Definition of Heisenberg Hamiltonian

[Petar Mali]

I have a question. What is the definition of Heisenberg hamiltonian?

$$\hat{H}=-\sum_{i,j}J_{i,j}\hat{\bfs{S}}_i\cdot \hat{\bfs{S}}_j$$

or

$$\hat{H}=-2\sum_{i,j}J_{i,j}\hat{\bfs{S}}_i\cdot \hat{\bfs{S}}_j$$

or

$$\hat{H}=\sum_{i,j}J_{i,j}\hat{\bfs{S}}_i\cdot \hat{\bfs{S}}_j$$

or

$$\hat{H}=2\sum_{i,j}J_{i,j}\hat{\bfs{S}}_i\cdot \hat{\bfs{S}}_j$$

You have all variation in different sources.

 

[Forestman]

I don't think that I have the whole answer, but I believe that it has to do with matrix mechanics which Heisenberg used to describe quantum theory. I think that it means that the rows and columns of the matrix have been inverted, and that the complex conjugate has been taken. I believe that this is known as the hamiltonian process in matrix math.

 

[element4]

@Forestman, I really don't think so!

@Petar Mail, all of them are Heisenberg Hamiltonians and the factors in front $$\pm 2$$ are just a matter of convention. But the sign can origin from the fact that the coupling constant can be positive or negative, depending on you have a anti-ferromagnetic or ferromagnetic system. The sign can be included in order to keep $$J_{ij}$$ positive.

The factor of 2 (sometimes $$\frac 12$$, depending on convention) can origin from the fact that some people sum over each pair of sites $$(ij)$$ once, while others prefer to sum over all sites individually. The last option over counts and a factor of 2 (or $$\frac 12$$, depending on which side you want to put it in) must be included.

In the end, you could put in any factor which is convinient, you just have to redefine $$J_{ij}$$ afterwards.

 

[Forestman]

Oh okay, thanks for correcting me.

I am new to the math of quantum physics. I am currently in the process of teaching myself calculus.

 

[Forestman]

If anyone could tell me in completely verbal terms I would like to know more about what the Heisenberg hamiltonian really means.

 

[Forestman]

Most of you probably already know this, but I post this for those who do not. Since I did not understand the Hamiltonian concept I looked it up for myself, and it just means the total kinetic and potential energy of a quantum system.

 

[kanato]

Forestman, the Heisenberg Hamiltonian is an effective model for interacting (three-dimensional) spins on a lattice.

The Atomic, Solid State, Comp. Physics forum is a forum for people with specific questions about solid state physics and related topics. If you have questions about basic quantum or classical physics (like what a Hamiltonian is) I would direct you to the classical physics and quantum physics forums.

 

[Forestman]

Thank you for answering that for me kanato.

I will make sure to use the other forums next time.