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Explanation(s) of Variance

  1. Oct 28, 2015 #1
    In a Quantum Mechanics Class, my tutor had shown me that
    the variance Δa2

    Δa2 = < ( a - < a > )2 >
    = < a2 - 2 a < a > + < a >2 >
    = < a2 > - 2 < a >< a > + < a >2
    = < a2 > - 2 < a >2 + < a >2
    = < a2 > - < a >2

    and my questions are

    i) why, in step 3,

    < x - y + z > will become < x > - < y > + < z > ??????

    ii)why, also in step 3, the middle term,

    < -2 a < a > > will finally become -2 < a >2 ?????

    Is it mean that < a < a > > = < a > < a > = < a >2 ??????? but why?

    Thanks!
     
    Last edited by a moderator: Oct 28, 2015
  2. jcsd
  3. Oct 28, 2015 #2

    Mark44

    Staff: Mentor

    I'm not familiar with your notation. One definition of the variance of a random variable X (from Wikipedia - see https://en.wikipedia.org/wiki/Variance) is:
    ##Var(X) = E[(X - \mu)^2]##
    Here E[X] means the expected value of X.

    The right side of the equation above can be expanded to yield
    ##Var(X) = E[(X - E[X])^2] \\
    = E[X^2 - 2XE[X] + (E[X])^2] \\
    = E[X^2 - 2E[X] E[X] +(E[X])^2] \\
    = E[X^2] - (E[X])^2##
     
  4. Oct 28, 2015 #3

    pwsnafu

    User Avatar
    Science Advisor

    In all three questions the answer is the expected value is a linear operator. For an observable ##A## the expected value is defined as
    ##\langle A \rangle _{\phi} := \langle \phi | \, A \,| \phi \rangle##. Suppose that ##A## and ##B## are observables (mathematically they are operators on some Hilbert space), and let ##\alpha## and ##\beta## be complex numbers. Then
    ##\langle \alpha A + \beta B \rangle_\phi = \langle \phi | \alpha A + \beta B | \phi \rangle = \langle \phi | \alpha A | \phi \rangle + \langle \phi | \beta B | \phi \rangle =\alpha \langle \phi | A | \phi \rangle + \beta \langle \phi | B | \phi \rangle = \alpha \langle A \rangle_\phi + \beta \langle A \rangle_\phi##
     
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