Lizwi Posted May 23, 2022 Share Posted May 23, 2022 (edited) Why is i =j in the steps highlighted in green? Edited May 23, 2022 by Lizwi Picture not attached Link to comment Share on other sites More sharing options...
swansont Posted May 23, 2022 Share Posted May 23, 2022 Because of the Kronecker delta in the first highlighted equation https://en.wikipedia.org/wiki/Kronecker_delta Which is there because <i|j> is in the preceding equation. Do you see why that is? 1 Link to comment Share on other sites More sharing options...
Lizwi Posted May 23, 2022 Author Share Posted May 23, 2022 Yeah, is Kronecker delta reaulting because of diagonal matrix? Link to comment Share on other sites More sharing options...
swansont Posted May 23, 2022 Share Posted May 23, 2022 8 minutes ago, Lizwi said: Yeah, is Kronecker delta reaulting because of diagonal matrix? What is <i|j> when i ≠ j ? (They are orthogonal; yes, it’s because you’ve got a diagonal matrix so you have eigenvalues and eigenvectors) Link to comment Share on other sites More sharing options...
joigus Posted May 25, 2022 Share Posted May 25, 2022 On 5/23/2022 at 2:28 PM, Lizwi said: Yeah, is Kronecker delta reaulting because of diagonal matrix? Kronecker delta results, as Swansont said, because of basis being orthogonal. I see you're studying quantum mechanics. The underlying physics of this is: States with different values of an observable must be orthogonal for reasons relating to probability interpretation of quantum mechanics, as products <i|j> are essentially transition amplitudes. Vectors must be normalised, so <i|i>=1 for same i. Orthonormal bases are central to the formalism of quantum mechanics. Link to comment Share on other sites More sharing options...
Lizwi Posted June 8, 2022 Author Share Posted June 8, 2022 Yes, I am doing QM. Thanks a lot. How do I write the hermitian conjugate of operator A. Please see the attachment. Quote Link to comment Share on other sites More sharing options...
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