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Quick Quantum Mechanics Q about basis

  1. Mar 23, 2013 #1

    bon

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    1. The problem statement, all variables and given/known data

    Suppose we have a system and that {|a>, |b>, ...} is a complete and orthonormal basis for the system

    Am i right in thinking Ʃ(j) <k|j><j|i> = <k|i> = 0 unless k=i?

    In other words, does the LHS expression equal the middle one because Ʃ(j) |j><j| is just the insertion of the identity and we can put it in anywhere?

    2. Relevant equations



    3. The attempt at a solution

    I've explained my attempt above.

    Thanks!
     
  2. jcsd
  3. Mar 23, 2013 #2

    Dick

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    You need to say what |k> and |i> are. If they are elements of the orthonormal basis then if k≠i, <k|i>=0, just because the basis is orthonormal and that's what the 'normal' part means. No need to insert the identity anywhere.
     
  4. Mar 23, 2013 #3

    bon

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    Thanks, sorry I wasn't clear. |k> and |i> are elements of the orthonormal basis. And I know that this means <k|i>=0 if k doesn't equal i. It's just that (as part of a larger calculation) I have arrived at the expression

    Ʃ(j)<k|j><j|i> (where |j> is also an element of the orthnormal basis) and just wanted to check this equals <k|i>. Am I correct in thinking it does?

    Thanks again
     
  5. Mar 23, 2013 #4

    Dick

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    Sure, if j sums over ALL of the elements of the orthonormal basis.
     
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