Quantum Mechanics

"Molecular dynamics is not primarily about making movies of molecules. More often it is about developing quantitative predictions of molecular size and shape, flexibilities, interactions with other molecules, behavior under pressure, and the relative frequency of one state or conformation compared to another. The complex nature of the force fields involved and the large size of typical molecular systems mean that molecular dynamics is almost always chaotic. The changes in a molecule that occur over time are important, but these should be understood in terms of changes in averaged quantities, structural forms, or families of “nearby” structures. Molecular dynamics relies on time-stepping to compute successive snapshots, but these are often used for sampling a probability distribution, or else a number of evolving paths are averaged to describe the likely sequence of changes that would be observed in a typical evolution of the molecule."

"Path integrals (the terminology for functional integrals, derived from their quantum mechanical origin) had been introduced by Dirac in the 1930s. However, their mathematical fuzziness (in contrast to the precise Euclidean Wiener integrals) had discouraged their serious application in quantum theory. Nonetheless, despite the absence of a mathematical definition it became apparent that path integrals in field theory were ideally suited to (i) implement the symmetries of the theory directly, (ii) incorporate constraints simply, (iii) explore field topology, (iv) isolate relevant dynamical variables, (v) describe non-zero temperature. They were key ingredients in model-making for unified theories, and by the late 1970s a working knowledge of functional integrals had become extremely useful for most field theorists."

"In the sixties, a new field of applications of functional integrals appeared — the quantization of gauge fields. The electromagnetic field, the Einstein gravitational field, the Yang-Mills field and the chiral field can serve as examples of gauge fields. The action functionals of those fields are invariant under gauge transformations which depend on one or several arbitrary functions. From a mathematical point of view, gauge fields are fields of geometrical origin which are connected with fibrations over four-dimensional space-time. The specificity of geometrical fields has to be taken into account when quantizing them; otherwise incorrect results may be obtained."

"The idea behind the Feynman path integral goes back to a paper by P. A. M. Dirac published in 1933 in Physikalische Zeitschrift der Sowjetunion. It formed the core of Richard Feynman’s space–time approach to quantum mechanics and quantum electrodynamics. Although the path integral was not mathematically well defined, it was widely used in quantum field theory, statistical mechanics, and string theory. Recently, path integrals have been the heuristic guide to spectacular developments in pure mathematics."

"The path integral is a formulation of quantum mechanics equivalent to the standard formulations, offering a new way of looking at the subject which is, arguably, more intuitive than the usual approaches. Applications of path integrals are as vast as those of quantum mechanics itself, including the quantum mechanics of a single particle, statistical mechanics, condensed matter physics and quantum field theory. ... It is in quantum field theory, both relativistic and nonrelativistic, that path integrals (functional integrals is a more accurate term) play a much more important role, for several reasons. They provide a relatively easy road to quantization and to expressions for Green’s functions, which are closely related to amplitudes for physical processes such as scattering and decays of particles. The path integral treatment of gauge field theories (non-abelian ones, in particular) is very elegant: gauge fixing and ghosts appear quite effortlessly. Also, there are a whole host of nonperturbative phenomena such as solitons and instantons that are most easily viewed via path integrals. Furthermore, the close relation between statistical mechanics and quantum mechanics, or statistical field theory and quantum field theory, is plainly visible via path integrals."

"You could not imagine the sum-over-histories picture being true for a part of nature and untrue for another part. You could not imagine it being true for electrons and untrue for gravity. It was a unifying principle that would either explain everything or explain nothing. And this made me profoundly skeptical. I knew how many great scientists had chased this will-o’-the-wisp of a unified theory. The ground of science was littered with the corpses of dead unified theories. Even Einstein had spent twenty years searching for a unified theory and had found nothing that satisfied him. I admired Dick tremendously, but I did not believe he could beat Einstein at his own game."

"The Feynman method has the virtue that it provides us with a vivid picture of nature’s quantum trickery at work. The idea is that the path of a particle through space is not generally well defined in quantum mechanics. … So when an electron arrives at a point in space—say a target screen—many different histories must be integrated together to create this one event. Feynman’s so-called path-integral, or sum-over-histories approach to quantum mechanics, set this remarkable concept out as a mathematical procedure. It remained more or less a curiosity for many years, but as physicists pushed quantum mechanics to its limits— applying it to gravitation and even cosmology—so the Feynman approach turned out to offer the best calculational tool for describing a quantum universe. History may well judge that, among his many outstanding contributions to physics, the path-integral formulation of quantum mechanics is the most significant."

