1D potential wells of the form Vx<a=−Vo1−xnan. (1st March 2021) - Corrigan, Abby; Shertzer, Janine
Abstract: We consider the finite potential well defined by V x < a = − V o 1 − x n a n, 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 = 2 m V o a / 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.