Abstract: The eigen-energy equation of a one-dimensional finite square potential well is subject to the tran-scendental equations,and therefore cannot be solved exactly.In this paper,we reduce the even-and the odd-parity equations into one equation,which can be solved consistently to obtain two approximated solutions,i.e.,the first-order Taylor-series solution and the quadratic approximation solution.From the validity of the two approximations and their error analysis,we find that the Taylor-series solution is useful to understand the numerical observation that the energy spectra increase with n2 (i.e.,the so-called n-square law),but fails for some specific values of R,where the parameter R is proportional to the width of the well multiplied by the square of the potential height.The quadratic approximation is applicable to all values of R.In the large R limit,the energy spectra reduce to the exactly solvable infinite-well case.For any R,the fidelity of the quadratic approximation wave function is always greater than 99.7%.

Key words: one-dimensional finite symmetric square well, the bound state, transcendental equation, energy spectrum, energy eigenwave function