"Since its inception in Richard Feynman’s 1942 doctoral thesis, the path integral has been a physicist’s dream and a mathematician’s nightmare. To a physicist, the path integral provides a powerful and intuitive way to understand quantum mechanics, building on the simple idea that quantum physics is fundamentally a theory of superposition and interference of probability amplitudes. The “sum over histories” offers a framework for tackling problems ranging from Feynman diagrams to lattice chromodynamics, from quantum cosmology to superfluid vortices to stock-option pricing. To a mathematician, the path integral is at best an ill-defined formal expression. It is some sort of vaguely integral-like object involving a “sum” over a badly specified collection of functions, having an undefined measure, and whose value is apparently determined by a group of unclear and perhaps incompatible limits that may or may not yield finite answers."

"How can quantum gravity help explain the ?"

"... it would be a bracing achievement, and major progress, to identify any concrete, observable phenomenon that brings in truly characteristic features of quantum gravity beyond the semiclassical approximation in common use. Actual observation would bring the subject to another level."

"Unfortunately, the formulation of general rules for the calculation of in the quantum theory of gravitation has only confirmed the presence of another difficulty: The theory contains infinities, arising from integrals over large virtual momenta. Quantum electrodynamics contains similar infinities, but only in three or four special cases, where they can be dealt with by a renormalization of , , and wave functions. ... In contrast, the quantum theory of gravitation contains an infinite variety of infinities, as can be seen by an elementary dimensional argument: The gravitational constant has dimension ћ/m2, so a term in a dimensionless probability amplitude of order G'm will diverge like a integral ʃ p'2n–1'dp. In this respect, the theory of gravitation is more like other nonrenormizable theories, such as the of , than it is like quantum electrodynamics."

"By now it is clear that consistent theories of quantum gravity can be constructed in the context of string theory. This can also be done in diverse dimensions by considering suitable compactifications. This diversity, impressive as it may be for a consistent theory to possess, poses a dilemma: The theory appears to be more permissive than desired!"

"There are (at least) two possible ways to formulate precisely (i.e. mathematically) elementary QM. The eldest one, historically speaking, is due to von Neumann in essence, and is formulated using the language of Hilbert spaces and the spectral theory of unbounded operators. A more recent and mature formulation was developed by several authors in the attempt to solve quantum field theory problems in mathematical physics. … The newer formulation can be considered an extension of the former one, in a very precise sense that we shall not go into here, also by virtue of the novel physical context it introduces and by the possibility of treating physical systems with infinitely many degrees of freedom, i.e. quantum fields. In particular, this second formulation makes precise sense of the demand for locality and covariance of relativistic quantum field theories, and allows to extend quantum field theories to a curved spacetime."

"The physicists didn't want to be bothered with the idea that maybe quantum theory is only provisional. A horn of plenty had been spilled before them, and every physicist could find something to apply quantum mechanics to. They were pleased to think that this great mathematician had shown it was so. Yet the Von Neumann proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It's not just flawed, it's silly. If you look at the assumptions made, it does not hold up for a moment. It's the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they're nonsense. You may quote me on that: The proof of Von Neumann is not merely false but foolish!"

"Thus the formal proof of von Neumann does not justify his informal conclusion: 'It is therefore not, as is often assumed, a question of reinterpretation of quantum mechanics - the present system of quantum mechanics would have to be objectively false in order that another description of the elementary process than the statistical one be possible.' It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made."

"As stated repeatedly in this book, John von Neumann's Mathematical Foundations of Quantum Mechanics was an extraordinarily influential work. It is important to recall that the language most commonly used to describe and discuss the measurement process in terms of a collapse or projection of the wave function essentially originates with this classic work. It was von Neumann who so clearly distinguished (in the mathematical sense) between the continuous time-symmetric quantum mechanical equations of motion and the discontinuous, time-asymmetric measurement process. Although much of his contribution to the development of the theory was made broadly within the boundaries of the Copenhagen view, he stepped beyond those boundaries in his interpretation of quantum measurement."

