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1D potential wells of the form Vx

We consider the finite potential well defined by Vx<a=−Vo1−xnan ,where n is an integer. The Schrödinger equation is recast into dimensionless form; the size of the well is characterized by a single parameter α=2mVoa/ℏ . We use three different mathematical techniques to solve for the energy eigenvalues and wave functions. Analytic solutions are given for the triangular well (n = 1), the parabolic well (n = 2), and the square well (n → ∞). The finite element method is used to obtain accurate numerical results for 1 ⩽ n ⩽ ∞ over a full range of α. Finally, both the energy and wave function are expanded in a Taylor series in order to study the analytic behavior of the ground state energy (as a function of n) as α → 0.