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How did physics operators come into being?

  1. Sep 22, 2015 #1


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    Now I am starting to learn Quantum Mechanics. In the class I am taught about operators, postulates and all other basic stuff.

    I understand operators to be +, -, /, etc; but quantum mechanical operators are entirely different; to understand them, I think, I need to know the historical development of the physics operators. So, I want to know how these operators were discovered/invented; some of the historical figures on this subject would also help along with some first text books from the original authors (modern texts are also okay).

    Thank you.
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  3. Sep 22, 2015 #2


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    Operators in quantum mechanics are indeed different to mathematical operations, but they aren't unique to quantum mechanics, but are useful for mapping any vector space to another. Have you tried


    operators in quantum mechanics are just special cases ("Hermitian operators")

  4. Sep 22, 2015 #3


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    I saw those two wiki pages; as I am just a beginner, I follow Lagrange's thought; the following passage has been extracted from the preface of Lagrange's "Mechanique Analyytique":

    "..lagrange preceded each part with an historical overview of the development of the subject. His study was motivated not simply by considerations of priority but also by genuine interest in the genesis of ideas....he suggested that althouh discussions of forgotten methods may seem of little value, they allow one to follow step by step the progress of analysis, and to see how simple and general methods are born from complicated and indirect procedures..."

    So, I felt little bit tough going through those wiki pages; instead I want to know how these quantum mechanical operators were invented in a step by step manner (as lagrange used to understand things)...so that it becomes easy for me to understand things..

    I want to know the sources from whom these quantum mechanical operators were born..
  5. Sep 22, 2015 #4


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    W. Heisenberg invented the matrix form of QM 'in the blink of an eye' ( his own words translated). He started by writing a matrix ( square array) of spectral emission intensities for transitions between electron shell m->n. He was looking for a system whose states were quantised. That first array evolved into the operator H.

    To get details search for 'Heisenberg - matrix mechanics - history'.

    'Inward Bound' by A. Pais has good coverage of this discovery. I highly recommend this book.


    There is a translation of Heisenbergs original paper in 'Sources of Quantum Mechanics'


    and the subsequent papers by Born and Jordan and (fanfare) the amazing paper by PAM Dirac.
    Last edited: Sep 22, 2015
  6. Sep 23, 2015 #5


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    This is a great question. I don't claim to have a complete historical picture, and I look forward to reading anything by anyone who has that kind of deep insight. However, my off-the-cuff reaction is the following. I think there was the Schrodinger picture, with a wave equation, and at some point people realized that the linearity of this equation meant that all of quantum mechanics could be expressed in terms of linear operations on a vector space, as in the Heisenberg picture. This would seem to lead naturally to the idea that if you're trying to calculate, e.g., the expectation value of something, it should be expressible as a matrix expression of the form<...|M|...>.
  7. Sep 23, 2015 #6


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    Heisenberg was working with infinite dimensional matrices. Schrödinger used differential equations. Von Neumann showed that the two theories were mathematically equivalent by expressing them both in terms of Hilbert spaces (which are just infinite dimensional vector spaces) and operators (which are just linear maps between vector spaces, ie in the finite dimensional case, just finite matrices). But going infinite dimensional brought in many subtle mathematical issues and von Neumann needed to develop a whole lot more mathematics in order to get this all to work.
  8. Sep 25, 2015 #7


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    Well, if you could make sense of Heisenberg's paper (the one with his discovery of matrix mechanics on the island of Helgoland) tell me. I'd be interested, because I never understood this paper (of course, having learnt quantum theory in the modern way you know, what Heisenberg gets out, but it's not clear to me how to understand it without this preknowledge). This is different with Schrödinger's series of papers concerning wave mechanics and of course Dirac's paper. I'd say that the full understanding of quantum theory came with Dirac's treatment and later with von Neumann's mathematical work on making this mathematically rigorous. On the other hand, it's easier to first learn the modern way and then indulge in historical studies on the development of quantum theory. A very good introduction to the history of quantum mechanics can be found in

    S. Weinberg, Lectures on Quantum Mechanics, Cambridge Uni. Press

    which is an excellent book, but perhaps not as a first read. In any case one should read it after same familiarity with QM has been gained from other sources. My favorite for QM1 is

    J. J. Sakurai, Modern Quantum Mechanics
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