"Hawking's intitial foray into quantum gravity was more modest than Wheeler's and other[s]... a sneak approach. He first wanted to know what the effect was of an ordinary, classic, curved-space gravitational field on a quantum system. He called this the semiclassical approach. Until that day, most quantum calculations had been done as if gravity didn't exist—they were hard enough without it in normal flat space-time... [Hawking accomplished this by] envisioning an "atom" whose nucleus was a catastrophically powerful black hole... Starobinsky ventured the opinion that rotating black holes would spray elementary particles. ...It was known from Penrose's work, among others, that you could extract energy from the spin of a black hole just like any other dynamo... in particles and radiation just like it did from a particle generator. ...But Hawking ...resolved to redo the calculation for himself ...he decided to warm up first, by calculating the rate of emission from a nonrotating quantum hole. He knew the answer should be no emission. ...his results were embarrassing. His imaginary black hole was spewing matter and radiation... he was reluctant to tell anybody but his closest friends; he was afraid Bekenstein would hear about it. ...It meant that holes had temperatures, just as Bekenstein's work implied."

"Gravity's weakness makes it very difficult to measure its quantum effects; as a result, we have no experimental data to guide theoretical physicists in the development of the missing theory. Detecting a “graviton" – the hypothetical particle making up part of a gravitational field – would require a particle collider the size of the Milky Way or a detector with a mass of the planet Jupiter. These experiments are so detached from our technological capabilities that physicists have focused on trying to remove the mathematical contradictions first, developing approaches like string theory, loop quantum gravity, and asymptotically safe gravity. But to know which theory describes physical reality, experimental tests must eventually be developed."

"... our most fundamental concept, that of spacetime, is itself being threatened as we probe the quantum nature of dynamical spacetime of quantum gravity."

"Bell's theorem, for which he is most famous, was more a triumph of character than of intellect. The difficult thing about it was the realization of what was understood and what was not understood in the discussion of hidden variables. Bell's honesty about his own understanding provided the impetus for his formulation and proof of the theorem."

"No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics."

"The experimental verification of violations of Bell’s inequality for randomly set measurements at space-like separation is the most astonishing result in the history of physics. Theoretical physics has yet to come to terms with what these results mean for our fundamental account of the world. Experimentalists, from Freedman and Clauser and Aspect forward, deserve their share of the credit for producing the necessary experimental conditions and for steadily closing the experimental loopholes available to the persistent skeptic. But the great achievement was Bell’s. It was he who understood the profound significance of these phenomena, the prediction of which can be derived easily even by a freshman physics student. Unfortunately, many physicists have not properly appreciated what Bell proved: they take the target of his theorem— what the theorem rules out as impossible—to be much narrower and more parochial than it is. Early on, Bell’s result was often reported as ruling out determinism, or hidden variables. Nowadays, it is sometimes reported as ruling out, or at least calling in question, realism. But these are all mistakes. What Bell’s theorem, together with the experimental results, proves to be impossible (subject to a few caveats we will attend to) is not determinism or hidden variables or realism but locality, in a perfectly clear sense. What Bell proved, and what theoretical physics has not yet properly absorbed, is that the physical world itself is non-local."

"There's an interesting scientific principle that a wrong answer can be much more stimulating to the field than just sort of finding the answer that's in the back of the book. A wrong result gets people excited. Worried. Obviously, you don't really want that to be happening—it's OK for a theorist to come up with a speculative new theory that gets shot down, but experimentalists are supposed to be very careful and their error limits are supposed to be realistic. Unfortunately, with this experiment, whenever you're looking for a stronger correlation, any kind of systematic error you can imagine typically weakens it and moves it toward the hidden-variable range. It was a hard experiment. In those days, at any rate, with the kind of equipment I had, and … well, what can I say? I screwed up."

"Physicists continue to debate whether Bell's theorem is airtight or not. However, the real question is not whether Bell can prove beyond doubt that reality is non-local, but whether the world is in fact non-local."

"Bell himself managed to devise such a proof which rejects all models of reality possessing the property of "locality". This proof has since become known as Bells theorem. It asserts that no local model of reality can underlie the quantum facts. Bell's theorem says that reality must be non-local."

"The gist of Bell's theorem is this: no local model of reality can explain the results of a particular experiment."

"Bell’s theorem is the most profound discovery of science."

"The main impact of Bell's theorem is from a philosophical-historical perspective: it reinforces, outside the physics of quantum entanglement, the incompatibility of hidden variable theories with quantum mechanics."

