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Generalized uncertainty principle

  1. Aug 31, 2013 #1
    1. The problem statement, all variables and given/known data
    A spin-half particle is in a known eigenstate of Sz. Show that the product [tex] <S^2_x> < S^2_y> [/tex] is consistent with the Uncertainty principle


    2. Relevant equations

    3. The attempt at a solution

    I know that the generalized uncertainty principle gives [tex] ΔS_x ΔS_y ≥ |<[S_x, S_y]>|

    => ΔS_x ΔS_y ≥ \frac{h^2}{16π^2} [/tex]
    but I'm stuck as to how to proceed to show the product in question is consistent with this.

    Thanks in advance for any help.
     
  2. jcsd
  3. Aug 31, 2013 #2

    TSny

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    Hello, DeltaFunction.

    I'm not quite sure what you are asking. Are you asking how to evaluate the product ##<S^2_x> < S^2_y> ##, or are you asking how the product ##<S^2_x> < S^2_y> ## is related to ##ΔS_x ΔS_y##?
     
  4. Aug 31, 2013 #3
    I'm a bit vague on this too. The problem simply states "Show that the product ##<S^2_x> < S^2_y> ## is consistent with the uncertainty principle"

    I assumed I'd need to link that to ##ΔS_x ΔS_y## to demonstrate this, but I'm stumped by this
     
  5. Aug 31, 2013 #4

    TSny

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    How is ##ΔS_x## defined in terms of ##<S^2_x>## and ##<S_x>^2##?
     
  6. Sep 1, 2013 #5
    I don't know I'm afraid :/
     
  7. Sep 1, 2013 #6

    TSny

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    You need to know what quantitative definition of "uncertainty" you are using. Generally, the uncertainty of an observable ##A## for a particular state is defined to be the standard deviation $$\Delta A=(\langle A^2\rangle-\langle A\rangle^2)^{1/2}$$
    So to determine the uncertainty ##\Delta S_x## for a state of definite z-component of spin, you will need to evaluate ##(\langle S_x^2\rangle-\langle S_x\rangle^2)^{1/2}## for that state.
     
    Last edited: Sep 1, 2013
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