I Matrix elements of Hamiltonian include Dirac delta term

MaestroBach
Messages
53
Reaction score
4
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:
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.
 
Physics news on Phys.org
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.
 
  • Like
Likes pines-demon
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!

Similar threads

Replies
2
Views
2K
Replies
4
Views
1K
Replies
12
Views
1K
Replies
1
Views
2K
Replies
7
Views
5K
Replies
1
Views
1K
Replies
0
Views
3K
Replies
4
Views
5K
Back
Top