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Linear superposition. Measurments.

  1. Mar 3, 2014 #1
    If in quantum physics some state is represented by
    ## \psi(x)=\sum_{k}C_k\psi_k(x)##
    ##C_m=\int \psi(x)\psi_m(x)dx##
    Why probability to measure ##\psi_m(x)## is ##|C_m|^2##?
     
  2. jcsd
  3. Mar 3, 2014 #2

    WannabeNewton

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  4. Mar 3, 2014 #3

    jfizzix

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    We can actually express [itex]|\psi\rangle[/itex] as a sum (integral) over all the position states:
    [itex]|\psi\rangle = \int dx |x\rangle\langle x|\psi\rangle = \int dx |x\rangle \psi(x)[/itex]

    [itex]\psi(x) = \langle x|\psi\rangle[/itex] is the component of [itex]|\psi\rangle[/itex] that overlaps with the position basis vector [itex]|x\rangle[/itex].
    It's an inner product, like with ordinary vectors. If you want to find the y-component of a vector [itex]\vec{v}[/itex], you take its inner product with the basis vector in the y-direction [itex]v_{y} = \vec{v}\cdot \hat{y}[/itex].


    We can also express these components [itex] \langle x|\psi\rangle[/itex] in other bases too. If we have some other (discrete) basis of states [itex]|k\rangle[/itex], we can express [itex]\langle x|\psi\rangle[/itex] as:

    [itex]\langle x|\psi\rangle = \sum_{k}\langle x|k\rangle\langle k|\psi\rangle = \sum_{k} \psi_{k}(x) C_{k}[/itex]
    where
    [itex]C_{k} = \langle k|\psi\rangle [/itex]

    On the other hand, we can also represent [itex]C_{k}[/itex] in terms of the position basis states [itex]|x\rangle[/itex], so that
    [itex]C_{k} = \int dx\; \langle k|x\rangle\langle x|\psi\rangle = \int dx\; \psi_{k}^{*}(x)\psi(x) [/itex]

    The probability to measure [itex]\psi_{m}(x)[/itex] is better thought of as just the probability of measuring [itex]k=m[/itex]. This probability is just [itex]|\langle m|\psi\rangle|^{2} = |C_{m}|^{2}[/itex].
     
  5. Mar 4, 2014 #4

    naima

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    I think that nobody knows from where Born rule comes. But we see how it goes to classical fields.
    When a monochromatic source of particles is in front of a double slit particles hit the screen at a point x with probability p(x) and give it an energy e.
    When there is a flux of particles this turns to be the intensity on the screen. For exemple with the electromagnetic field the density of intensity is E² + B²
     
    Last edited: Mar 4, 2014
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