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

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In summary, the index j serves as a way to keep track of which basis you are using in eq 1.48a. It is similar to the basis i, but it is not separate from it.
  • #1
compchemrulez
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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.
 
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  • #2
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
 
  • #3
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
 
  • #4
was |a> in the basis {| j >} to begin with?
 
  • #5
or perhaps the better question is, are i and j two separate bases? Is j just another index over the basis i ?
 
  • #6
oh wait, is aj just the jth component of ket | a > ?
 
  • #7
compchemrulez said:
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##)
 
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  • #8
ok I think this is making more sense now

Thank you Malawi_glenn, you will see me posting many more questions in this forum : )
 
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  • #9
compchemrulez said:
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.
 
  • #10
How should I cite an equation from a book?
 

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

1. What is the significance of the equation in Szabo & Ostlund's book?

The equation in Szabo & Ostlund's book is a fundamental equation in quantum chemistry known as the Hartree-Fock equation. It is used to calculate the electronic structure of atoms and molecules, providing valuable insights into their properties and behavior.

2. How do I interpret the terms in the equation?

The terms in the equation represent different physical quantities such as the kinetic energy of electrons, the potential energy of the electron-nucleus interaction, and the electron-electron repulsion energy. These terms are combined to form the total energy of the system.

3. What is the role of the wavefunction in this equation?

The wavefunction, represented by the Greek letter psi (ψ), is a mathematical function that describes the probability of finding an electron at a specific location in space. In the Hartree-Fock equation, the wavefunction is used to calculate the energy of the system.

4. How is this equation derived?

The Hartree-Fock equation is derived from the Schrödinger equation, which is the fundamental equation of quantum mechanics. It involves a series of mathematical approximations and techniques to solve the many-body problem of electrons in a system. The full derivation can be found in Szabo & Ostlund's book.

5. Can this equation be applied to any system?

The Hartree-Fock equation is a general equation that can be applied to any system of atoms and molecules. However, it does have its limitations and may not accurately describe systems with strong electron correlation or relativistic effects. In such cases, more advanced equations and methods are needed.

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