Proof of second quantization operators

[Tilde90]

Please, can somebody show me why a Hamiltonian like $\sum_nh(x_n)$ can be written as $\sum_{i,j}t_{i,j}a^+_ia_j$, with $t_{i,j}=\int f^*_i(x)h(x)f_j(x)dx$?

Thank you.

 

[nonequilibrium]

Can you define your h, x and f?

 

[DimReg]

I think what you are looking for is that you can show that the raising and lowering operators are enough to create any operator on fock space. Basically, the proof of that, is by using raising and lowering operators as a "basis", you have enough freedom to make the operator matrix elements have any value you want. Weinberg's QFT book has a description of this in his "Cluster Decomposition Principle" chapter.

 

[cgk]

OP, these proofs are often omitted because they can become very messy. I think there was one in ``Molecular electronic structure theory'' by Helgaker, Jorgensen, Olsen (``the purple book''). You might want to have a look if your university library has one of those (it's also tremendously useful for lots of other quantum many body things if you really want to know what is going on).

 

[Tilde90]

Thank you all for your replies. I apologise for the lack of details in my first post, but that evening I was at the brink of desperation trying to prove that result. :-) Finally, I found a proof online, but as you say it's not something which books usually demonstrate (despite its importance).

 

[espen180]

I think what you are looking for is that you can show that the raising and lowering operators are enough to create any operator on fock space. Basically, the proof of that, is by using raising and lowering operators as a "basis", you have enough freedom to make the operator matrix elements have any value you want. Weinberg's QFT book has a description of this in his "Cluster Decomposition Principle" chapter.

For the sake of claification, do you mean taking linear combinations of finite products of raising and lowering operators?

 

[B-80]

Thank you all for your replies. I apologise for the lack of details in my first post, but that evening I was at the brink of desperation trying to prove that result. :-) Finally, I found a proof online, but as you say it's not something which books usually demonstrate (despite its importance).

This is also relevant to my interests... Where did you find the proof if you don't mind me asking?

 

[DimReg]

For the sake of claification, do you mean taking linear combinations of finite products of raising and lowering operators?

It's been a while since I saw the precise statement of the theorem, but I believe that is the case. Other interpretations don't seem to be powerful enough

 

[Tilde90]

This is also relevant to my interests... Where did you find the proof if you don't mind me asking?

Sure. It's an Italian paper, http://www.dcci.unipi.it/~ivo/didattica/dispense.chimteo/secquant.pdf , pages 11-13 (and following pages for two particle Hamiltonians). The formalism is very clear.