"That's all. That's the difficulty. That's why quantum mechanics can't seem to be imitable by a local classical computer. I've entertained myself always by squeezing the difficulty of quantum mechanics into a smaller and smaller place, so as to get more and more worried about this particular item. It seems to be almost ridiculous that you can squeeze it to a numerical question that one thing is bigger than another. But there you are—it is bigger than any logical argument can produce, if you have this kind of logic."

"The theorem tells you that maybe there must be something happening faster than light, although it pains me even to say that much. The theorem certainly implies that Einstein's concept of space and time, neatly divided up into separate regions by light velocity, is not tenable. But then, to say that there's something going faster than light is to say more than I know."

"Again, part of that psychohistorical study I would like to see is why it did not impress the Copenhagen people, especially Bohr. But in the end it turns out that these other people were, in a way, right, because what I am notorious for, the so-called Bell's theorem, is just for showing that Einstein's explanation doesn't work. Einstein's explanation works so long as you have perfect correlations, which means measuring the same component of spin on the two sides [spin is a measure of a property similar but not identical to the rotation of a particle on its axis]. But as soon as you are measuring in a nonparallel direction, you get results that cannot be explained by Einstein's idea that the answers existed before the experiment."

"One of these articles, written by N. David Mermin, gave me a tremendous shock. Mermin described the results of experiments that had been carried out as recently as 1982 to test something called Bell's theorem using two-photon 'cascade' emission from excited calcium atoms. Put simply, Bell's theorem says that my idea of naive realism is in conflict with the predictions of quantum theory in a way that can be tested in the laboratory in special experiments on pairs of quantum particles. These experiments had been done: quantum theory had been proved right and naive realism wrong! There in a montage was a pictorial history of the debate about reality and the experiments that had been done to test it (reproduced opposite). This work struck me as desperately important to my understanding of physical reality, something that as a scientist I felt I ought to know about. This discovery also made me feel rather embarrassed. Here I was, proud of my scientific qualifications and with almost 10 years' experience in chemical physics research at various prestigious institutions around the world, and I had been going around with a conception of physical reality that was completely wrong! Why hadn't somebody told me about this before?"

"The purpose of the first part is to convince the reader that the formalism leading to Bell's inequalities is very general and reasonable. What is surprising is that such a reasonable formalism conflicts with quantum mechanics. In fact, situations exhibiting a conflict are very rare, and quantum optics is the domain where the most significant tests of this conflict have been carried out"

"I’ve had experts in quantum field theory – people who’ve spent years calculating path integrals of mind-boggling complexity – ask me to explain the Bell inequality to them, or other simple conceptual things like Grover’s algorithm. I felt as if Andrew Wiles had asked me to explain the Pythagorean Theorem."

"At the very least, Bell's Theorem prevents us from interpreting quantum amplitudes as probability in the obvious way. You cannot point at a single configuration, with probability proportional to the squared modulus, and say, "This is what the universe looked like all along.""

"In the same paper, Bell also discussed two rather unwelcome properties of hidden-variables theories. The first was contextuality. This tells us that, except in trivial cases, any hidden-variable theory must be such that the result of measuring a particular observable will depend on which other observable) are measured simultaneously. The second was nonlocality. All me hidden-variable models that Bell examined, including Bohm's, had the unpleasant feature that the behaviour of a particular particle depended on the properties of all others, however far away they were. In the EPR case, the measurement result obtained on one particle would depend on what measurement is performed on the second. As Bell said, this was the resolution of the EPR problem that Einstein would have liked least, and it is in this sense that it may be said that Bell proved Einstein wrong."

"In this connection the "classical object" is usually called apparatus, and its interaction with the electron is spoken of as measurement. However, it must be emphasized that we are here not discussing a process of measurement in which the physicist-observer takes part. By measurement, in quantum mechanics, we understand any process of interaction between classical and quantum objects, occurring apart from and independently of any observer. The importance of the concept of measurement in quantum mechanics was elucidated by N. Bohr. We have defined "apparatus" as a physical object which is governed, with sufficient accuracy, by classical mechanics. Such, for instance, is a body of large enough mass. However, it must not be supposed that apparatus is necessarily macroscopic. Under certain conditions, the part of apparatus may also be taken by an object which is microscopic, since the idea of "with sufficient accuracy" depends on the actual problem proposed. Thus, the motion of an electron in a Wilson chamber is observed by means of the cloudy track which it leaves, and the thickness of this is large compared with atomic dimensions; when the path is determined with such low accuracy, the electron is an entirely classical object. Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation."

