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Suppose I want to study a quantum mechanical quantity of a single particle. I have designed an appropriate apparatus, accuracy of which is limited by relevant laws of quantum theory. I have obtained a number of (let's say 10) readings of that single particle for corresponding quantity. As per quantum theory these are eigenvalues of operator corresponding to that quantity being studied. Now how would I construct hermitian operator for that quantity when

i) that quantity can have only those 10 values which I have obtained &

ii) that quantity can also have values other than those 10 values I have obtained?

auden
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Anupam Nath
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    You can't construct an operator from experiment. You can create a mathematical model, including your proposed operator, then compare the predictions of your model with experiment. However that doesn't prove your operator is correct, only that it is not incorrect. – John Rennie Jul 10 '16 at 14:36
  • @JohnRennie & to others I want to know then how are quantum operators created at the first place(i.e. from scratch)?How were operators like energy or angular momentum created? & also how are operators created when a new quantum property of a particle is found(properties like charm,truth etc.)? – Anupam Nath Jul 10 '16 at 14:45
  • See http://physics.stackexchange.com/questions/153672/how-did-the-operators-come-about/154001#154001 and possible duplicate of http://physics.stackexchange.com/questions/153497/experimentally-what-categorizes-a-measurement-as-corresponding-to-a-certain-obs – Martin Jul 10 '16 at 14:49
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    Operators like energy or angular momentum were created by developing a mathematical model to describe quantum mechanics. That is, physicists did a lot of head scratching and scribbling to come up with a mathematical model that correctly predicted the results of experiments. The results came from experiment, but the model (and the operators) came out of physicists brains. – John Rennie Jul 10 '16 at 14:50
  • @JohnRennie If I find a quantum quantity previously undiscovered then what would be my steps to create an operator for that quantity?Can you please tell me some good text resource on this topic(operator construction)?Also, as per your response there is a fair chance that popular operators used today can be wrong, am I right? – Anupam Nath Jul 10 '16 at 15:02
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    Our current mathematical model to describe the quantum world is the Standard Model. At the time of writing there have been no (accepted) experimental measurements that contradict the Standard Model. You would have to either measure something not predicted by the Standard Model or measure a value that contradicted a prediction from the Standard Model. Either way your first step should be to write off to Stockholm to apply for your Nobel prize. – John Rennie Jul 10 '16 at 15:09
  • Can anyone please suggest some books focusing on creation of quantum operators for students? – Anupam Nath Jul 10 '16 at 15:16
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    Any basic quantum mechanics book should take you through the quantization process, of turning classical equations into their quantum equivalents. It's often done by replacing concepts such as momentum by a momentum operator. –  Jul 10 '16 at 15:42
  • Did you study the equivalent problem in classical mechanics? If not, that may explain your confusion. You need to understand, first, why we study position, momentum, angular momentum, energy etc. in classical mechanics. At the end of the day it comes down to symmetry properties of spacetime which give meaning to these quantities or even make them conserved quantities. – CuriousOne Jul 10 '16 at 21:23
  • @AnupamNath: Please also see the questions I link. There is a rough description of the formalism of quantum mechanics with a short discussion of how operators come from classical mechanics (and how those that don't are derived by guessing and analogy). In any case, you have to study classical mechanics first. – Martin Jul 10 '16 at 23:21

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