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I am an electronics and communication engineer, specializing in signal processing. I have some touch with the mathematics concerning communication systems and also with signal processing. I want to utilize this knowledge to study and understand Quantum Mechanics from the perspective of an engineer. I am not interested in reading about the historical development of QM and i am also not interested in the particle formalism. I know things have started from the wave-particle duality but my current interests are not study QM from that angle. What I am interested is to start studying a treatment of QM from very abstract notions such as, 'what is an observable ? (without referring to any particular physical system)' and 'what is meant by incompatibility observables ?' and then go on with what is a state vector and its mathematical properties. I am okay to deal with the mathematics and abstract notions but I some how do not like the notion of a particle, velocity and momentum and such physical things as they directly contradict my intuition which is based on classical mechanics ( basic stuff and not the mathematical treatment involving phase space as i am not much aware of it).

I request you to give some suggestions on advantages and pitfalls in venturing into such a thing. I also request you to provide me good reference books or text books which give such a treatment of QM without assuming any previous knowledge of QM.

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Why not sakurai? –  ramanujan_dirac Feb 3 '13 at 12:08
    
Possible duplicate: physics.stackexchange.com/q/5014/2451 –  Qmechanic May 23 '13 at 19:51

7 Answers 7

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Try "Mathematical Foundations of Quantum Mechanics" by George Mackey. It is about 130 pages with a chapter on classical mechanics. The author is a very well know mathematician, and I think the books is what you are looking for. Also on higher level is the book "Quantum mechanics for mathematicians" by Leon Takhtadzhian. The book by Folland on Quantum Field Theory has a chapter on Quantum Mechanics, which can be read independently of the rest of the book.

Edit: Since this came up on the first page, I'll add one more. F.Strocchi "An Introduction to the Mathematical Structures of Quantum Mechanics. A Short Course for Mathematicians."

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An unconventional approach would be to study quantum computation or quantum information theory first.

What is 'unusual' about quantum mechanics is the mathematical underpinnings, which is essentially a generalization of probability theory. (I have heard more than one colleague say that quantum mechanics is simply physics which involves 'non-commutative probability', i.e. in testing whether some collection of events are realized for some sample space, there is a pertinent sense of the order in which one tests those events.) To the extent that this is true, it is not important to be learning the actual physics alongside that mathematical underpinning, so long as you can learn about something evolving e.g. under the Schrödinger equation or collapsing under measurement.

Studying quantum information evolving under a computational process is one way you could achieve that. Because the narrative of the field is less about the crisis in physics in the 20s–40s, and more about physicists and computer scientists struggling to find a common language, the development is clearer and there is a better record of justifying the elements of the formalism from a fundamental standpoint. By studying quantum information and/or quantum computation, you will be able to decouple the learning of the underpinnings from the learning of the physics, and thereby get to the heart of any conceptual troubles you may have; and it will give you a tidier sandbox in which to play with ideas.

To this end, I recommend "Nielsen & Chuang", which is the standard introductory text of the field. It's suitable as an introduction both for those coming from a computer science background, and from a quantum physics background; so apart from learning some of the formalism, you can get some exposure to some of the physics as well. There are other texts which I have not read, though; and about a bazillion pages of lecture notes floating around on the web.

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I strongly advise you Quantum Theory: Concepts and Methods by Asher Peres.

I think this book is answering the questions you're asking. Like 'what is an observable ?'

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That's quite an unusual request i.e. to start with an abstract formulation of quantum mechanics, especially for someone in a profession so closely connected with the "real world". However, to answer your question, I think what you're looking for is an axiomatic approach to quantum mechanics. Such treatments will keep the physical examples to the minimum and skip to the mathematics straight away.

You could start with this reference (just for a chatty treatment of what the postulates look like !), and maybe search for quantum mechanics textbooks with "axiomatic" in the title.

For many people, the fact that the quantum mechanical predictions of physical quantities are sometimes counterintuitive is precisely what gives the subject its appeal.

Hope this helps.

Edit: It appears not so easy to find "Axiomatic Quantum Mechanics" in textbook titles ! However Google returns a few articles featuring those words.

Of course there is also Von Neumann's "Mathematical Foundations of Quantum Mechanics".

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You might find articles by Leon Cohen of interest. He has considered the relationship between classical and quantum theory from a signal processing since the 1960s. For example, PROCEEDINGS OF THE IEEE, VOL. 77, NO. 7, JULY 1989, "Time-Frequency Distributions-A Review". This concentrates on the relationship between the Wigner function in quantum theory and various concepts in signal processing.

This might not answer your question so much as point to something that you might find more broadly helpful because of its signal processing provenance. The mathematics of Hilbert spaces only enters into a small fraction of the signal processing literature, but the vast majority of it could be put into such mathematical terms (signal processing is, after all, preoccupied with Fourier and other integral transforms).

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I always recommend Tony Sudbery's Quantum Mechanics and the Particles of Nature, don't be put off by the bad word in the title: he is fairly axiomatic and has both the abstract part and the concrete part. I recommend them more highly than either Mackey, already cited, or Varadarajan, both of which are idiosyncratic. Prof. Sudbery is an expert in Quantum Information Theory but does not take a biassed or idiosyncratic approach in his text.

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Here is a little book by a physicist trained as an engineer in fluid dynamics applied to aircraft:

"Foundations of Quantum Physics" by Toyoki Koga (1912-2010), Wood and Jones, Pasadena, CA, 1980.

This book has forewords by Henry Margenau and Karl Popper.

Another book by him is "Inquiries into Foundations of Quantum Physics", 1983.

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