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Probability Density or Expectation Value?

  1. Apr 16, 2010 #1


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    In a paper in Physical Review A, the author discusses a wave function for one particle, Ψ(r,t), where r is the position vector.

    He writes "The probability distribution for one-particle detection at a point r is given by

    |<r|Ψ >|2 ".

    Is that correct? The above expression looks, to me, more like the expectation value for r.

    Shouldn't the probability distribution be |<Ψ|Ψ >|2?

    Thanks in advance.
  2. jcsd
  3. Apr 16, 2010 #2


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    No, that is correct. The position representation of the wavefunction is given by <r|Ψ > ... you can derive this from the resolution of the identity for the continuous distribution of eigenstates ... see the first chapter (I think) of Cohen-Tannoudji for a detailed derivation. Since <r|Ψ > is the wavefunction, then of course |<r|Ψ >|2 is the probability density.

    For a normalized wavefunction, <Ψ|Ψ >=1, so that certainly isn't correct.
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