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Is ##<i (\Omega  \omega I)V>=0 ## the same as ##<i\Omega  \omega IV>=0 ## ?
Why dotting both sides with a basis bra <i?
Hi!
I am studying Shankar's "Principles of QM" and the first chapter is all about linear algebra with Dirac's notation and I have reached the section "The Characteristic Equation and the Solution to the Eigenvalue Problem" which says that starting from the eigenvalue problem and equation 1.8.3:
$$ (\Omega  \omega I)V> = 0> $$ (where ##V>## is any ket in ## \mathbb V^n(C) ## ).
We operate both sides with ## (\Omega  \omega I)^{1} ## and we get:
$$ V> = (\Omega  \omega I)^{1} 0> $$ But since the inverse only exists when the determinant is nonzero and we don't want the trivial solution, we need to consider the condition: ##det(\Omega  \omega I)=0## to determine the eigenvalues ##\omega##. So, to find them, we (I quote directly from the book) "project Eq. 1.8.3 onto a basis. Dotting both sides with a basis bra ##<i##, we get:"
$$<i\Omega  \omega IV>=0$$ And that's where I am stuck! Everything seems to go the same as in linear algebra with the matrix notation, except that at this point what I would normally do would be to explicitly write and solve ##det(\Omega  \omega I)=0## to find the eigenvalues and then solve 1.8.3 for each of them to get the eigenvectors. Why do we now need to "dot both sides with a basis bra"?
Also, as this is my first encounter with Dirac's notation and despite I think the book does a decent job introducing it, I still think it does not explain some things properly which may result confusing to me, which leads me to my second question:
Is "##<i (\Omega  \omega I)V>=0 ##" the same as "##<i\Omega  \omega IV>=0 ##" ? They seem to be (following the logic of the book) the same, but (to me) the second expression looks more like a subtraction of a bra times an operator minus a scalar times an identity operator times a ket while the first expression looks more like "one entity". If they indeed are the same, then why disposing the parenthesis? If they are not the same, is there an intuitive way to differentiate them that I am missing?
By the way, if anyone knows a good resource to learn Dirac's notation in a clearer manner, I would highly appreciate if you let me know it because that's with what I am struggling the most.
I am studying Shankar's "Principles of QM" and the first chapter is all about linear algebra with Dirac's notation and I have reached the section "The Characteristic Equation and the Solution to the Eigenvalue Problem" which says that starting from the eigenvalue problem and equation 1.8.3:
$$ (\Omega  \omega I)V> = 0> $$ (where ##V>## is any ket in ## \mathbb V^n(C) ## ).
We operate both sides with ## (\Omega  \omega I)^{1} ## and we get:
$$ V> = (\Omega  \omega I)^{1} 0> $$ But since the inverse only exists when the determinant is nonzero and we don't want the trivial solution, we need to consider the condition: ##det(\Omega  \omega I)=0## to determine the eigenvalues ##\omega##. So, to find them, we (I quote directly from the book) "project Eq. 1.8.3 onto a basis. Dotting both sides with a basis bra ##<i##, we get:"
$$<i\Omega  \omega IV>=0$$ And that's where I am stuck! Everything seems to go the same as in linear algebra with the matrix notation, except that at this point what I would normally do would be to explicitly write and solve ##det(\Omega  \omega I)=0## to find the eigenvalues and then solve 1.8.3 for each of them to get the eigenvectors. Why do we now need to "dot both sides with a basis bra"?
Also, as this is my first encounter with Dirac's notation and despite I think the book does a decent job introducing it, I still think it does not explain some things properly which may result confusing to me, which leads me to my second question:
Is "##<i (\Omega  \omega I)V>=0 ##" the same as "##<i\Omega  \omega IV>=0 ##" ? They seem to be (following the logic of the book) the same, but (to me) the second expression looks more like a subtraction of a bra times an operator minus a scalar times an identity operator times a ket while the first expression looks more like "one entity". If they indeed are the same, then why disposing the parenthesis? If they are not the same, is there an intuitive way to differentiate them that I am missing?
By the way, if anyone knows a good resource to learn Dirac's notation in a clearer manner, I would highly appreciate if you let me know it because that's with what I am struggling the most.