Noncommutative Quantum Mechanics

We investigate the effect of Noncommutative Quantum Mechanics for a particle in Earth’s gravity using the Hamiltonian in two dimensions. We solve the Schrödinger equation in detail, we then split the Schrödinger equation into x component and y component. For y component, we further solve the equation and obtain a Simple Harmonic Oscillator (SHO) equation. For x component, we come across an imaginary term that makes it a complex equation, after we solve it, we obtain a SHO equation again. The two equations for components of x and y are solved independently, and we obtain their energy levels, normalized stationary states with Hermite polynomials. A perturbation term appears due to noncommutativity in the Schrödinger equation which we dealt with subsequently. Our results contain corrections of which the most important here are the energy levels. Finally, we are able to combine energy levels from both components of x and y, normalized stationary states as our full solutions. Keywords: Quantum Mechanics, Noncommutativity, Schrödinger equation, Hamiltonian, Harmonic Oscillator, energy levels.