Date: Jan 26, 2013 10:02 PM
Author: Will Janoschka
Subject: Re: Susskind's proof of orthogonality of eigenvectors
On Sat, 26 Jan 2013 21:01:30, Hetware <hattons@speakyeasy.net> wrote:

> http://www.youtube.com/watch?v=CaTF4QZ94Fk&list=ECA27CEA1B8B27EB67

>

> Lecture 3, beginning around 1:03:20.

>

> This is what I believe he intended:

>

> Begin with the assumption that we have two unique eigenvalues for a 2X2

> Hermitian matrix.

>

> M|a> = lambda_a|a>

>

> M|b> = lambda_b|b>

>

> Multiply the first by the conjugate of the second and the second by the

> conjugate of the first.

>

> <b|M|a> = lambda_a<b|a>

>

> <a|M|b> = lambda_b<a|b>

>

> Observe that:

>

> <a|M|b> = <b|M|a>*

>

> <a|b> = <b|a>*

>

> So, as I understand it:

>

> <a|M|b> = lambda_b<a|b> = <b|M|a>* = lambda_b<b|a>*

>

> Notice this is different from what Susskind presents. I have not

> conjugated lambda_b, whereas he did. I know he has already stated that

> the eigenvalues are real, so lambda_b*=lambda_b. Therefore, there is no

> difference in bedeutung (denotation). There is a difference in

> sinn(sense), however.

>

> I don't see the motivation for conjugating lambda_b where he did so. He

> isn't really conjugating both sides of the equation:

>

> <a|M|b> = lambda_b<a|b>

>

> That would result in:

>

> <a|M|b>* = (lambda_b<a|b>)* = <b|M|a> = lambda_b*<b|a>,

>

> if I'm not mistaken.

>

> One comment on the YouTube page says that he screwed up the presentation

> at that point. It certainly made me do a double-take, but if he had

> said something like "Now we rewrite <a|M|b> = lambda_b<a|b> in it's

> equivalent complex conjugate form by replacing all terms by equivalent

> complex conjugate terms." I believe his development would be

> procedurally valid.

>

> Does that make sense?

No! Nor do you! Complex conjugate in how many terms?

Four is not enough!