Understanding Slater Determinants Using He 1s2 & 1s12s1

[fricke]

I am really really really confused how to read and construct Slater determinants :(
Can someone please explain it using He at the ground state (1s²) and He at excited state (1s¹2s¹) ?

 

[atyy]

The basic idea is that the wave function for identical fermions must be anti-symmetric.

The Slater determinant simply makes use of the definition of a determinant to notate the above idea.

https://en.wikipedia.org/wiki/Determinant

(For quantum mechanics, you can think of it as the determinant as just notation and not otherwise meaningful. However, in mathematics or probability, the determinant has the meaning of an area or a volume: http://mathinsight.org/relationship_determinants_area_volume. It is very useful in the change of variables formula in integration using Jacobians, which makes sense because the integral is a sum over area or volume elements: https://www.stat.wisc.edu/courses/st309-larget/jacobian.pdf.)

 

[fricke]

For quantum mechanics, you can think of it as the determinant as just notation and not otherwise meaningful

I understand how to calculate the determinant. But how to construct determinant for electrons spin in an atom?

 

[atyy]

I understand how to calculate the determinant. But how to construct determinant for electrons spin in an atom?

http://jila.colorado.edu/Chem4541/Indistinguishable%20particles,%20Pauli%20Principle,%20Slater%20determinants%20p%2091-98.pdf

 

[fricke]

http://jila.colorado.edu/Chem4541/Indistinguishable%20particles,%20Pauli%20Principle,%20Slater%20determinants%20p%2091-98.pdf

thank you!
I have another question.
Prof taught us one row represents one electron, so each electron is represented in a row in Slater Determinant. But when I googled it, some represents one electron in one column (not in one row!). Mathematically, it seems to be not wrong (since I didn't do any calculation to prove these two are the same) but how can representing electrons in column rather than row be the same in quantum electron spin?

 

[atyy]

Prof taught us one row represents one electron, so each electron is represented in a row in Slater Determinant. But when I googled it, some represents one electron in one column (not in one row!). Mathematically, it seems to be not wrong (since I didn't do any calculation to prove these two are the same) but how can representing electrons in column rather than row be the same in quantum electron spin?

I'm not entirely sure of this part, but I believe it is because the determinant is the same whether rows and columns are transposed.

https://www.khanacademy.org/math/li...-determinant-of-transpose?_escaped_fragment_=

But you should check the wave function is the same whether you write each electron as a row or a column.

 

[fricke]

I'm not entirely sure of this part, but I believe it is because the determinant is the same whether rows and columns are transposed.

https://www.khanacademy.org/math/li...-determinant-of-transpose?_escaped_fragment_=

But you should check the wave function is the same whether you write each electron as a row or a column.

thank you. I have found out a really easy way to construct Slater determinant for electron spin of an atom :D