Quick Quantum Mechanics Q about basis

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Homework Help Overview

The discussion revolves around a quantum mechanics problem involving a complete and orthonormal basis for a system. The original poster questions the validity of an expression involving inner products and the insertion of the identity operator in the context of this basis.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to confirm if the expression Ʃ(j) equals under the assumption that |k> and |i> are elements of the orthonormal basis. Some participants suggest that the orthonormality condition implies equals zero when k does not equal i, questioning the need for the identity insertion.

Discussion Status

Participants are exploring the implications of the orthonormality of the basis and the correctness of the expression in question. There is acknowledgment of the original poster's clarification regarding |k> and |i>, and some guidance is provided regarding the summation over the complete basis.

Contextual Notes

There is an emphasis on the need for clarity regarding the elements of the basis being discussed, as well as the implications of the orthonormality condition on the inner products involved.

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!
 
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bon said:

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.
 
Dick said:
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
 
bon said:
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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