We develop the formalism of quantum mechanics on three dimensional fuzzy space and solve the Schrodinger equation for a free particle, finite and infinite fuzzy wells. We show that all results reduce to the appropriate commutative limits. A high ener
gy cut-off is found for the free particle spectrum, which also results in the modification of the high energy dispersion relation. An ultra-violet/infra-red duality is manifest in the free particle spectrum. The finite well also has an upper bound on the possible energy eigenvalues. The phase shifts due to scattering around the finite fuzzy potential well have been calculated.
An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal di
stances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the $su(2)$ algebra. This has been computed for both the discrete, as well as for the Perelemovs $SU(2)$ coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by $ninmathbb{Z}/2$.
Thermal correlation functions and the associated effective statistical potential are computed in two- and three-dimensional non-commutative space using an operator formulation that makes no reference to a star product. The corresponding results for t
he Moyal and Voros star products are then easily obtained by taking the corresponding overlap with Moyal and Voros bases. The forms of the correlation function and the effective potential are found to be the same, except that in the Voros case the thermal length undergoes a non-commutative deformation, ensuring that it has a lower bound of the order of $sqrt{theta}$. It is shown that in a suitable basis (called here quasi-commutative) in the multi-particle sector the thermal correlation function coincides with the commutative result both in the Moyal and Voros cases, with an appropriate non-commutative correction to the thermal length in the Voros case, and that the Pauli principle is restored.
