Quick Quantum Mechanics Q about basis

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  • #1
bon
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Homework Statement



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?

Homework Equations





The Attempt at a Solution



I've explained my attempt above.

Thanks!
 

Answers and Replies

  • #2
Dick
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Homework Statement



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?

Homework Equations





The Attempt at a Solution



I've explained my attempt above.

Thanks!

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.
 
  • #3
bon
559
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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.

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
 
  • #4
Dick
Science Advisor
Homework Helper
26,263
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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

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

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