How to Derive Exchange Energy from Ground State Equation?

[supernano]

I've calculated the eigenstates of the Hubbard Hamiltonian for two fermions.
The ground state is (U² - (U² + 16t²)^1/2)/2
For U = infty, I get 0.
For U >> t, I should get the exchange energy J = -4t²/U
How do I get from the ground state equation to J?

 

[DrClaude]

Use the Taylor series expansion $\sqrt{1 + x} \approx 1 + x/2$ for small x.

 

[supernano]

Too easy, thanks!

 

[PeroK]

When I saw a thread called "limit of large but finite" under "calculus" I wasn't expecting something about the Hubbard Hamiltonian of two fermions. But, clearly, @DrClaude was prepared for anything!

 

[DrClaude]

When I saw a thread called "limit of large but finite" under "calculus" I wasn't expecting something about the Hubbard Hamiltonian of two fermions. But, clearly, @DrClaude was prepared for anything!

I was actually more prepared for the Hubbard Hamiltonian than for regular calculus

I hesitated to move the thread. @supernano, note that while your question was a mathematical one, you have a higher probability of getting an answer for such a question in the QM forum. It's more a question of knowing the tricks than knowing maths.

 

[Mark44]

I've calculated the eigenstates of the Hubbard Hamiltonian for two fermions.
The ground state is (U² - (U² + 16t²)^1/2)/2
For U = infty, I get 0.

For one thing, you can't just substitute $\infty$ into the expression. For another thing, the expression above has the form $[\infty - \infty]$, which is one of several indeterminate forms, along with $[\frac0 0]$, $[\frac{\infty}{\infty}]$ and a few others.

For U >> t, I should get the exchange energy J = -4t²/U
How do I get from the ground state equation to J?