# Hamilton-Jacobi equation for interacting spins ?

1. Oct 8, 2004

### lalbatros

The Schrodinder equation leads to the classical Hamilton-Jacobi
equation with an additional term called "quantum potential". This is
the start of the Bohmian interpretation of quantum mechanics. The
derivation is straigthforward by assuming psi = R exp(iS/hb). The
vanishing of the additional term for the classical limit hb->0
illustrates nicely the link between QM and CM.

I would like to know if a similar find could be done for a system of
interacting spins (or qbits !). I take interacting spins as an
example of 'discrete system' (no space coordinate). Is there also an
Hamilton-Jacobi to be derived ? If yes, is there also a classical
limit to be seen ?

I would be interrested in ideas, suggestions and eventually
references related to this topic.

2. Oct 8, 2004

### seratend

Beware that taking simply hbar=0 is simple in H-J formulation of QM, but it is far from giving you the link between QM and CM when hbar is different from 0 (you only look at a space point with such approximation).
Have a look at the WBK and optical approximations for a better understanding.

And, yes you have equivalent formulations of QM in the H-J formalism with the spin space.
There's a lot of papers in arxiv concerning this topic. I've found one, but beware as it is not the most accurate one. At least, it will give you additional paper pointers (Bogan, quant-ph0212110) .

Seratend.

3. Oct 8, 2004

### slyboy

The Bogan paper is not a "spin-only" formualtion. I know of several attempts to incorporate spin into Bohmian mechanics, but they all add it onto the usual position approach. It would be interesting to see a spin-only formulation, not necessarily for any foundational reason, but because it might provide a method of simulating some quantum computations on a classical computer. I would suggest that one could approach this by picking a privelliged spin direction to have a definite value, playing the role that position plays in the usual Bohmian formalism.