About the spin-orbit coupling in atoms

[wdlang]

i am now reading the book by woodgate: elementary atomic strucure, 2nd edition.

on page 62, he discusses the spin-orbit coupling in hydrogen atom, and calculates the first order shift of the energy due to this effect.

I have some doubt about his procedure from eq.4.25 to eq.4.27

the spin-orbit coupling is f(r) s \dot l

its average is then <f(r) s\dot l>

why can he take the equality <f(r) s \dot l>= <f(r)> <s \dot l> ?

this greatly simplifes the calculation!

 

[wdlang]

i am now reading the book by woodgate: elementary atomic strucure, 2nd edition.

on page 62, he discusses the spin-orbit coupling in hydrogen atom, and calculates the first order shift of the energy due to this effect.

I have some doubt about his procedure from eq.4.25 to eq.4.27

the spin-orbit coupling is f(r) s \dot l

its average is then <f(r) s\dot l>

why can he take the equality <f(r) s \dot l>= <f(r)> <s \dot l> ?

this greatly simplifes the calculation!

attached is the image of that page

 

[Vanadium 50]

why can he take the equality <f(r) s \dot l>= <f(r)> <s \dot l> ?

That's not what he is doing. He's substituting in for f(r).

 

[wdlang]

That's not what he is doing. He's substituting in for f(r).

Thank you. I have figured it out.

Yes, he actually makes some approximations.

Note that f(r) is a function invariant under an arbitrary rotation; therefore it preserve the angular momentum of the electron and also, its spin. This can also be understood using wigner-eckart theorem.

But f(r) do couple different n's

He drops the coupling between different n's, only taking the diagonal term

If you are interested, you can see the book by condon and shortley, page 121.