes hears expressed is that Bohmian mechanics, for which particles with well-defined positions in space play a central role, cannot easily accommodate a relational framework, as most prominently advocated by Julian Barbour With such a framework, location in space is not a fundamental physical notion.

Rather it is shapes formed by arrangements of particles, shapes determined by relative positions, that are physical. Bohmian mechanics can naturally be extended to a relational framework, which also leads to a relational notion of time as well.

Thus, rather than being a regression to outdated modes of physics, a Bohmian perspective suggests the possibility that much of what we regard as fundamental in physics might in fact be imposed by us, through our choice of gauge. Bohmian mechanics has never been widely accepted in the mainstream of the physics community.

Since it is not part of the standard physics curriculum, many physicists—probably the majority—are simply unfamiliar with the theory and how it works. Sometimes the theory is rejected without explicit discussion of reasons for rejection. Such objections will not be dealt with here, as the reply to them will be obvious to those who understand the theory. In what follows only objections that are not based on elementary misunderstandings will be discussed. A common objection is that Bohmian mechanics is too complicated or inelegant.

To evaluate this objection one must compare the axioms of Bohmian mechanics with those of standard quantum mechanics. And, as noted by Hilary Putnam,. Happily, I did not give that reason in Putnam [] , but in any case it is not true.

The formula for the velocity field is extremely simple: you have the probability current in the theory anyway, and you take the velocity vector to be proportional to the current. There is nothing particularly inelegant about that; if anything, it is remarkably elegant! One frequent objection is that Bohmian mechanics, since it makes precisely the same predictions as standard quantum mechanics insofar as the predictions of standard quantum mechanics are unambiguous , is not a distinct theory but merely a reformulation of standard quantum theory.

In this vein, Heisenberg wrote,. Heisenberg More recently, Sir Anthony Leggett has echoed this charge. And in connection with the double-slit experiment, he writes,. No experimental consequences are drawn from [the assumption of definite particle trajectories] other than the standard predictions of the QM formalism, so whether one regards it as a substantive resolution of the apparent paradox or as little more than a reformulation of it is no doubt a matter of personal taste the present author inclines towards the latter point of view.

Leggett R Now Bohmian mechanics and standard quantum mechanics provide clearly different descriptions of what is happening on the microscopic quantum level. So it is only with a purely instrumental attitude towards scientific theories that Bohmian mechanics and standard quantum mechanics can possibly be regarded as different formulations of exactly the same theory.

But even if they were, why would this be an objection to Bohmian mechanics? Even if they were, we should still ask which of the two formulations is superior. Supporters of Bohmian mechanics give more weight to its greater simplicity and clarity. The position of Leggett, however, is very difficult to understand. There should be no measurement problem for a physicist with a purely instrumentalist understanding of quantum mechanics. But for more than thirty years Leggett has forcefully argued that quantum mechanics indeed suffers from the measurement problem.

For Leggett the problem is so serious that it has led him to suggest that quantum mechanics might fail on the macroscopic level. Sir Roger Penrose also seems to have doubts as to whether Bohmian mechanics indeed resolves the measurement problem. He writes that. But contrary to what he writes, his real concern seems to be with the emergence of classical behavior, and not with the measurement problem per se. Under normal circumstances this condition will be satisfied for the center of mass motion of a macroscopic object.

Among these are dwell and tunneling times Leavens , escape times and escape positions Daumer et al. Especially problematical from an orthodox perspective is quantum cosmology, for which the relevant quantum system is the entire universe, and hence there is no observer outside the system to cause collapse of the wave function upon measurement. The idea is that Bohmians, like Everettians, must take the wave-function as physically real.

Moreover, since Bohmian mechanics involves no wave-function collapse for the wave function of the universe , all of the branches of the wave function, and not just the one that happens to be occupied by the actual particle configuration, persist. These branches are those that Everettians regard as representing parallel worlds. As David Deutsch expresses the charge,.

But that is just another way of saying that they are universes too. Deutsch See Brown and Wallace for an extended version of this argument. Not surprisingly, Bohmians do not agree that the branches of the wave function should be construed as representing worlds. For one Bohmian response, see Maudlin Other Bohmian responses have been given by Lewis and Valentini b. The claim of Deutsch, Brown, and Wallace is of a novel character that we should perhaps pause to examine. On the one hand, for anyone who, like Wallace, accepts the viability of a functionalist many-worlds understanding of quantum mechanics—and in particular accepts that it follows as a matter of functional and structural analysis that when the wave function develops suitable complex patterns these ipso facto describe what we should regard as worlds—the claim should be compelling.

