Polar decomposition

[Lizwi]

Why is i =j in the steps highlighted in green?

Edited May 23, 2022 by Lizwi
Picture not attached

[swansont]

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?

[Lizwi]

Yeah, is Kronecker delta reaulting because of diagonal matrix?

[swansont]

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)

[joigus]

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. 

[Lizwi]

Yes,  I  am doing QM. 

 

Thanks a lot. 

 

How do I write the hermitian conjugate of operator  A. 

 

Please see the attachment. 

Quote