"It is often stated that of all the theories proposed in this century, the silliest is quantum theory. In fact, some say that the only thing that quantum theory has going for it is that it is unquestionably correct."

"Erwin with his psi can do Calculations quite a few. But one thing has not been seen: Just what does psi really mean?"

"Of course, the apparent disarray could have stemmed entirely from my own ignorance. But when I revealed my impression of confusion and dissonance to one of the attendees, he reassured me that my perception was accurate. “It’s a mess,” he said of the conference (and, by implication, the whole business of interpreting quantum mechanics). The problem, he noted, arose because, for the most part, the different interpretations of quantum mechanics cannot be empirically distinguished from one another; philosophers and physicists favor one interpretation over another for aesthetic and philosophical—that is, subjective—reasons."

"Physicists do not believe quantum mechanics because it explains the world, but because it predicts the outcome of experiments with almost miraculous accuracy. Theorists kept predicting new particles and other phenomena, and experiments kept bearing out those predictions."

"After these conversations with Tagore some of the ideas that had seemed so crazy suddenly made much more sense. That was a great help for me."

"My attitude — I would paraphrase Goering—is that when I hear of Schrödinger's cat, I reach for my gun."

"Einstein was confused, not the quantum theory."

"Only around the end of the nineteenth century did scientists come across a few observations that did not fit well with Newton's laws, and these led to the net revolution in physics - the theory of relativity and quantum mechanics."

"Quantum physics, as our new subject is called, answers such questions as: Why do the stars shine? Why do the elements exhibit the order that is so apparent in the periodic table? How do transistors and other microelectronic devices work? Why does copper conduct electricity but glass does not? In fact, scientists and engineers have applied quantum physics in almost every aspect of everyday life, from medical instrumentation to transportation systems to entertainment industries. Indeed, because quantum physics accounts for all of chemistry, including biochemistry, we need to understand it if we are to understand life itself. Some of the predictions of quantum physics seem strange even to the physicists and philosophers who study its foundations. Still, experiment after experiment has proved the theory correct, and many have exposed even stranger aspects of the theory.The quantum world is an amusement park full of wonderful rides that are guaranteed to shake up the commonsense world view you have developed since childhood."

"Quantum mechanics is clearly superior to classical mechanics for the description of microscopic phenomena, and in principle works equally well for macroscopic phenomena. Hence it is at least plausible that the mathematical and logical structure of quantum mechanics better reflect physical reality than do their classical counter parts. If this reasoning is accepted, quantum theory requires various changes in our view of physical reality relative to what was widely accepted before the quantum era, among them the following:1. Physical objects never possess a completely precise position or momentum. 2. The fundamental dynamical laws of physics are stochastic and not deterministic, so from the present state of the world one cannot infer a unique future (or past) course of events. 3. The principle of unicity does not hold: there is not a unique exhaustive description of a physical system or a physical process. Instead, reality is such that it can be described in various alternative, incompatible ways, using descriptions which cannot be combined or compared."

"Unlike Newton's mechanics, or Maxwell's electrodynamics, or Einstein's relativity, quantum theory was not created—or even definitively packaged—by one individual, and it retains to this day some of the scars of its exhilarating but traumatic youth. There is no general consensus as to what its fundamental principles are, how it should be taught, or what it really "means." Every competent physicist can "do" quantum mechanics, but the stories we tell ourselves about what we are doing are as various as the tales of Scheherazade, and almost as implausible."

"Just a few months after de Broglie's suggestion, Schrödinger took the decisive step... by determining an equation that governs the shape and the evolution of probability waves, or as they became known, s. It was not long before Schrödinger's equation and the probabilistic interpretation were being used to make wonderfully accurate predictions. By 1927, therefore, classical innocence had been lost. Gone were the days of a whose individual constituents were set in motion at some moment in the past and obediently fulfilled their inescapable, uniquely determined destiny. According to quantum mechanics, the universe evolves according to a rigorous and precise mathematical formalism, but this framework determines only the probability that any particular function will happen—not which future actually ensues."

"Quantum mechanics, that mysterious, confusing discipline, which none of us really understands but which we know how to use. It works perfectly, as far as we can tell, in describing physical reality, but it is a ‘counter-intuitive discipline’, as social scientists would say. Quantum mechanics is not a theory, but rather a framework, within which we believe any correct theory must fit."