On the other hand, for those who reject the functional analysis and regard many worlds as ontologically inadequate see Maudlin , or who, like Vaidman see the SEP entry on the many-worlds interpretation of quantum mechanics , accepts many worlds on non-functionalist grounds, the claim should seem empty.

In other words, one has basically to have already accepted a strong version of many worlds and already rejected Bohm in order to feel the force of the claim. Another interesting aspect of the claim is this: It seems that one could consider, at least as a logical possibility, a world consisting of particles moving according to some well-defined equations of motion, and in particular according to the equations of Bohmian mechanics.

It seems entirely implausible that there should be a logical problem with doing so. We should be extremely sceptical of any argument, like the claim of Deutsch, Brown, and Wallace, that suggests that there is. Thus what, in defense of many worlds, Deutsch, Brown, and Wallace present as an objection to Bohmian mechanics should perhaps be regarded instead as an objection to many worlds itself.

This is regarded by some Bohmians, not as an objectionable feature of the theory, but as an important clue about the meaning of the quantum-mechanical wave function. Bohmian mechanics does not account for phenomena such as particle creation and annihilation characteristic of quantum field theory. This is not an objection to Bohmian mechanics but merely a recognition that quantum field theory explains a great deal more than does nonrelativistic quantum mechanics, whether in orthodox or Bohmian form.

It does, however, underline the need to find an adequate, if not compelling, Bohmian version of quantum field theory, and of gauge theories in particular. A crucial issue is whether a quantum field theory is fundamentally about fields or particles—or something else entirely. For a general discussion of this issue, and of the point and value of Bohmian mechanics, see the exchange of letters between Goldstein and Weinberg by following the link provided in the Other Internet Resources section below.

Inspired by the structure of Bell-type quantum field theories, Tumulka has developed a novel approach to the problem of the ultraviolet divergences of quantum field theory based on what he calls interior boundary conditions. See D.

For a brief introduction to Bohmian mechanics see Tumulka For longer accessible presentations see Bricmont , Norsen , Bricmont , and Maudlin The Completeness of the Quantum Mechanical Description 2. History 4. The Defining Equations of Bohmian Mechanics 5.

The Quantum Potential 6. The Two-Slit Experiment 7. The Measurement Problem 8. The Collapse of the Wave Function 9. Quantum Randomness Quantum Observables Spin Contextuality Nonlocality Lorentz Invariance Identical Particles Quantum Motion on Shape Space The Completeness of the Quantum Mechanical Description Conceptual difficulties have plagued quantum mechanics since its inception, despite its extraordinary predictive successes.

His difficulty had little to do with the novelty of the wave function: That it is an abstract, unintuitive mathematical construct is a scruple that almost always surfaces against new aids to thought and that carries no great message. They concluded with this observation: While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists.

In relation to a theory incorporating a more complete description, Einstein remarked that the statistical quantum theory would … take an approximately analogous position to the statistical mechanics within the framework of classical mechanics. Einstein We note here, and show below, that Bohmian mechanics exactly fits this description. He concluded that It is therefore not, as is often assumed, a question of a re-interpretation of quantum mechanics—the present system of quantum mechanics would have to be objectively false, in order that another description of the elementary processes than the statistical one be possible.

For example, Max Born, who formulated the statistical interpretation of the wave function, assured us that No concealed parameters can be introduced with the help of which the indeterministic description could be transformed into a deterministic one. Born Bohmian mechanics is a counterexample to the claims of von Neumann and Born.

We still find, a quarter of a century after the rediscovery of Bohmian mechanics in , statements such as these: The proof he [von Neumann] published …, though it was made much more convincing later on by Kochen and Specker , still uses assumptions which, in my opinion, can quite reasonably be questioned. Wigner [] Now there are many more statements of a similar character that we could cite. Bell , reprinted in c: Wigner to the contrary notwithstanding, Bell did not establish the impossibility of a deterministic reformulation of quantum theory, nor did he ever claim to have done so.

In the course of investigating Bohmian mechanics, he observed that: in this theory an explicit causal mechanism exists whereby the disposition of one piece of apparatus affects the results obtained with a distant piece.

Bell a, reprinted c: Despite my insistence that the determinism was inferred rather than assumed, you might still suspect somehow that it is a preoccupation with determinism that creates the problem.

History The pilot-wave approach to quantum theory was initiated by Einstein, even before the discovery of quantum mechanics itself.

Bibliography Albert, David Z. Becker, Adam, , What is Real? Bell, John S. Reprinted in Bell c: 14—21 and in Wheeler and Zurek — Isham, Roger Penrose, D.