How to interpret this equation in Szabo & Ostlund's book

[compchemrulez]

TL;DR Summary

Szabo & Ostlund

I am trying to interpret equation 1.48a on page 11 in Szabo & Ostlund's "Modern Quantum Chemistry".

What purpose does the index j serve? Is j another basis? Why do we need j?


Reference:
Szabo, A., & Ostlund, N. S. (1996). Modern quantum chemistry: Introduction to advanced electronic structure theory. Mineola, N.Y: Dover Publications.

 

[malawi_glenn]

it uses eq 1.47 the orthogonality property.
If you know linear algebra, you can think of these indices as you would do for basis vectors in say R³

The <j|a> is similar as you would do in linear algebra, for a vector a, what its compnents are in a certain basis

 

[compchemrulez]

yes, but where does j even come from? if we are trying to find the components of |a> with respect to the basis {|i>}, why do we need j? I do not understand

 

[compchemrulez]

was |a> in the basis {| j >} to begin with?

 

[compchemrulez]

or perhaps the better question is, are i and j two separate bases? Is j just another index over the basis i ?

 

[compchemrulez]

oh wait, is a_j just the jth component of ket | a > ?

 

[malawi_glenn]

or perhaps the better question is, are i and j two separate bases? Is j just another index over the basis i ?

You can use any letter you want, you will get the kronecker delta anyway.
Yes they are in the same basis, well $\langle i |$ is the dual-basis of $| i \rangle$. If this was "regular vectors" $\langle i |$ would be the "row vector" of $| i \rangle$ so to say.

Review the linear algebra chapter again. Bras and kets at this point are just representations of vectors in a complex vector space. Eq. 1.48a is the "same" as eq 1.8 but you write $|a\rangle$ instead of $\vec a$ and $\sum |i\rangle a_i$ instead of $\vec a = a_1 \hat e_1 + a_2 \hat e_2 + a_3 \hat e_3$ and $|i \rangle$ instead of $\hat e_i$ and $\langle i |$ would be $(\hat e_i)^T$ (the row vector form of $\hat e_i$)

 

[compchemrulez]

ok I think this is making more sense now

Thank you Malawi_glenn, you will see me posting many more questions in this forum : )

 

[malawi_glenn]

Thank you Malawi_glenn, you will see me posting many more questions in this forum : )

You need to post the relevant equations here, and your own effort in trying to understand.
For this, I recommend that you learn some basic LaTeX, there is a nice guide here https://www.physicsforums.com/help/latexhelp/

A good title is also needed "help with an equation in book X" is not very useful.

A question like this, I would have reported, but I decided to cut some slack here since you are new and I have the book pretty close to me in my little library.

 

[compchemrulez]

How should I cite an equation from a book?