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Might be a basic question

  1. Apr 27, 2010 #1

    From the ordinary courses in QM it is known that the momentum kets satisfy the completeness relation

    [tex] \int d^3 p \mid p \rangle \langle p \mid = 1[/tex]

    Knowing this, how can you calculate

    [tex] \int d^3 p \mid p \rangle p \langle p \mid [/tex]


    Thanks a lot!
    Last edited: Apr 28, 2010
  2. jcsd
  3. Apr 28, 2010 #2


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    What do you want to calculate? It's an operator you can apply to bras and kets

    I prefer

    [tex]\int d^3p |p\rangle\langle p| p [/tex]
    Last edited: Apr 28, 2010
  4. Apr 28, 2010 #3
    What is the p outside the bra/ket exactly? Do you want it to be a momentum operator? or an eigenvalue for the p-ket? If it's the former I think you have an illegal product so there is no such thing as calculating it. If it's a eigenvalue (ie a real number) then it's just the identity operator multiplied by the scalar p.
  5. Apr 28, 2010 #4


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    That's why I prefer

    [tex]\int d^3p |p\rangle\langle p| \hat{p}^\dagger = \int d^3p |p\rangle\langle p| p [/tex]
  6. Apr 28, 2010 #5
    operator times operator
  7. Apr 28, 2010 #6
    That p stands for the momentum operator. That's why I'm not sure about how to calculate it. I think the result has something to do with a delta function.
  8. Apr 29, 2010 #7


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    your p cannot be an operator as it is in the wrong position, whereas my p (with the hat on top) is an operator; in the second step it is replaced by the eigenvalue as it acts (to the left) on the state vector
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