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Matrix representing projection operators

  1. Jan 28, 2009 #1
    Hey everyone! I have a question regarding the matrix representation of a projection operator. Specifically, does the wavefunction have to be normalized before determining the projection operator? For example:

    |Ψ1> = 1/3|u1> + i/3|u2> + 1/3|u3>

    |Ψ2> = 1/3|u1> + i/3|u3>

    Ψ1 is obviously normalized and Ψ2 isnt

    Now to calculate the matrix that represents the projection operator, just make row and column matrices and multiply out. For Ψ1:

    | 1/3|u1> |
    | i/3|u2> | * (| 1/3|u1> i/3|u2> 1/3|u3> |)
    | 1/3|u3> |

    Which gives a 3x3 Hermitian matrix.

    My question is do I have to normalize Ψ2 in order to do the same process? The result is a Hermitian operator either way, I believe, but I just want to know if 'technically' it needs to be normalized first.

    thanks for any input!

    Timmy
     
  2. jcsd
  3. Jan 28, 2009 #2
    ps: I realize this is a little 'homeworky' and if it is too much so, I apologize for posting it in the wrong place.
     
  4. Jan 28, 2009 #3
    yes if you want to build a projection operator out of a wavefunction then the wavefunction has to be normalized, but it's also true that it's Hermitian either way. A projection operator P should obey PP=P, so |[tex]\psi[/tex]><[tex]\psi[/tex]|[tex]\psi[/tex]><[tex]\psi[/tex]|=|[tex]\psi[/tex]><[tex]\psi[/tex]| requires <[tex]\psi[/tex]|[tex]\psi[/tex]>=1
     
  5. Jan 28, 2009 #4
    Alright! Thanks for the help--I appreciate it!

    Timmy
     
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