I Matrix elements of Hamiltonian include Dirac delta term

MaestroBach
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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:
Screenshot 2024-07-01 145152.png

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.
 
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Please provide the textbook.

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