1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Ket Notation - Effects of the Projection Operator

  1. Jan 31, 2012 #1
    Ket Notation -- Effects of the Projection Operator

    1. The problem statement, all variables and given/known data
    From Sakurai's Modern Quantum Mechanics (Revised Edition), it is just deriving equation 1.3.12.


    2. Relevant equations
    [tex] \begin{eqnarray*}\langle \alpha |\cdot (\sum_{a'}^N |a'\rangle \langle a'|) \cdot|\alpha \rangle \end{eqnarray*} [/tex]
    3. The attempt at a solution
    The summation can be moved to the left, so everything is being summed from a' to N, but does an alpha bra inner product with a' (or <α|a'>) does the sum of this from all a' to N equal Ʃ<a'|α>? maybe this is simple and I just can't see it?
     
  2. jcsd
  3. Jan 31, 2012 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: Ket Notation -- Effects of the Projection Operator

    Check (1.2.12), which is a fundamental property of the inner product. That property holds even when the bras and kets correspond to different bases.
     
  4. Jan 31, 2012 #3
    Re: Ket Notation -- Effects of the Projection Operator

    So, is it in this particular case that they are equal because we are considering the eigenkets of A, a hermitian operator? Because these are the eigenkets of A, does that mean that all operators on it are real? Even though it is the operator <α| acting on it and not A?
     
  5. Jan 31, 2012 #4

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: Ket Notation -- Effects of the Projection Operator

    There's no assumption here that [itex]\langle \alpha | a'\rangle[/itex] is real, just that [itex]\langle \alpha | a'\rangle = \langle a'| \alpha\rangle^*.[/itex] That is why the absolute value appears in the formula.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Ket Notation - Effects of the Projection Operator
Loading...