Matrix elements of Hamiltonian include Dirac delta term

[MaestroBach]

TL;DR Summary

Currently reading a textbook on non-equilibrium green's functions and I'm stuck in chapter 1 where it recaps just general quantum mechanics because of dirac deltas included in the matrix elements of a generalized hamiltonian.

Currently reading a textbook on non-equilibrium green's functions and I'm stuck in chapter 1 where it recaps just general quantum mechanics because of dirac deltas included in the matrix elements of a generalized hamiltonian.

The textbook gives this:

I just don't understand how to think about the dirac deltas in this case, given that as far as I understand, it'll be a function that's 0 everywhere except where r = r' at which point it's equal to infinity.

For whatever reason, I can't get latex to work right now, but x = r * sigma, where sigma is the spin quantum number.

 

[pines-demon]

Please provide the textbook.

Also what is $\hat{h}$, $\mathbf S$, the different $\nabla$? Please define your variables.

 

[MaestroBach]

Sorry, I should have provided a lot more context.

The book is Nonequilibrium many-body theory of quantum systems by Stefanucci,

$\hat{h}$ is any generalized hamiltonian $\hat{h} = h(\hat{r},~\hat{p},~\hat{S})$ where "$\hat{r}$ is the position operator, $\hat{p}$ is the momentum operator, and $\hat{S}$ isi the spin operator.

$\mathbf S$ is "the matrix of the spin operator with elements $<\sigma | \hat{S} | \sigma' >$", where $\sigma$ is the eigenvalue of $S_z$ so as far as I understand, it's essentially the same as $\hat{S}$

The primed $\nabla$ applies to the primed variable.