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Express moment / expectation value in lower order expectation values

  1. Jan 6, 2012 #1
    Hello everybody,

    I'm looking for a proof of the following equation:

    <x6> = <x>6+15s2<x>4

    where the brackets denote an expectationvalue and s is the standard deviation.

    Thanks in advance!
  2. jcsd
  3. Jan 6, 2012 #2


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    It is not true in general. For example if <x> = 0, <x6> will not = 0 unless x = 0 itself.
  4. Jan 6, 2012 #3
    Sorry, I forgot to mention. It's a first order Taylor approximation.

    Thanks for the reply,

  5. Jan 6, 2012 #4
    We have for example:

    <x2> = <x>2+s2

    <x4> - <x2>2 ≈ 4s2<x>2 (to first order)

    Now combining both equations yields:

    <x4> = <x>4+6s2<x>2

    Unfortunately this doesn't work that easily for <x6>
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