Non-associative quantum mechanics and jordan algebras

We present a Jordan algebraic formulation of the non-commutative Landau problem coupled to a harmonic potential. To achieve this, we use a new formulation of the Hilbert space version of quantum mechanics. Using this construction, the Hilbert space corresponding to the non-commutative Landau problem is obtained. For that problem, non-commutative parameters are described in terms of an associator in the Jordan algebraic setting. Pure states and density matrices arising from this problem are characterized. This in turn leads us to the Jordan-Schrödinger time-evolution equation for the state vectors for this specific problem.