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Standard deviation of expectation values

  1. Apr 18, 2007 #1


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    Very basic question which has confused me:

    if the variance of an expectation value <A> is:

    uncertainty of [tex]A=<(A-<A>)^2>^0.5 [/tex]

    how is this equal to:

    [tex](<A^2>-<A>^2)^0.5 [/tex]

    Last edited: Apr 18, 2007
  2. jcsd
  3. Apr 18, 2007 #2

    Doc Al

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    Staff: Mentor

    Expand it out:
    [tex]<(A-<A>)^2> = < A^2 - 2A<A> + <A>^2 > = < A^2 - <A>^2 >[/tex]
  4. Apr 18, 2007 #3


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    Staff: Mentor

    Start by expanding the squared term in parentheses:

    [itex](A - <A>)^2 = A^2 - 2<A>A + <A>^2[/itex]

    Note that <A> is simply a number and can be manipulated like any other numeric constant. Simplify the resulting expectation value by taking advantage of general properties of expectation values, i.e.

    [itex]<A+B> = <A> + <B>[/itex]

    [itex]<cA> = c<A>[/itex]

    where c is a numeric constant.
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