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.