Based on the separation-of-variables method and Mathieu function theory, we rigorously solved the Schrödinger equation for a two-dimensional infinite-depth elliptical quantum well, obtaining the energy level distributions and corresponding wave functions for various eccentricities e_r and quantum states. By separating variables in elliptical coordinates, the wave function is expressed as the product of radial Mathieu functions and angular Mathieu functions. Combining the angular periodicity boundary condition with the infinite-well boundary condition, we derived the characteristic equation that determines the discrete energy eigenvalues. Through analysis of the zero-distribution patterns of Mathieu functions under different parities and angular quantum numbers m, the energy spectra were numerically computed, and the dependence of the energy levels on the eccentricity was investigated in detail. Using the analytical expressions for the wave functions, the properties of the wave functions in different quantum states were thoroughly analyzed.

By solving the Schrodinger equations in three-dimensional (3D) and two-dimensional (2D) finite depth center spherically symmetric square potential well, the eigenequations of bounded energy levels and their distributions are obtained, and the analytical solution for any bound state wave function is given. Both the eigenequations of the energy levels as well as the stationary state wave functions are closely related to the (spherical) Bessel functions. It is found that for the 3D case, there exists a bound solution when 2μa²V₀/ħ²>π²/4 is present. While for the 2D case, there exists a bound state if 2μa²V₀/ħ²>0.1993. Finally, numerical methods are used to analyze the characteristics of probability density distribution of different quantum states. It is found that in both three-dimensional and two-dimensional situations, the maximum number of probability density distributions inside the well corresponds exactly to the quantum number n. The larger n or the smaller α is, the higher the probability of discovering particles outside the well. In addition, the angular momentum quantum number l or m can characterize the degree to which particles deviate from the center or the uniform distribution.