<p>We study the quantum mechanics of a particle confined to a parallelogram-shaped domain using a non-orthogonal (oblique) coordinate system adapted to the geometry. By employing a suitable Ansatz for the eigenfunctions, we derive a general expression for the energy spectrum in terms of the geometric parameters. The physically admissible spectrum and corresponding exact wave functions are obtained by imposing Dirichlet boundary conditions, yielding a subset of eigenvalues and eigenfunctions that can be obtained in closed form.</p>