Probability Density or Expectation Value?

[LarryS]

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.

 

[SpectraCat]

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.

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.