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General Uncertainty Principle

  1. May 26, 2015 #1
    The general Uncertainty Principle is written in Griffiths' Intro to Quantum Mechanics 2nd Ed. Section 3.4, Page 109, Eq. (3.139) without dependence on the wave function itself. While it is written in R. Shankar's Principles of Quantum Mechanics 2nd Ed., Section 9.2, Page 239, Eq. (9.2.12) with a dependence on the wave function.

    I can't understand the difference between the two equations. Is the one written in Shankar more general? Or they are the same equation but in a different formulation?
     
  2. jcsd
  3. May 26, 2015 #2
    Would it be too much to ask for you to actually show these two equations? No offense, but if you expect everyone to have those books and to look it up I'm not sure I even want to help you.
     
  4. May 27, 2015 #3

    Demystifier

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    A) Trivial stuff

    First, the page and equation number you gave are from the 1st edition, not the 2nd.
    Second, equation (3.139) does depend on the wave function, because the symbols <, > depend on the wave function. See eq. (3.116) where this dependence is more explicit.
     
    Last edited: May 27, 2015
  5. May 27, 2015 #4

    Demystifier

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    B) Non-trivial stuff

    The uncertainty relation (UR) in Shankar is not equivalent to the UR in Griffiths, even though they both depend on the wave function. The UR in Griffiths is what we usually call Heisenberg UR (even though he was not the first who derived it rigorously), while the UR in Shankar was first derived by Schrodinger. The Heisenberg UR follows from the Schrodinger UR, but the Schrodinger UR does not follow from the Heisenberg UR. In this sense the Schrodinger UR is more "general", but in most practical cases the Heisenberg UR is more useful.

    See also
    http://en.wikipedia.org/wiki/Uncert...2.80.93Schr.C3.B6dinger_uncertainty_relations
    http://lanl.arxiv.org/abs/physics/0510275
     
    Last edited: May 27, 2015
  6. May 27, 2015 #5
    Many Thanks :)
    The confsion is solved. But, what kind of generality does the Schrodinger UR has? In other words, what distinguishes it from Heisenberg's UR?
     
  7. May 27, 2015 #6

    Demystifier

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    I have put "general" in quotation marks. It is not really about generality, but about strength of an inequality. For instance, the inequality
    ##x\geq 2##
    is stronger than
    ##x\geq 1##,
    even if they are both simultaneously true. The stronger inequality implies the weaker inequality, but the weaker inequality does not imply the stronger inequality.
     
  8. May 28, 2015 #7
    Got it. Thank you for your help :)
     
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